[{"data":1,"prerenderedAt":20591},["ShallowReactive",2],{"post-\u002Fblog\u002F2023\u002F2023-12-18-the-master-theroy-for-time-complexity":3},{"id":4,"title":5,"body":6,"description":12,"draft":20576,"enableComment":20577,"extension":20578,"image":20579,"meta":20580,"navigation":20577,"onday":20581,"path":20582,"seo":20583,"stem":20584,"summary":20585,"tags":20586,"__hash__":20590},"blog\u002Fblog\u002F2023\u002F2023-12-18-The-Master-Theroy-For-Time-Complexity.md","The Master Theorem For Time Complexity",{"type":7,"value":8,"toc":20567},"minimark",[9,13,18,21,271,274,722,755,801,805,808,1140,1770,1963,2844,3496,3551,4574,4805,5312,5315,5319,5490,5493,5636,5639,5645,5932,5935,5940,6430,6433,8947,8950,9240,9243,9247,9250,9253,9499,9606,9711,9717,10182,10649,11011,11014,12105,12108,12156,13376,13379,13385,13504,13925,13930,14047,14695,14700,14837,15757,16415,16641,16646,16783,17452,17897,18272,18275,18278,18282,18744,18962,19169,19347,19857,20341,20555,20559],[10,11,12],"p",{},"When we navigate through the ocean of divide and conquer, there's a question that must be addressed—how is the time complexity of divide and conquer algorithms calculated?",[14,15,17],"h2",{"id":16},"from-divide-and-conquer-to-recurrence-relation","From Divide and Conquer to Recurrence Relation",[10,19,20],{},"Divide and conquer involves breaking down a large problem into smaller subproblems and then merging the solutions of these subproblems to obtain the solution to the original problem. Recurrence relation is a mathematical way of modeling the time cost of this process.",[10,22,23,24,95,96,125,126,167,168,224,225,270],{},"Starting with a concrete example makes it easier to understand. Taking merge sort as an example, let ",[25,26,30,64],"span",{"className":27,"translate":29},[28],"katex","no",[25,31,34],{"className":32},[33],"katex-mathml",[35,36,38],"math",{"xmlns":37},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[39,40,41,59],"semantics",{},[42,43,44,48,53,56],"mrow",{},[45,46,47],"mi",{},"T",[49,50,52],"mo",{"stretchy":51},"false","(",[45,54,55],{},"n",[49,57,58],{"stretchy":51},")",[60,61,63],"annotation",{"encoding":62},"application\u002Fx-tex","T(n)",[25,65,69],{"className":66,"ariaHidden":68},[67],"katex-html","true",[25,70,73,78,84,88,91],{"className":71},[72],"base",[25,74],{"className":75,"style":77},[76],"strut","height:1em;vertical-align:-0.25em;",[25,79,47],{"className":80,"style":83},[81,82],"mord","mathnormal","margin-right:0.13889em;",[25,85,52],{"className":86},[87],"mopen",[25,89,55],{"className":90},[81,82],[25,92,58],{"className":93},[94],"mclose"," represent the time complexity of the original problem, where ",[25,97,99,112],{"className":98,"translate":29},[28],[25,100,102],{"className":101},[33],[35,103,104],{"xmlns":37},[39,105,106,110],{},[42,107,108],{},[45,109,55],{},[60,111,55],{"encoding":62},[25,113,115],{"className":114,"ariaHidden":68},[67],[25,116,118,122],{"className":117},[72],[25,119],{"className":120,"style":121},[76],"height:0.4306em;",[25,123,55],{"className":124},[81,82]," is the size of the input data. In the first split, the original data is divided into two equal parts, each of size ",[25,127,129,151],{"className":128,"translate":29},[28],[25,130,132],{"className":131},[33],[35,133,134],{"xmlns":37},[39,135,136,148],{},[42,137,138,140,144],{},[45,139,55],{},[45,141,143],{"mathvariant":142},"normal","\u002F",[145,146,147],"mn",{},"2",[60,149,150],{"encoding":62},"n\u002F2",[25,152,154],{"className":153,"ariaHidden":68},[67],[25,155,157,160,163],{"className":156},[72],[25,158],{"className":159,"style":77},[76],[25,161,55],{"className":162},[81,82],[25,164,166],{"className":165},[81],"\u002F2",". Then, recursive calls to merge sort are made on these two parts, and the time complexity of this part is ",[25,169,171,197],{"className":170,"translate":29},[28],[25,172,174],{"className":173},[33],[35,175,176],{"xmlns":37},[39,177,178,194],{},[42,179,180,182,184,186,188,190,192],{},[145,181,147],{},[45,183,47],{},[49,185,52],{"stretchy":51},[45,187,55],{},[45,189,143],{"mathvariant":142},[145,191,147],{},[49,193,58],{"stretchy":51},[60,195,196],{"encoding":62},"2 T(n\u002F2)",[25,198,200],{"className":199,"ariaHidden":68},[67],[25,201,203,206,209,212,215,218,221],{"className":202},[72],[25,204],{"className":205,"style":77},[76],[25,207,147],{"className":208},[81],[25,210,47],{"className":211,"style":83},[81,82],[25,213,52],{"className":214},[87],[25,216,55],{"className":217},[81,82],[25,219,166],{"className":220},[81],[25,222,58],{"className":223},[94],". Finally, the two sorted subarrays are merged, requiring a time complexity of ",[25,226,228,249],{"className":227,"translate":29},[28],[25,229,231],{"className":230},[33],[35,232,233],{"xmlns":37},[39,234,235,246],{},[42,236,237,240,242,244],{},[45,238,239],{"mathvariant":142},"Θ",[49,241,52],{"stretchy":51},[45,243,55],{},[49,245,58],{"stretchy":51},[60,247,248],{"encoding":62},"\\Theta(n)",[25,250,252],{"className":251,"ariaHidden":68},[67],[25,253,255,258,261,264,267],{"className":254},[72],[25,256],{"className":257,"style":77},[76],[25,259,239],{"className":260},[81],[25,262,52],{"className":263},[87],[25,265,55],{"className":266},[81,82],[25,268,58],{"className":269},[94],".",[10,272,273],{},"Based on the analysis above, we can write the recurrence relation for merge sort as follows:",[25,275,278],{"className":276,"translate":29},[277],"katex-display",[25,279,281,417],{"className":280,"translate":29},[28],[25,282,284],{"className":283},[33],[35,285,287],{"xmlns":37,"display":286},"block",[39,288,289,414],{},[42,290,291,293,295,297,299,302],{},[45,292,47],{},[49,294,52],{"stretchy":51},[45,296,55],{},[49,298,58],{"stretchy":51},[49,300,301],{},"=",[42,303,304,307],{},[49,305,306],{"fence":68},"{",[308,309,313,358],"mtable",{"rowspacing":310,"columnalign":311,"columnspacing":312},"0.25em","right left right","0em 1em",[314,315,316,325,342],"mtr",{},[317,318,319],"mtd",{},[320,321,323],"mstyle",{"scriptlevel":322,"displaystyle":68},"0",[42,324],{},[317,326,327],{},[320,328,329],{"scriptlevel":322,"displaystyle":68},[42,330,331,333,335,337,340],{},[42,332],{},[45,334,239],{"mathvariant":142},[49,336,52],{"stretchy":51},[145,338,339],{},"1",[49,341,58],{"stretchy":51},[317,343,344],{},[320,345,346],{"scriptlevel":322,"displaystyle":68},[42,347,348,352,354,356],{},[349,350,351],"mtext",{},"if ",[45,353,55],{},[49,355,301],{},[145,357,339],{},[314,359,360,366,399],{},[317,361,362],{},[320,363,364],{"scriptlevel":322,"displaystyle":68},[42,365],{},[317,367,368],{},[320,369,370],{"scriptlevel":322,"displaystyle":68},[42,371,372,374,376,378,380,382,384,386,388,391,393,395,397],{},[42,373],{},[145,375,147],{},[45,377,47],{},[49,379,52],{"stretchy":51},[45,381,55],{},[45,383,143],{"mathvariant":142},[145,385,147],{},[49,387,58],{"stretchy":51},[49,389,390],{},"+",[45,392,239],{"mathvariant":142},[49,394,52],{"stretchy":51},[45,396,55],{},[49,398,58],{"stretchy":51},[317,400,401],{},[320,402,403],{"scriptlevel":322,"displaystyle":68},[42,404,405,407,409,412],{},[349,406,351],{},[45,408,55],{},[49,410,411],{},">",[145,413,339],{},[60,415,416],{"encoding":62},"T(n) = \\left\\{ \\begin{aligned} &\\Theta(1) & \\text{if } n=1\\\\ &2 T(n\u002F2) + \\Theta(n) & \\text{if } n>1 \\end{aligned} \\right.",[25,418,420,450],{"className":419,"ariaHidden":68},[67],[25,421,423,426,429,432,435,438,443,447],{"className":422},[72],[25,424],{"className":425,"style":77},[76],[25,427,47],{"className":428,"style":83},[81,82],[25,430,52],{"className":431},[87],[25,433,55],{"className":434},[81,82],[25,436,58],{"className":437},[94],[25,439],{"className":440,"style":442},[441],"mspace","margin-right:0.2778em;",[25,444,301],{"className":445},[446],"mrel",[25,448],{"className":449,"style":442},[441],[25,451,453,457],{"className":452},[72],[25,454],{"className":455,"style":456},[76],"height:3em;vertical-align:-1.25em;",[25,458,461,471,718],{"className":459},[460],"minner",[25,462,466],{"className":463,"style":465},[87,464],"delimcenter","top:0em;",[25,467,306],{"className":468},[469,470],"delimsizing","size4",[25,472,474],{"className":473},[81],[25,475,477,529,631,636],{"className":476},[308],[25,478,481],{"className":479},[480],"col-align-r",[25,482,486,520],{"className":483},[484,485],"vlist-t","vlist-t2",[25,487,490,515],{"className":488},[489],"vlist-r",[25,491,495,506],{"className":492,"style":494},[493],"vlist","height:1.75em;",[25,496,498,503],{"style":497},"top:-3.75em;",[25,499],{"className":500,"style":502},[501],"pstrut","height:2.84em;",[25,504],{"className":505},[81],[25,507,509,512],{"style":508},"top:-2.25em;",[25,510],{"className":511,"style":502},[501],[25,513],{"className":514},[81],[25,516,519],{"className":517},[518],"vlist-s","​",[25,521,523],{"className":522},[489],[25,524,527],{"className":525,"style":526},[493],"height:1.25em;",[25,528],{},[25,530,533],{"className":531},[532],"col-align-l",[25,534,536,623],{"className":535},[484,485],[25,537,539,620],{"className":538},[489],[25,540,542,567],{"className":541,"style":494},[493],[25,543,545,549],{"style":544},"top:-3.91em;",[25,546],{"className":547,"style":548},[501],"height:3em;",[25,550,552,555,558,561,564],{"className":551},[81],[25,553],{"className":554},[81],[25,556,239],{"className":557},[81],[25,559,52],{"className":560},[87],[25,562,339],{"className":563},[81],[25,565,58],{"className":566},[94],[25,568,570,573],{"style":569},"top:-2.41em;",[25,571],{"className":572,"style":548},[501],[25,574,576,579,582,585,588,591,594,597,601,605,608,611,614,617],{"className":575},[81],[25,577],{"className":578},[81],[25,580,147],{"className":581},[81],[25,583,47],{"className":584,"style":83},[81,82],[25,586,52],{"className":587},[87],[25,589,55],{"className":590},[81,82],[25,592,166],{"className":593},[81],[25,595,58],{"className":596},[94],[25,598],{"className":599,"style":600},[441],"margin-right:0.2222em;",[25,602,390],{"className":603},[604],"mbin",[25,606],{"className":607,"style":600},[441],[25,609,239],{"className":610},[81],[25,612,52],{"className":613},[87],[25,615,55],{"className":616},[81,82],[25,618,58],{"className":619},[94],[25,621,519],{"className":622},[518],[25,624,626],{"className":625},[489],[25,627,629],{"className":628,"style":526},[493],[25,630],{},[25,632],{"className":633,"style":635},[634],"arraycolsep","width:1em;",[25,637,639],{"className":638},[480],[25,640,642,710],{"className":641},[484,485],[25,643,645,707],{"className":644},[489],[25,646,648,678],{"className":647,"style":494},[493],[25,649,650,653],{"style":544},[25,651],{"className":652,"style":548},[501],[25,654,656,663,666,669,672,675],{"className":655},[81],[25,657,660],{"className":658},[81,659],"text",[25,661,351],{"className":662},[81],[25,664,55],{"className":665},[81,82],[25,667],{"className":668,"style":442},[441],[25,670,301],{"className":671},[446],[25,673],{"className":674,"style":442},[441],[25,676,339],{"className":677},[81],[25,679,680,683],{"style":569},[25,681],{"className":682,"style":548},[501],[25,684,686,692,695,698,701,704],{"className":685},[81],[25,687,689],{"className":688},[81,659],[25,690,351],{"className":691},[81],[25,693,55],{"className":694},[81,82],[25,696],{"className":697,"style":442},[441],[25,699,411],{"className":700},[446],[25,702],{"className":703,"style":442},[441],[25,705,339],{"className":706},[81],[25,708,519],{"className":709},[518],[25,711,713],{"className":712},[489],[25,714,716],{"className":715,"style":526},[493],[25,717],{},[25,719],{"className":720},[94,721],"nulldelimiter",[10,723,724,725,754],{},"It can be imagined that for any divide and conquer algorithm, its time complexity can be expressed as a recurrence relation. The right-hand side of the recurrence relation usually consists of two parts: the total cost of the divided subproblems and the total cost of dividing and merging subproblems. The first part remains a function of ",[25,726,728,741],{"className":727,"translate":29},[28],[25,729,731],{"className":730},[33],[35,732,733],{"xmlns":37},[39,734,735,739],{},[42,736,737],{},[45,738,47],{},[60,740,47],{"encoding":62},[25,742,744],{"className":743,"ariaHidden":68},[67],[25,745,747,751],{"className":746},[72],[25,748],{"className":749,"style":750},[76],"height:0.6833em;",[25,752,47],{"className":753,"style":83},[81,82],", while the second part represents the actual asymptotic complexity.",[10,756,757,758,270],{},"Having a recurrence relation is not sufficient to determine the exact time complexity of the algorithm. We need to further calculate the actual solution ",[25,759,761,780],{"className":760,"translate":29},[28],[25,762,764],{"className":763},[33],[35,765,766],{"xmlns":37},[39,767,768,778],{},[42,769,770,772,774,776],{},[45,771,47],{},[49,773,52],{"stretchy":51},[45,775,55],{},[49,777,58],{"stretchy":51},[60,779,63],{"encoding":62},[25,781,783],{"className":782,"ariaHidden":68},[67],[25,784,786,789,792,795,798],{"className":785},[72],[25,787],{"className":788,"style":77},[76],[25,790,47],{"className":791,"style":83},[81,82],[25,793,52],{"className":794},[87],[25,796,55],{"className":797},[81,82],[25,799,58],{"className":800},[94],[14,802,804],{"id":803},"substitution-method","Substitution Method",[10,806,807],{},"The first method for solving recurrence relations is called the substitution method. We need to guess the form of the solution first and then use mathematical induction to prove that the guess is correct.",[10,809,810,811,921,922,1067,1068,1139],{},"For Equation (1), let's assume that our guess for the solution is ",[25,812,814,853],{"className":813,"translate":29},[28],[25,815,817],{"className":816},[33],[35,818,819],{"xmlns":37},[39,820,821,850],{},[42,822,823,825,827,829,831,833,836,838,840,843,846,848],{},[45,824,47],{},[49,826,52],{"stretchy":51},[45,828,55],{},[49,830,58],{"stretchy":51},[49,832,301],{},[45,834,835],{},"O",[49,837,52],{"stretchy":51},[45,839,55],{},[45,841,842],{},"log",[49,844,845],{},"⁡",[45,847,55],{},[49,849,58],{"stretchy":51},[60,851,852],{"encoding":62},"T(n) = O(n \\log n)",[25,854,856,883],{"className":855,"ariaHidden":68},[67],[25,857,859,862,865,868,871,874,877,880],{"className":858},[72],[25,860],{"className":861,"style":77},[76],[25,863,47],{"className":864,"style":83},[81,82],[25,866,52],{"className":867},[87],[25,869,55],{"className":870},[81,82],[25,872,58],{"className":873},[94],[25,875],{"className":876,"style":442},[441],[25,878,301],{"className":879},[446],[25,881],{"className":882,"style":442},[441],[25,884,886,889,893,896,899,903,912,915,918],{"className":885},[72],[25,887],{"className":888,"style":77},[76],[25,890,835],{"className":891,"style":892},[81,82],"margin-right:0.02778em;",[25,894,52],{"className":895},[87],[25,897,55],{"className":898},[81,82],[25,900],{"className":901,"style":902},[441],"margin-right:0.1667em;",[25,904,907,908],{"className":905},[906],"mop","lo",[25,909,911],{"style":910},"margin-right:0.01389em;","g",[25,913],{"className":914,"style":902},[441],[25,916,55],{"className":917},[81,82],[25,919,58],{"className":920},[94],". If we remove the asymptotic notation, this equation is equivalent to ",[25,923,925,964],{"className":924,"translate":29},[28],[25,926,928],{"className":927},[33],[35,929,930],{"xmlns":37},[39,931,932,961],{},[42,933,934,936,938,940,942,945,953,955,957,959],{},[45,935,47],{},[49,937,52],{"stretchy":51},[45,939,55],{},[49,941,58],{"stretchy":51},[49,943,944],{},"≤",[946,947,948,951],"msub",{},[45,949,950],{},"c",[145,952,339],{},[45,954,55],{},[45,956,842],{},[49,958,845],{},[45,960,55],{},[60,962,963],{"encoding":62},"T(n) \\le c_1 n \\log n",[25,965,967,994],{"className":966,"ariaHidden":68},[67],[25,968,970,973,976,979,982,985,988,991],{"className":969},[72],[25,971],{"className":972,"style":77},[76],[25,974,47],{"className":975,"style":83},[81,82],[25,977,52],{"className":978},[87],[25,980,55],{"className":981},[81,82],[25,983,58],{"className":984},[94],[25,986],{"className":987,"style":442},[441],[25,989,944],{"className":990},[446],[25,992],{"className":993,"style":442},[441],[25,995,997,1001,1050,1053,1056,1061,1064],{"className":996},[72],[25,998],{"className":999,"style":1000},[76],"height:0.8889em;vertical-align:-0.1944em;",[25,1002,1004,1007],{"className":1003},[81],[25,1005,950],{"className":1006},[81,82],[25,1008,1011],{"className":1009},[1010],"msupsub",[25,1012,1014,1041],{"className":1013},[484,485],[25,1015,1017,1038],{"className":1016},[489],[25,1018,1021],{"className":1019,"style":1020},[493],"height:0.3011em;",[25,1022,1024,1028],{"style":1023},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[25,1025],{"className":1026,"style":1027},[501],"height:2.7em;",[25,1029,1035],{"className":1030},[1031,1032,1033,1034],"sizing","reset-size6","size3","mtight",[25,1036,339],{"className":1037},[81,1034],[25,1039,519],{"className":1040},[518],[25,1042,1044],{"className":1043},[489],[25,1045,1048],{"className":1046,"style":1047},[493],"height:0.15em;",[25,1049],{},[25,1051,55],{"className":1052},[81,82],[25,1054],{"className":1055,"style":902},[441],[25,1057,907,1059],{"className":1058},[906],[25,1060,911],{"style":910},[25,1062],{"className":1063,"style":902},[441],[25,1065,55],{"className":1066},[81,82],", where ",[25,1069,1071,1089],{"className":1070,"translate":29},[28],[25,1072,1074],{"className":1073},[33],[35,1075,1076],{"xmlns":37},[39,1077,1078,1086],{},[42,1079,1080],{},[946,1081,1082,1084],{},[45,1083,950],{},[145,1085,339],{},[60,1087,1088],{"encoding":62},"c_1",[25,1090,1092],{"className":1091,"ariaHidden":68},[67],[25,1093,1095,1099],{"className":1094},[72],[25,1096],{"className":1097,"style":1098},[76],"height:0.5806em;vertical-align:-0.15em;",[25,1100,1102,1105],{"className":1101},[81],[25,1103,950],{"className":1104},[81,82],[25,1106,1108],{"className":1107},[1010],[25,1109,1111,1131],{"className":1110},[484,485],[25,1112,1114,1128],{"className":1113},[489],[25,1115,1117],{"className":1116,"style":1020},[493],[25,1118,1119,1122],{"style":1023},[25,1120],{"className":1121,"style":1027},[501],[25,1123,1125],{"className":1124},[1031,1032,1033,1034],[25,1126,339],{"className":1127},[81,1034],[25,1129,519],{"className":1130},[518],[25,1132,1134],{"className":1133},[489],[25,1135,1137],{"className":1136,"style":1047},[493],[25,1138],{}," is any positive constant.",[10,1141,1142,1143,1171,1172,1224,1225,1067,1371,1441,1442,1591,1592,1661,1662,1769],{},"According to mathematical induction, we first need to establish that the guess holds for small values of ",[25,1144,1146,1159],{"className":1145,"translate":29},[28],[25,1147,1149],{"className":1148},[33],[35,1150,1151],{"xmlns":37},[39,1152,1153,1157],{},[42,1154,1155],{},[45,1156,55],{},[60,1158,55],{"encoding":62},[25,1160,1162],{"className":1161,"ariaHidden":68},[67],[25,1163,1165,1168],{"className":1164},[72],[25,1166],{"className":1167,"style":121},[76],[25,1169,55],{"className":1170},[81,82],". When ",[25,1173,1175,1193],{"className":1174,"translate":29},[28],[25,1176,1178],{"className":1177},[33],[35,1179,1180],{"xmlns":37},[39,1181,1182,1190],{},[42,1183,1184,1186,1188],{},[45,1185,55],{},[49,1187,301],{},[145,1189,339],{},[60,1191,1192],{"encoding":62},"n=1",[25,1194,1196,1214],{"className":1195,"ariaHidden":68},[67],[25,1197,1199,1202,1205,1208,1211],{"className":1198},[72],[25,1200],{"className":1201,"style":121},[76],[25,1203,55],{"className":1204},[81,82],[25,1206],{"className":1207,"style":442},[441],[25,1209,301],{"className":1210},[446],[25,1212],{"className":1213,"style":442},[441],[25,1215,1217,1221],{"className":1216},[72],[25,1218],{"className":1219,"style":1220},[76],"height:0.6444em;",[25,1222,339],{"className":1223},[81],", according to Equation (1), ",[25,1226,1228,1267],{"className":1227,"translate":29},[28],[25,1229,1231],{"className":1230},[33],[35,1232,1233],{"xmlns":37},[39,1234,1235,1264],{},[42,1236,1237,1239,1241,1243,1245,1247,1249,1251,1253,1255,1257],{},[45,1238,47],{},[49,1240,52],{"stretchy":51},[145,1242,339],{},[49,1244,58],{"stretchy":51},[49,1246,301],{},[45,1248,239],{"mathvariant":142},[49,1250,52],{"stretchy":51},[145,1252,339],{},[49,1254,58],{"stretchy":51},[49,1256,301],{},[946,1258,1259,1262],{},[45,1260,1261],{},"d",[145,1263,339],{},[60,1265,1266],{"encoding":62},"T(1) = \\Theta(1) = d_1",[25,1268,1270,1297,1324],{"className":1269,"ariaHidden":68},[67],[25,1271,1273,1276,1279,1282,1285,1288,1291,1294],{"className":1272},[72],[25,1274],{"className":1275,"style":77},[76],[25,1277,47],{"className":1278,"style":83},[81,82],[25,1280,52],{"className":1281},[87],[25,1283,339],{"className":1284},[81],[25,1286,58],{"className":1287},[94],[25,1289],{"className":1290,"style":442},[441],[25,1292,301],{"className":1293},[446],[25,1295],{"className":1296,"style":442},[441],[25,1298,1300,1303,1306,1309,1312,1315,1318,1321],{"className":1299},[72],[25,1301],{"className":1302,"style":77},[76],[25,1304,239],{"className":1305},[81],[25,1307,52],{"className":1308},[87],[25,1310,339],{"className":1311},[81],[25,1313,58],{"className":1314},[94],[25,1316],{"className":1317,"style":442},[441],[25,1319,301],{"className":1320},[446],[25,1322],{"className":1323,"style":442},[441],[25,1325,1327,1331],{"className":1326},[72],[25,1328],{"className":1329,"style":1330},[76],"height:0.8444em;vertical-align:-0.15em;",[25,1332,1334,1337],{"className":1333},[81],[25,1335,1261],{"className":1336},[81,82],[25,1338,1340],{"className":1339},[1010],[25,1341,1343,1363],{"className":1342},[484,485],[25,1344,1346,1360],{"className":1345},[489],[25,1347,1349],{"className":1348,"style":1020},[493],[25,1350,1351,1354],{"style":1023},[25,1352],{"className":1353,"style":1027},[501],[25,1355,1357],{"className":1356},[1031,1032,1033,1034],[25,1358,339],{"className":1359},[81,1034],[25,1361,519],{"className":1362},[518],[25,1364,1366],{"className":1365},[489],[25,1367,1369],{"className":1368,"style":1047},[493],[25,1370],{},[25,1372,1374,1392],{"className":1373,"translate":29},[28],[25,1375,1377],{"className":1376},[33],[35,1378,1379],{"xmlns":37},[39,1380,1381,1389],{},[42,1382,1383],{},[946,1384,1385,1387],{},[45,1386,1261],{},[145,1388,339],{},[60,1390,1391],{"encoding":62},"d_1",[25,1393,1395],{"className":1394,"ariaHidden":68},[67],[25,1396,1398,1401],{"className":1397},[72],[25,1399],{"className":1400,"style":1330},[76],[25,1402,1404,1407],{"className":1403},[81],[25,1405,1261],{"className":1406},[81,82],[25,1408,1410],{"className":1409},[1010],[25,1411,1413,1433],{"className":1412},[484,485],[25,1414,1416,1430],{"className":1415},[489],[25,1417,1419],{"className":1418,"style":1020},[493],[25,1420,1421,1424],{"style":1023},[25,1422],{"className":1423,"style":1027},[501],[25,1425,1427],{"className":1426},[1031,1032,1033,1034],[25,1428,339],{"className":1429},[81,1034],[25,1431,519],{"className":1432},[518],[25,1434,1436],{"className":1435},[489],[25,1437,1439],{"className":1438,"style":1047},[493],[25,1440],{}," is some constant greater than 0. According to the guess, we hope ",[25,1443,1445,1483],{"className":1444,"translate":29},[28],[25,1446,1448],{"className":1447},[33],[35,1449,1450],{"xmlns":37},[39,1451,1452,1480],{},[42,1453,1454,1456,1458,1460,1462,1464,1470,1472,1474,1476,1478],{},[45,1455,47],{},[49,1457,52],{"stretchy":51},[145,1459,339],{},[49,1461,58],{"stretchy":51},[49,1463,944],{},[946,1465,1466,1468],{},[45,1467,950],{},[145,1469,339],{},[45,1471,842],{},[49,1473,845],{},[145,1475,339],{},[49,1477,301],{},[145,1479,322],{},[60,1481,1482],{"encoding":62},"T(1) \\le c_1 \\log 1 = 0",[25,1484,1486,1513,1582],{"className":1485,"ariaHidden":68},[67],[25,1487,1489,1492,1495,1498,1501,1504,1507,1510],{"className":1488},[72],[25,1490],{"className":1491,"style":77},[76],[25,1493,47],{"className":1494,"style":83},[81,82],[25,1496,52],{"className":1497},[87],[25,1499,339],{"className":1500},[81],[25,1502,58],{"className":1503},[94],[25,1505],{"className":1506,"style":442},[441],[25,1508,944],{"className":1509},[446],[25,1511],{"className":1512,"style":442},[441],[25,1514,1516,1519,1559,1562,1567,1570,1573,1576,1579],{"className":1515},[72],[25,1517],{"className":1518,"style":1000},[76],[25,1520,1522,1525],{"className":1521},[81],[25,1523,950],{"className":1524},[81,82],[25,1526,1528],{"className":1527},[1010],[25,1529,1531,1551],{"className":1530},[484,485],[25,1532,1534,1548],{"className":1533},[489],[25,1535,1537],{"className":1536,"style":1020},[493],[25,1538,1539,1542],{"style":1023},[25,1540],{"className":1541,"style":1027},[501],[25,1543,1545],{"className":1544},[1031,1032,1033,1034],[25,1546,339],{"className":1547},[81,1034],[25,1549,519],{"className":1550},[518],[25,1552,1554],{"className":1553},[489],[25,1555,1557],{"className":1556,"style":1047},[493],[25,1558],{},[25,1560],{"className":1561,"style":902},[441],[25,1563,907,1565],{"className":1564},[906],[25,1566,911],{"style":910},[25,1568],{"className":1569,"style":902},[441],[25,1571,339],{"className":1572},[81],[25,1574],{"className":1575,"style":442},[441],[25,1577,301],{"className":1578},[446],[25,1580],{"className":1581,"style":442},[441],[25,1583,1585,1588],{"className":1584},[72],[25,1586],{"className":1587,"style":1220},[76],[25,1589,322],{"className":1590},[81],", but no matter how we choose ",[25,1593,1595,1612],{"className":1594,"translate":29},[28],[25,1596,1598],{"className":1597},[33],[35,1599,1600],{"xmlns":37},[39,1601,1602,1610],{},[42,1603,1604],{},[946,1605,1606,1608],{},[45,1607,950],{},[145,1609,339],{},[60,1611,1088],{"encoding":62},[25,1613,1615],{"className":1614,"ariaHidden":68},[67],[25,1616,1618,1621],{"className":1617},[72],[25,1619],{"className":1620,"style":1098},[76],[25,1622,1624,1627],{"className":1623},[81],[25,1625,950],{"className":1626},[81,82],[25,1628,1630],{"className":1629},[1010],[25,1631,1633,1653],{"className":1632},[484,485],[25,1634,1636,1650],{"className":1635},[489],[25,1637,1639],{"className":1638,"style":1020},[493],[25,1640,1641,1644],{"style":1023},[25,1642],{"className":1643,"style":1027},[501],[25,1645,1647],{"className":1646},[1031,1032,1033,1034],[25,1648,339],{"className":1649},[81,1034],[25,1651,519],{"className":1652},[518],[25,1654,1656],{"className":1655},[489],[25,1657,1659],{"className":1658,"style":1047},[493],[25,1660],{},", this equation cannot hold because ",[25,1663,1665,1693],{"className":1664,"translate":29},[28],[25,1666,1668],{"className":1667},[33],[35,1669,1670],{"xmlns":37},[39,1671,1672,1690],{},[42,1673,1674,1676,1678,1680,1682,1684],{},[45,1675,47],{},[49,1677,52],{"stretchy":51},[145,1679,339],{},[49,1681,58],{"stretchy":51},[49,1683,301],{},[946,1685,1686,1688],{},[45,1687,1261],{},[145,1689,339],{},[60,1691,1692],{"encoding":62},"T(1) = d_1",[25,1694,1696,1723],{"className":1695,"ariaHidden":68},[67],[25,1697,1699,1702,1705,1708,1711,1714,1717,1720],{"className":1698},[72],[25,1700],{"className":1701,"style":77},[76],[25,1703,47],{"className":1704,"style":83},[81,82],[25,1706,52],{"className":1707},[87],[25,1709,339],{"className":1710},[81],[25,1712,58],{"className":1713},[94],[25,1715],{"className":1716,"style":442},[441],[25,1718,301],{"className":1719},[446],[25,1721],{"className":1722,"style":442},[441],[25,1724,1726,1729],{"className":1725},[72],[25,1727],{"className":1728,"style":1330},[76],[25,1730,1732,1735],{"className":1731},[81],[25,1733,1261],{"className":1734},[81,82],[25,1736,1738],{"className":1737},[1010],[25,1739,1741,1761],{"className":1740},[484,485],[25,1742,1744,1758],{"className":1743},[489],[25,1745,1747],{"className":1746,"style":1020},[493],[25,1748,1749,1752],{"style":1023},[25,1750],{"className":1751,"style":1027},[501],[25,1753,1755],{"className":1754},[1031,1032,1033,1034],[25,1756,339],{"className":1757},[81,1034],[25,1759,519],{"className":1760},[518],[25,1762,1764],{"className":1763},[489],[25,1765,1767],{"className":1766,"style":1047},[493],[25,1768],{}," is always greater than 0. The mathematical induction fails before it even starts.",[10,1771,1772,1773,1836,1837,1889,1890,1933,1934,1962],{},"However, there is no need to worry; this is just a special case caused by ",[25,1774,1776,1798],{"className":1775,"translate":29},[28],[25,1777,1779],{"className":1778},[33],[35,1780,1781],{"xmlns":37},[39,1782,1783,1795],{},[42,1784,1785,1787,1789,1791,1793],{},[45,1786,842],{},[49,1788,845],{},[145,1790,339],{},[49,1792,301],{},[145,1794,322],{},[60,1796,1797],{"encoding":62}," \\log 1 = 0",[25,1799,1801,1827],{"className":1800,"ariaHidden":68},[67],[25,1802,1804,1807,1812,1815,1818,1821,1824],{"className":1803},[72],[25,1805],{"className":1806,"style":1000},[76],[25,1808,907,1810],{"className":1809},[906],[25,1811,911],{"style":910},[25,1813],{"className":1814,"style":902},[441],[25,1816,339],{"className":1817},[81],[25,1819],{"className":1820,"style":442},[441],[25,1822,301],{"className":1823},[446],[25,1825],{"className":1826,"style":442},[441],[25,1828,1830,1833],{"className":1829},[72],[25,1831],{"className":1832,"style":1220},[76],[25,1834,322],{"className":1835},[81],". We can place the initial state of mathematical induction at ",[25,1838,1840,1858],{"className":1839,"translate":29},[28],[25,1841,1843],{"className":1842},[33],[35,1844,1845],{"xmlns":37},[39,1846,1847,1855],{},[42,1848,1849,1851,1853],{},[45,1850,55],{},[49,1852,411],{},[145,1854,339],{},[60,1856,1857],{"encoding":62},"n>1",[25,1859,1861,1880],{"className":1860,"ariaHidden":68},[67],[25,1862,1864,1868,1871,1874,1877],{"className":1863},[72],[25,1865],{"className":1866,"style":1867},[76],"height:0.5782em;vertical-align:-0.0391em;",[25,1869,55],{"className":1870},[81,82],[25,1872],{"className":1873,"style":442},[441],[25,1875,411],{"className":1876},[446],[25,1878],{"className":1879,"style":442},[441],[25,1881,1883,1886],{"className":1882},[72],[25,1884],{"className":1885,"style":1220},[76],[25,1887,339],{"className":1888},[81]," without affecting the results of mathematical induction. We are only interested in the asymptotic properties of ",[25,1891,1893,1912],{"className":1892,"translate":29},[28],[25,1894,1896],{"className":1895},[33],[35,1897,1898],{"xmlns":37},[39,1899,1900,1910],{},[42,1901,1902,1904,1906,1908],{},[45,1903,47],{},[49,1905,52],{"stretchy":51},[45,1907,55],{},[49,1909,58],{"stretchy":51},[60,1911,63],{"encoding":62},[25,1913,1915],{"className":1914,"ariaHidden":68},[67],[25,1916,1918,1921,1924,1927,1930],{"className":1917},[72],[25,1919],{"className":1920,"style":77},[76],[25,1922,47],{"className":1923,"style":83},[81,82],[25,1925,52],{"className":1926},[87],[25,1928,55],{"className":1929},[81,82],[25,1931,58],{"className":1932},[94]," when ",[25,1935,1937,1950],{"className":1936,"translate":29},[28],[25,1938,1940],{"className":1939},[33],[35,1941,1942],{"xmlns":37},[39,1943,1944,1948],{},[42,1945,1946],{},[45,1947,55],{},[60,1949,55],{"encoding":62},[25,1951,1953],{"className":1952,"ariaHidden":68},[67],[25,1954,1956,1959],{"className":1955},[72],[25,1957],{"className":1958,"style":121},[76],[25,1960,55],{"className":1961},[81,82]," is sufficiently large, not in the initial stage.",[10,1964,1965,1966,2017,2018,2298,2299,2426,2427,2632,2633,2702,2703,2773,2774,2843],{},"Now, let ",[25,1967,1969,1987],{"className":1968,"translate":29},[28],[25,1970,1972],{"className":1971},[33],[35,1973,1974],{"xmlns":37},[39,1975,1976,1984],{},[42,1977,1978,1980,1982],{},[45,1979,55],{},[49,1981,301],{},[145,1983,147],{},[60,1985,1986],{"encoding":62},"n = 2",[25,1988,1990,2008],{"className":1989,"ariaHidden":68},[67],[25,1991,1993,1996,1999,2002,2005],{"className":1992},[72],[25,1994],{"className":1995,"style":121},[76],[25,1997,55],{"className":1998},[81,82],[25,2000],{"className":2001,"style":442},[441],[25,2003,301],{"className":2004},[446],[25,2006],{"className":2007,"style":442},[441],[25,2009,2011,2014],{"className":2010},[72],[25,2012],{"className":2013,"style":1220},[76],[25,2015,147],{"className":2016},[81],", then ",[25,2019,2021,2079],{"className":2020,"translate":29},[28],[25,2022,2024],{"className":2023},[33],[35,2025,2026],{"xmlns":37},[39,2027,2028,2076],{},[42,2029,2030,2032,2034,2036,2038,2040,2042,2044,2046,2048,2050,2052,2058,2060,2062,2068,2070],{},[45,2031,47],{},[49,2033,52],{"stretchy":51},[145,2035,147],{},[49,2037,58],{"stretchy":51},[49,2039,301],{},[145,2041,147],{},[45,2043,47],{},[49,2045,52],{"stretchy":51},[145,2047,339],{},[49,2049,58],{"stretchy":51},[49,2051,390],{},[946,2053,2054,2056],{},[45,2055,1261],{},[145,2057,147],{},[49,2059,301],{},[145,2061,147],{},[946,2063,2064,2066],{},[45,2065,1261],{},[145,2067,339],{},[49,2069,390],{},[946,2071,2072,2074],{},[45,2073,1261],{},[145,2075,147],{},[60,2077,2078],{"encoding":62},"T(2) = 2T(1) + d_2 = 2d_1 + d_2",[25,2080,2082,2109,2139,2194,2252],{"className":2081,"ariaHidden":68},[67],[25,2083,2085,2088,2091,2094,2097,2100,2103,2106],{"className":2084},[72],[25,2086],{"className":2087,"style":77},[76],[25,2089,47],{"className":2090,"style":83},[81,82],[25,2092,52],{"className":2093},[87],[25,2095,147],{"className":2096},[81],[25,2098,58],{"className":2099},[94],[25,2101],{"className":2102,"style":442},[441],[25,2104,301],{"className":2105},[446],[25,2107],{"className":2108,"style":442},[441],[25,2110,2112,2115,2118,2121,2124,2127,2130,2133,2136],{"className":2111},[72],[25,2113],{"className":2114,"style":77},[76],[25,2116,147],{"className":2117},[81],[25,2119,47],{"className":2120,"style":83},[81,82],[25,2122,52],{"className":2123},[87],[25,2125,339],{"className":2126},[81],[25,2128,58],{"className":2129},[94],[25,2131],{"className":2132,"style":600},[441],[25,2134,390],{"className":2135},[604],[25,2137],{"className":2138,"style":600},[441],[25,2140,2142,2145,2185,2188,2191],{"className":2141},[72],[25,2143],{"className":2144,"style":1330},[76],[25,2146,2148,2151],{"className":2147},[81],[25,2149,1261],{"className":2150},[81,82],[25,2152,2154],{"className":2153},[1010],[25,2155,2157,2177],{"className":2156},[484,485],[25,2158,2160,2174],{"className":2159},[489],[25,2161,2163],{"className":2162,"style":1020},[493],[25,2164,2165,2168],{"style":1023},[25,2166],{"className":2167,"style":1027},[501],[25,2169,2171],{"className":2170},[1031,1032,1033,1034],[25,2172,147],{"className":2173},[81,1034],[25,2175,519],{"className":2176},[518],[25,2178,2180],{"className":2179},[489],[25,2181,2183],{"className":2182,"style":1047},[493],[25,2184],{},[25,2186],{"className":2187,"style":442},[441],[25,2189,301],{"className":2190},[446],[25,2192],{"className":2193,"style":442},[441],[25,2195,2197,2200,2203,2243,2246,2249],{"className":2196},[72],[25,2198],{"className":2199,"style":1330},[76],[25,2201,147],{"className":2202},[81],[25,2204,2206,2209],{"className":2205},[81],[25,2207,1261],{"className":2208},[81,82],[25,2210,2212],{"className":2211},[1010],[25,2213,2215,2235],{"className":2214},[484,485],[25,2216,2218,2232],{"className":2217},[489],[25,2219,2221],{"className":2220,"style":1020},[493],[25,2222,2223,2226],{"style":1023},[25,2224],{"className":2225,"style":1027},[501],[25,2227,2229],{"className":2228},[1031,1032,1033,1034],[25,2230,339],{"className":2231},[81,1034],[25,2233,519],{"className":2234},[518],[25,2236,2238],{"className":2237},[489],[25,2239,2241],{"className":2240,"style":1047},[493],[25,2242],{},[25,2244],{"className":2245,"style":600},[441],[25,2247,390],{"className":2248},[604],[25,2250],{"className":2251,"style":600},[441],[25,2253,2255,2258],{"className":2254},[72],[25,2256],{"className":2257,"style":1330},[76],[25,2259,2261,2264],{"className":2260},[81],[25,2262,1261],{"className":2263},[81,82],[25,2265,2267],{"className":2266},[1010],[25,2268,2270,2290],{"className":2269},[484,485],[25,2271,2273,2287],{"className":2272},[489],[25,2274,2276],{"className":2275,"style":1020},[493],[25,2277,2278,2281],{"style":1023},[25,2279],{"className":2280,"style":1027},[501],[25,2282,2284],{"className":2283},[1031,1032,1033,1034],[25,2285,147],{"className":2286},[81,1034],[25,2288,519],{"className":2289},[518],[25,2291,2293],{"className":2292},[489],[25,2294,2296],{"className":2295,"style":1047},[493],[25,2297],{},". We hope that ",[25,2300,2302,2336],{"className":2301,"translate":29},[28],[25,2303,2305],{"className":2304},[33],[35,2306,2307],{"xmlns":37},[39,2308,2309,2333],{},[42,2310,2311,2313,2315,2317,2319,2321,2327,2329,2331],{},[45,2312,47],{},[49,2314,52],{"stretchy":51},[145,2316,147],{},[49,2318,58],{"stretchy":51},[49,2320,944],{},[946,2322,2323,2325],{},[45,2324,950],{},[145,2326,339],{},[45,2328,842],{},[49,2330,845],{},[145,2332,147],{},[60,2334,2335],{"encoding":62},"T(2) \\le c_1 \\log 2",[25,2337,2339,2366],{"className":2338,"ariaHidden":68},[67],[25,2340,2342,2345,2348,2351,2354,2357,2360,2363],{"className":2341},[72],[25,2343],{"className":2344,"style":77},[76],[25,2346,47],{"className":2347,"style":83},[81,82],[25,2349,52],{"className":2350},[87],[25,2352,147],{"className":2353},[81],[25,2355,58],{"className":2356},[94],[25,2358],{"className":2359,"style":442},[441],[25,2361,944],{"className":2362},[446],[25,2364],{"className":2365,"style":442},[441],[25,2367,2369,2372,2412,2415,2420,2423],{"className":2368},[72],[25,2370],{"className":2371,"style":1000},[76],[25,2373,2375,2378],{"className":2374},[81],[25,2376,950],{"className":2377},[81,82],[25,2379,2381],{"className":2380},[1010],[25,2382,2384,2404],{"className":2383},[484,485],[25,2385,2387,2401],{"className":2386},[489],[25,2388,2390],{"className":2389,"style":1020},[493],[25,2391,2392,2395],{"style":1023},[25,2393],{"className":2394,"style":1027},[501],[25,2396,2398],{"className":2397},[1031,1032,1033,1034],[25,2399,339],{"className":2400},[81,1034],[25,2402,519],{"className":2403},[518],[25,2405,2407],{"className":2406},[489],[25,2408,2410],{"className":2409,"style":1047},[493],[25,2411],{},[25,2413],{"className":2414,"style":902},[441],[25,2416,907,2418],{"className":2417},[906],[25,2419,911],{"style":910},[25,2421],{"className":2422,"style":902},[441],[25,2424,147],{"className":2425},[81]," holds. Simplifying, we get ",[25,2428,2430,2469],{"className":2429,"translate":29},[28],[25,2431,2433],{"className":2432},[33],[35,2434,2435],{"xmlns":37},[39,2436,2437,2466],{},[42,2438,2439,2445,2448,2454,2456,2462,2464],{},[946,2440,2441,2443],{},[45,2442,950],{},[145,2444,339],{},[49,2446,2447],{},"≥",[946,2449,2450,2452],{},[45,2451,1261],{},[145,2453,339],{},[49,2455,390],{},[946,2457,2458,2460],{},[45,2459,1261],{},[145,2461,147],{},[45,2463,143],{"mathvariant":142},[145,2465,147],{},[60,2467,2468],{"encoding":62},"c_1 \\ge d_1 + d_2\u002F2",[25,2470,2472,2528,2583],{"className":2471,"ariaHidden":68},[67],[25,2473,2475,2479,2519,2522,2525],{"className":2474},[72],[25,2476],{"className":2477,"style":2478},[76],"height:0.786em;vertical-align:-0.15em;",[25,2480,2482,2485],{"className":2481},[81],[25,2483,950],{"className":2484},[81,82],[25,2486,2488],{"className":2487},[1010],[25,2489,2491,2511],{"className":2490},[484,485],[25,2492,2494,2508],{"className":2493},[489],[25,2495,2497],{"className":2496,"style":1020},[493],[25,2498,2499,2502],{"style":1023},[25,2500],{"className":2501,"style":1027},[501],[25,2503,2505],{"className":2504},[1031,1032,1033,1034],[25,2506,339],{"className":2507},[81,1034],[25,2509,519],{"className":2510},[518],[25,2512,2514],{"className":2513},[489],[25,2515,2517],{"className":2516,"style":1047},[493],[25,2518],{},[25,2520],{"className":2521,"style":442},[441],[25,2523,2447],{"className":2524},[446],[25,2526],{"className":2527,"style":442},[441],[25,2529,2531,2534,2574,2577,2580],{"className":2530},[72],[25,2532],{"className":2533,"style":1330},[76],[25,2535,2537,2540],{"className":2536},[81],[25,2538,1261],{"className":2539},[81,82],[25,2541,2543],{"className":2542},[1010],[25,2544,2546,2566],{"className":2545},[484,485],[25,2547,2549,2563],{"className":2548},[489],[25,2550,2552],{"className":2551,"style":1020},[493],[25,2553,2554,2557],{"style":1023},[25,2555],{"className":2556,"style":1027},[501],[25,2558,2560],{"className":2559},[1031,1032,1033,1034],[25,2561,339],{"className":2562},[81,1034],[25,2564,519],{"className":2565},[518],[25,2567,2569],{"className":2568},[489],[25,2570,2572],{"className":2571,"style":1047},[493],[25,2573],{},[25,2575],{"className":2576,"style":600},[441],[25,2578,390],{"className":2579},[604],[25,2581],{"className":2582,"style":600},[441],[25,2584,2586,2589,2629],{"className":2585},[72],[25,2587],{"className":2588,"style":77},[76],[25,2590,2592,2595],{"className":2591},[81],[25,2593,1261],{"className":2594},[81,82],[25,2596,2598],{"className":2597},[1010],[25,2599,2601,2621],{"className":2600},[484,485],[25,2602,2604,2618],{"className":2603},[489],[25,2605,2607],{"className":2606,"style":1020},[493],[25,2608,2609,2612],{"style":1023},[25,2610],{"className":2611,"style":1027},[501],[25,2613,2615],{"className":2614},[1031,1032,1033,1034],[25,2616,147],{"className":2617},[81,1034],[25,2619,519],{"className":2620},[518],[25,2622,2624],{"className":2623},[489],[25,2625,2627],{"className":2626,"style":1047},[493],[25,2628],{},[25,2630,166],{"className":2631},[81],". Since ",[25,2634,2636,2653],{"className":2635,"translate":29},[28],[25,2637,2639],{"className":2638},[33],[35,2640,2641],{"xmlns":37},[39,2642,2643,2651],{},[42,2644,2645],{},[946,2646,2647,2649],{},[45,2648,1261],{},[145,2650,339],{},[60,2652,1391],{"encoding":62},[25,2654,2656],{"className":2655,"ariaHidden":68},[67],[25,2657,2659,2662],{"className":2658},[72],[25,2660],{"className":2661,"style":1330},[76],[25,2663,2665,2668],{"className":2664},[81],[25,2666,1261],{"className":2667},[81,82],[25,2669,2671],{"className":2670},[1010],[25,2672,2674,2694],{"className":2673},[484,485],[25,2675,2677,2691],{"className":2676},[489],[25,2678,2680],{"className":2679,"style":1020},[493],[25,2681,2682,2685],{"style":1023},[25,2683],{"className":2684,"style":1027},[501],[25,2686,2688],{"className":2687},[1031,1032,1033,1034],[25,2689,339],{"className":2690},[81,1034],[25,2692,519],{"className":2693},[518],[25,2695,2697],{"className":2696},[489],[25,2698,2700],{"className":2699,"style":1047},[493],[25,2701],{}," and ",[25,2704,2706,2724],{"className":2705,"translate":29},[28],[25,2707,2709],{"className":2708},[33],[35,2710,2711],{"xmlns":37},[39,2712,2713,2721],{},[42,2714,2715],{},[946,2716,2717,2719],{},[45,2718,1261],{},[145,2720,147],{},[60,2722,2723],{"encoding":62},"d_2",[25,2725,2727],{"className":2726,"ariaHidden":68},[67],[25,2728,2730,2733],{"className":2729},[72],[25,2731],{"className":2732,"style":1330},[76],[25,2734,2736,2739],{"className":2735},[81],[25,2737,1261],{"className":2738},[81,82],[25,2740,2742],{"className":2741},[1010],[25,2743,2745,2765],{"className":2744},[484,485],[25,2746,2748,2762],{"className":2747},[489],[25,2749,2751],{"className":2750,"style":1020},[493],[25,2752,2753,2756],{"style":1023},[25,2754],{"className":2755,"style":1027},[501],[25,2757,2759],{"className":2758},[1031,1032,1033,1034],[25,2760,147],{"className":2761},[81,1034],[25,2763,519],{"className":2764},[518],[25,2766,2768],{"className":2767},[489],[25,2769,2771],{"className":2770,"style":1047},[493],[25,2772],{}," are constants, such ",[25,2775,2777,2794],{"className":2776,"translate":29},[28],[25,2778,2780],{"className":2779},[33],[35,2781,2782],{"xmlns":37},[39,2783,2784,2792],{},[42,2785,2786],{},[946,2787,2788,2790],{},[45,2789,950],{},[145,2791,339],{},[60,2793,1088],{"encoding":62},[25,2795,2797],{"className":2796,"ariaHidden":68},[67],[25,2798,2800,2803],{"className":2799},[72],[25,2801],{"className":2802,"style":1098},[76],[25,2804,2806,2809],{"className":2805},[81],[25,2807,950],{"className":2808},[81,82],[25,2810,2812],{"className":2811},[1010],[25,2813,2815,2835],{"className":2814},[484,485],[25,2816,2818,2832],{"className":2817},[489],[25,2819,2821],{"className":2820,"style":1020},[493],[25,2822,2823,2826],{"style":1023},[25,2824],{"className":2825,"style":1027},[501],[25,2827,2829],{"className":2828},[1031,1032,1033,1034],[25,2830,339],{"className":2831},[81,1034],[25,2833,519],{"className":2834},[518],[25,2836,2838],{"className":2837},[489],[25,2839,2841],{"className":2840,"style":1047},[493],[25,2842],{}," exists, and the initial condition holds.",[10,2845,2846,2847,2882,2883,3363,3364,3495],{},"Next, mathematical induction assumes that the solution holds at ",[25,2848,2850,2867],{"className":2849,"translate":29},[28],[25,2851,2853],{"className":2852},[33],[35,2854,2855],{"xmlns":37},[39,2856,2857,2865],{},[42,2858,2859,2861,2863],{},[45,2860,55],{},[45,2862,143],{"mathvariant":142},[145,2864,147],{},[60,2866,150],{"encoding":62},[25,2868,2870],{"className":2869,"ariaHidden":68},[67],[25,2871,2873,2876,2879],{"className":2872},[72],[25,2874],{"className":2875,"style":77},[76],[25,2877,55],{"className":2878},[81,82],[25,2880,166],{"className":2881},[81],", i.e., ",[25,2884,2886,2966],{"className":2885,"translate":29},[28],[25,2887,2889],{"className":2888},[33],[35,2890,2891],{"xmlns":37},[39,2892,2893,2963],{},[42,2894,2895,2897,2899,2901,2903,2905,2907,2909,2915,2922,2924,2926,2932,2934,2940,2946,2948,2950,2952,2954,2956,2959,2961],{},[45,2896,47],{},[49,2898,52],{"stretchy":51},[45,2900,55],{},[45,2902,143],{"mathvariant":142},[145,2904,147],{},[49,2906,58],{"stretchy":51},[49,2908,944],{},[946,2910,2911,2913],{},[45,2912,950],{},[145,2914,339],{},[2916,2917,2918,2920],"mfrac",{},[45,2919,55],{},[145,2921,147],{},[45,2923,842],{},[49,2925,845],{},[2916,2927,2928,2930],{},[45,2929,55],{},[145,2931,147],{},[49,2933,301],{},[2916,2935,2936,2938],{},[145,2937,339],{},[145,2939,147],{},[946,2941,2942,2944],{},[45,2943,950],{},[145,2945,339],{},[45,2947,55],{},[49,2949,52],{"stretchy":51},[45,2951,842],{},[49,2953,845],{},[45,2955,55],{},[49,2957,2958],{},"−",[145,2960,339],{},[49,2962,58],{"stretchy":51},[60,2964,2965],{"encoding":62},"T(n\u002F2) \\le c_1 \\frac{n}{2} \\log\\frac{n}{2} = \\frac{1}{2}c_1n(\\log n - 1)",[25,2967,2969,2999,3209,3351],{"className":2968,"ariaHidden":68},[67],[25,2970,2972,2975,2978,2981,2984,2987,2990,2993,2996],{"className":2971},[72],[25,2973],{"className":2974,"style":77},[76],[25,2976,47],{"className":2977,"style":83},[81,82],[25,2979,52],{"className":2980},[87],[25,2982,55],{"className":2983},[81,82],[25,2985,166],{"className":2986},[81],[25,2988,58],{"className":2989},[94],[25,2991],{"className":2992,"style":442},[441],[25,2994,944],{"className":2995},[446],[25,2997],{"className":2998,"style":442},[441],[25,3000,3002,3006,3046,3121,3124,3129,3132,3200,3203,3206],{"className":3001},[72],[25,3003],{"className":3004,"style":3005},[76],"height:1.0404em;vertical-align:-0.345em;",[25,3007,3009,3012],{"className":3008},[81],[25,3010,950],{"className":3011},[81,82],[25,3013,3015],{"className":3014},[1010],[25,3016,3018,3038],{"className":3017},[484,485],[25,3019,3021,3035],{"className":3020},[489],[25,3022,3024],{"className":3023,"style":1020},[493],[25,3025,3026,3029],{"style":1023},[25,3027],{"className":3028,"style":1027},[501],[25,3030,3032],{"className":3031},[1031,1032,1033,1034],[25,3033,339],{"className":3034},[81,1034],[25,3036,519],{"className":3037},[518],[25,3039,3041],{"className":3040},[489],[25,3042,3044],{"className":3043,"style":1047},[493],[25,3045],{},[25,3047,3049,3052,3118],{"className":3048},[81],[25,3050],{"className":3051},[87,721],[25,3053,3055],{"className":3054},[2916],[25,3056,3058,3109],{"className":3057},[484,485],[25,3059,3061,3106],{"className":3060},[489],[25,3062,3065,3080,3091],{"className":3063,"style":3064},[493],"height:0.6954em;",[25,3066,3068,3071],{"style":3067},"top:-2.655em;",[25,3069],{"className":3070,"style":548},[501],[25,3072,3074],{"className":3073},[1031,1032,1033,1034],[25,3075,3077],{"className":3076},[81,1034],[25,3078,147],{"className":3079},[81,1034],[25,3081,3083,3086],{"style":3082},"top:-3.23em;",[25,3084],{"className":3085,"style":548},[501],[25,3087],{"className":3088,"style":3090},[3089],"frac-line","border-bottom-width:0.04em;",[25,3092,3094,3097],{"style":3093},"top:-3.394em;",[25,3095],{"className":3096,"style":548},[501],[25,3098,3100],{"className":3099},[1031,1032,1033,1034],[25,3101,3103],{"className":3102},[81,1034],[25,3104,55],{"className":3105},[81,82,1034],[25,3107,519],{"className":3108},[518],[25,3110,3112],{"className":3111},[489],[25,3113,3116],{"className":3114,"style":3115},[493],"height:0.345em;",[25,3117],{},[25,3119],{"className":3120},[94,721],[25,3122],{"className":3123,"style":902},[441],[25,3125,907,3127],{"className":3126},[906],[25,3128,911],{"style":910},[25,3130],{"className":3131,"style":902},[441],[25,3133,3135,3138,3197],{"className":3134},[81],[25,3136],{"className":3137},[87,721],[25,3139,3141],{"className":3140},[2916],[25,3142,3144,3189],{"className":3143},[484,485],[25,3145,3147,3186],{"className":3146},[489],[25,3148,3150,3164,3172],{"className":3149,"style":3064},[493],[25,3151,3152,3155],{"style":3067},[25,3153],{"className":3154,"style":548},[501],[25,3156,3158],{"className":3157},[1031,1032,1033,1034],[25,3159,3161],{"className":3160},[81,1034],[25,3162,147],{"className":3163},[81,1034],[25,3165,3166,3169],{"style":3082},[25,3167],{"className":3168,"style":548},[501],[25,3170],{"className":3171,"style":3090},[3089],[25,3173,3174,3177],{"style":3093},[25,3175],{"className":3176,"style":548},[501],[25,3178,3180],{"className":3179},[1031,1032,1033,1034],[25,3181,3183],{"className":3182},[81,1034],[25,3184,55],{"className":3185},[81,82,1034],[25,3187,519],{"className":3188},[518],[25,3190,3192],{"className":3191},[489],[25,3193,3195],{"className":3194,"style":3115},[493],[25,3196],{},[25,3198],{"className":3199},[94,721],[25,3201],{"className":3202,"style":442},[441],[25,3204,301],{"className":3205},[446],[25,3207],{"className":3208,"style":442},[441],[25,3210,3212,3216,3285,3325,3328,3331,3336,3339,3342,3345,3348],{"className":3211},[72],[25,3213],{"className":3214,"style":3215},[76],"height:1.1901em;vertical-align:-0.345em;",[25,3217,3219,3222,3282],{"className":3218},[81],[25,3220],{"className":3221},[87,721],[25,3223,3225],{"className":3224},[2916],[25,3226,3228,3274],{"className":3227},[484,485],[25,3229,3231,3271],{"className":3230},[489],[25,3232,3235,3249,3257],{"className":3233,"style":3234},[493],"height:0.8451em;",[25,3236,3237,3240],{"style":3067},[25,3238],{"className":3239,"style":548},[501],[25,3241,3243],{"className":3242},[1031,1032,1033,1034],[25,3244,3246],{"className":3245},[81,1034],[25,3247,147],{"className":3248},[81,1034],[25,3250,3251,3254],{"style":3082},[25,3252],{"className":3253,"style":548},[501],[25,3255],{"className":3256,"style":3090},[3089],[25,3258,3259,3262],{"style":3093},[25,3260],{"className":3261,"style":548},[501],[25,3263,3265],{"className":3264},[1031,1032,1033,1034],[25,3266,3268],{"className":3267},[81,1034],[25,3269,339],{"className":3270},[81,1034],[25,3272,519],{"className":3273},[518],[25,3275,3277],{"className":3276},[489],[25,3278,3280],{"className":3279,"style":3115},[493],[25,3281],{},[25,3283],{"className":3284},[94,721],[25,3286,3288,3291],{"className":3287},[81],[25,3289,950],{"className":3290},[81,82],[25,3292,3294],{"className":3293},[1010],[25,3295,3297,3317],{"className":3296},[484,485],[25,3298,3300,3314],{"className":3299},[489],[25,3301,3303],{"className":3302,"style":1020},[493],[25,3304,3305,3308],{"style":1023},[25,3306],{"className":3307,"style":1027},[501],[25,3309,3311],{"className":3310},[1031,1032,1033,1034],[25,3312,339],{"className":3313},[81,1034],[25,3315,519],{"className":3316},[518],[25,3318,3320],{"className":3319},[489],[25,3321,3323],{"className":3322,"style":1047},[493],[25,3324],{},[25,3326,55],{"className":3327},[81,82],[25,3329,52],{"className":3330},[87],[25,3332,907,3334],{"className":3333},[906],[25,3335,911],{"style":910},[25,3337],{"className":3338,"style":902},[441],[25,3340,55],{"className":3341},[81,82],[25,3343],{"className":3344,"style":600},[441],[25,3346,2958],{"className":3347},[604],[25,3349],{"className":3350,"style":600},[441],[25,3352,3354,3357,3360],{"className":3353},[72],[25,3355],{"className":3356,"style":77},[76],[25,3358,339],{"className":3359},[81],[25,3361,58],{"className":3362},[94],". Then we prove that ",[25,3365,3367,3402],{"className":3366,"translate":29},[28],[25,3368,3370],{"className":3369},[33],[35,3371,3372],{"xmlns":37},[39,3373,3374,3400],{},[42,3375,3376,3378,3380,3382,3384,3386,3392,3394,3396,3398],{},[45,3377,47],{},[49,3379,52],{"stretchy":51},[45,3381,55],{},[49,3383,58],{"stretchy":51},[49,3385,944],{},[946,3387,3388,3390],{},[45,3389,950],{},[145,3391,339],{},[45,3393,55],{},[45,3395,842],{},[49,3397,845],{},[45,3399,55],{},[60,3401,963],{"encoding":62},[25,3403,3405,3432],{"className":3404,"ariaHidden":68},[67],[25,3406,3408,3411,3414,3417,3420,3423,3426,3429],{"className":3407},[72],[25,3409],{"className":3410,"style":77},[76],[25,3412,47],{"className":3413,"style":83},[81,82],[25,3415,52],{"className":3416},[87],[25,3418,55],{"className":3419},[81,82],[25,3421,58],{"className":3422},[94],[25,3424],{"className":3425,"style":442},[441],[25,3427,944],{"className":3428},[446],[25,3430],{"className":3431,"style":442},[441],[25,3433,3435,3438,3478,3481,3484,3489,3492],{"className":3434},[72],[25,3436],{"className":3437,"style":1000},[76],[25,3439,3441,3444],{"className":3440},[81],[25,3442,950],{"className":3443},[81,82],[25,3445,3447],{"className":3446},[1010],[25,3448,3450,3470],{"className":3449},[484,485],[25,3451,3453,3467],{"className":3452},[489],[25,3454,3456],{"className":3455,"style":1020},[493],[25,3457,3458,3461],{"style":1023},[25,3459],{"className":3460,"style":1027},[501],[25,3462,3464],{"className":3463},[1031,1032,1033,1034],[25,3465,339],{"className":3466},[81,1034],[25,3468,519],{"className":3469},[518],[25,3471,3473],{"className":3472},[489],[25,3474,3476],{"className":3475,"style":1047},[493],[25,3477],{},[25,3479,55],{"className":3480},[81,82],[25,3482],{"className":3483,"style":902},[441],[25,3485,907,3487],{"className":3486},[906],[25,3488,911],{"style":910},[25,3490],{"className":3491,"style":902},[441],[25,3493,55],{"className":3494},[81,82]," also holds.",[10,3497,3498,3499,3550],{},"Substituting ",[25,3500,3502,3526],{"className":3501,"translate":29},[28],[25,3503,3505],{"className":3504},[33],[35,3506,3507],{"xmlns":37},[39,3508,3509,3523],{},[42,3510,3511,3513,3515,3517,3519,3521],{},[45,3512,47],{},[49,3514,52],{"stretchy":51},[45,3516,55],{},[45,3518,143],{"mathvariant":142},[145,3520,147],{},[49,3522,58],{"stretchy":51},[60,3524,3525],{"encoding":62},"T(n\u002F2)",[25,3527,3529],{"className":3528,"ariaHidden":68},[67],[25,3530,3532,3535,3538,3541,3544,3547],{"className":3531},[72],[25,3533],{"className":3534,"style":77},[76],[25,3536,47],{"className":3537,"style":83},[81,82],[25,3539,52],{"className":3540},[87],[25,3542,55],{"className":3543},[81,82],[25,3545,166],{"className":3546},[81],[25,3548,58],{"className":3549},[94]," into Equation (1), we get",[25,3552,3554],{"className":3553,"translate":29},[277],[25,3555,3557,3788],{"className":3556,"translate":29},[28],[25,3558,3560],{"className":3559},[33],[35,3561,3562],{"xmlns":37,"display":286},[39,3563,3564,3785],{},[308,3565,3568],{"rowspacing":3566,"columnspacing":3567},"0.16em","1em",[314,3569,3570,3574,3778,3781],{},[317,3571],{"className":3572},[3573],"mtr-glue",[317,3575,3576],{},[320,3577,3578],{"scriptlevel":322,"displaystyle":68},[308,3579,3582,3634,3692,3746],{"rowspacing":310,"columnalign":3580,"columnspacing":3581},"right left","0em",[314,3583,3584,3598],{},[317,3585,3586],{},[320,3587,3588],{"scriptlevel":322,"displaystyle":68},[42,3589,3590,3592,3594,3596],{},[45,3591,47],{},[49,3593,52],{"stretchy":51},[45,3595,55],{},[49,3597,58],{"stretchy":51},[317,3599,3600],{},[320,3601,3602],{"scriptlevel":322,"displaystyle":68},[42,3603,3604,3606,3608,3610,3612,3624,3626,3628,3630,3632],{},[42,3605],{},[49,3607,301],{},[145,3609,147],{},[45,3611,47],{},[42,3613,3614,3616,3622],{},[49,3615,52],{"fence":68},[2916,3617,3618,3620],{},[45,3619,55],{},[145,3621,147],{},[49,3623,58],{"fence":68},[49,3625,390],{},[45,3627,239],{"mathvariant":142},[49,3629,52],{"stretchy":51},[45,3631,55],{},[49,3633,58],{"stretchy":51},[314,3635,3636,3642],{},[317,3637,3638],{},[320,3639,3640],{"scriptlevel":322,"displaystyle":68},[42,3641],{},[317,3643,3644],{},[320,3645,3646],{"scriptlevel":322,"displaystyle":68},[42,3647,3648,3650,3652,3654,3660,3666,3668,3670,3672,3674,3676,3678,3680,3682,3684,3690],{},[42,3649],{},[49,3651,944],{},[145,3653,147],{},[2916,3655,3656,3658],{},[145,3657,339],{},[145,3659,147],{},[946,3661,3662,3664],{},[45,3663,950],{},[145,3665,339],{},[45,3667,55],{},[49,3669,52],{"stretchy":51},[45,3671,842],{},[49,3673,845],{},[45,3675,55],{},[49,3677,2958],{},[145,3679,339],{},[49,3681,58],{"stretchy":51},[49,3683,390],{},[946,3685,3686,3688],{},[45,3687,1261],{},[145,3689,147],{},[45,3691,55],{},[314,3693,3694,3700],{},[317,3695,3696],{},[320,3697,3698],{"scriptlevel":322,"displaystyle":68},[42,3699],{},[317,3701,3702],{},[320,3703,3704],{"scriptlevel":322,"displaystyle":68},[42,3705,3706,3708,3710,3716,3718,3720,3722,3724,3726,3728,3734,3736,3742,3744],{},[42,3707],{},[49,3709,301],{},[946,3711,3712,3714],{},[45,3713,950],{},[145,3715,339],{},[45,3717,55],{},[45,3719,842],{},[49,3721,845],{},[45,3723,55],{},[49,3725,390],{},[49,3727,52],{"stretchy":51},[946,3729,3730,3732],{},[45,3731,1261],{},[145,3733,147],{},[49,3735,2958],{},[946,3737,3738,3740],{},[45,3739,950],{},[145,3741,339],{},[49,3743,58],{"stretchy":51},[45,3745,55],{},[314,3747,3748,3754],{},[317,3749,3750],{},[320,3751,3752],{"scriptlevel":322,"displaystyle":68},[42,3753],{},[317,3755,3756],{},[320,3757,3758],{"scriptlevel":322,"displaystyle":68},[42,3759,3760,3762,3764,3770,3772,3774,3776],{},[42,3761],{},[49,3763,944],{},[946,3765,3766,3768],{},[45,3767,950],{},[145,3769,339],{},[45,3771,55],{},[45,3773,842],{},[49,3775,845],{},[45,3777,55],{},[317,3779],{"className":3780},[3573],[317,3782],{"className":3783},[3784],"mml-eqn-num",[60,3786,3787],{"encoding":62},"\\begin{equation} \\begin{aligned} T(n) & = 2 T\\left(\\frac{n}{2}\\right) + \\Theta(n) \\\\ & \\le 2 \\frac{1}{2}c_1n(\\log n - 1) + d_2n \\\\ & = c_1 n \\log n + (d_2 - c_1) n \\\\ & \\le c_1 n \\log n \\end{aligned} 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0",[25,4612,4614,4669,4725],{"className":4613,"ariaHidden":68},[67],[25,4615,4617,4620,4660,4663,4666],{"className":4616},[72],[25,4618],{"className":4619,"style":1330},[76],[25,4621,4623,4626],{"className":4622},[81],[25,4624,1261],{"className":4625},[81,82],[25,4627,4629],{"className":4628},[1010],[25,4630,4632,4652],{"className":4631},[484,485],[25,4633,4635,4649],{"className":4634},[489],[25,4636,4638],{"className":4637,"style":1020},[493],[25,4639,4640,4643],{"style":1023},[25,4641],{"className":4642,"style":1027},[501],[25,4644,4646],{"className":4645},[1031,1032,1033,1034],[25,4647,147],{"className":4648},[81,1034],[25,4650,519],{"className":4651},[518],[25,4653,4655],{"className":4654},[489],[25,4656,4658],{"className":4657,"style":1047},[493],[25,4659],{},[25,4661],{"className":4662,"style":600},[441],[25,4664,2958],{"className":4665},[604],[25,4667],{"className":4668,"style":600},[441],[25,4670,4672,4676,4716,4719,4722],{"className":4671},[72],[25,4673],{"className":4674,"style":4675},[76],"height:0.6891em;vertical-align:-0.15em;",[25,4677,4679,4682],{"className":4678},[81],[25,4680,950],{"className":4681},[81,82],[25,4683,4685],{"className":4684},[1010],[25,4686,4688,4708],{"className":4687},[484,485],[25,4689,4691,4705],{"className":4690},[489],[25,4692,4694],{"className":4693,"style":1020},[493],[25,4695,4696,4699],{"style":1023},[25,4697],{"className":4698,"style":1027},[501],[25,4700,4702],{"className":4701},[1031,1032,1033,1034],[25,4703,339],{"className":4704},[81,1034],[25,4706,519],{"className":4707},[518],[25,4709,4711],{"className":4710},[489],[25,4712,4714],{"className":4713,"style":1047},[493],[25,4715],{},[25,4717],{"className":4718,"style":442},[441],[25,4720,4605],{"className":4721},[446],[25,4723],{"className":4724,"style":442},[441],[25,4726,4728,4731],{"className":4727},[72],[25,4729],{"className":4730,"style":1220},[76],[25,4732,322],{"className":4733},[81],", obviously such ",[25,4736,4738,4755],{"className":4737,"translate":29},[28],[25,4739,4741],{"className":4740},[33],[35,4742,4743],{"xmlns":37},[39,4744,4745,4753],{},[42,4746,4747],{},[946,4748,4749,4751],{},[45,4750,950],{},[145,4752,339],{},[60,4754,1088],{"encoding":62},[25,4756,4758],{"className":4757,"ariaHidden":68},[67],[25,4759,4761,4764],{"className":4760},[72],[25,4762],{"className":4763,"style":1098},[76],[25,4765,4767,4770],{"className":4766},[81],[25,4768,950],{"className":4769},[81,82],[25,4771,4773],{"className":4772},[1010],[25,4774,4776,4796],{"className":4775},[484,485],[25,4777,4779,4793],{"className":4778},[489],[25,4780,4782],{"className":4781,"style":1020},[493],[25,4783,4784,4787],{"style":1023},[25,4785],{"className":4786,"style":1027},[501],[25,4788,4790],{"className":4789},[1031,1032,1033,1034],[25,4791,339],{"className":4792},[81,1034],[25,4794,519],{"className":4795},[518],[25,4797,4799],{"className":4798},[489],[25,4800,4802],{"className":4801,"style":1047},[493],[25,4803],{}," exists, so the proof is complete.",[10,4806,4807,4808,4908,4909,5010,5011,5113,5114,5246,5247,5282,5283,5311],{},"Thus, we have proved that ",[25,4809,4811,4846],{"className":4810,"translate":29},[28],[25,4812,4814],{"className":4813},[33],[35,4815,4816],{"xmlns":37},[39,4817,4818,4844],{},[42,4819,4820,4822,4824,4826,4828,4830,4832,4834,4836,4838,4840,4842],{},[45,4821,47],{},[49,4823,52],{"stretchy":51},[45,4825,55],{},[49,4827,58],{"stretchy":51},[49,4829,301],{},[45,4831,835],{},[49,4833,52],{"stretchy":51},[45,4835,55],{},[45,4837,842],{},[49,4839,845],{},[45,4841,55],{},[49,4843,58],{"stretchy":51},[60,4845,852],{"encoding":62},[25,4847,4849,4876],{"className":4848,"ariaHidden":68},[67],[25,4850,4852,4855,4858,4861,4864,4867,4870,4873],{"className":4851},[72],[25,4853],{"className":4854,"style":77},[76],[25,4856,47],{"className":4857,"style":83},[81,82],[25,4859,52],{"className":4860},[87],[25,4862,55],{"className":4863},[81,82],[25,4865,58],{"className":4866},[94],[25,4868],{"className":4869,"style":442},[441],[25,4871,301],{"className":4872},[446],[25,4874],{"className":4875,"style":442},[441],[25,4877,4879,4882,4885,4888,4891,4894,4899,4902,4905],{"className":4878},[72],[25,4880],{"className":4881,"style":77},[76],[25,4883,835],{"className":4884,"style":892},[81,82],[25,4886,52],{"className":4887},[87],[25,4889,55],{"className":4890},[81,82],[25,4892],{"className":4893,"style":902},[441],[25,4895,907,4897],{"className":4896},[906],[25,4898,911],{"style":910},[25,4900],{"className":4901,"style":902},[441],[25,4903,55],{"className":4904},[81,82],[25,4906,58],{"className":4907},[94]," is the solution to the recurrence relation (1). However, this is not the end; this asymptotic notation is not a tight bound. Can we further guess that ",[25,4910,4912,4948],{"className":4911,"translate":29},[28],[25,4913,4915],{"className":4914},[33],[35,4916,4917],{"xmlns":37},[39,4918,4919,4945],{},[42,4920,4921,4923,4925,4927,4929,4931,4933,4935,4937,4939,4941,4943],{},[45,4922,47],{},[49,4924,52],{"stretchy":51},[45,4926,55],{},[49,4928,58],{"stretchy":51},[49,4930,301],{},[45,4932,239],{"mathvariant":142},[49,4934,52],{"stretchy":51},[45,4936,55],{},[45,4938,842],{},[49,4940,845],{},[45,4942,55],{},[49,4944,58],{"stretchy":51},[60,4946,4947],{"encoding":62},"T(n) = \\Theta(n \\log n)",[25,4949,4951,4978],{"className":4950,"ariaHidden":68},[67],[25,4952,4954,4957,4960,4963,4966,4969,4972,4975],{"className":4953},[72],[25,4955],{"className":4956,"style":77},[76],[25,4958,47],{"className":4959,"style":83},[81,82],[25,4961,52],{"className":4962},[87],[25,4964,55],{"className":4965},[81,82],[25,4967,58],{"className":4968},[94],[25,4970],{"className":4971,"style":442},[441],[25,4973,301],{"className":4974},[446],[25,4976],{"className":4977,"style":442},[441],[25,4979,4981,4984,4987,4990,4993,4996,5001,5004,5007],{"className":4980},[72],[25,4982],{"className":4983,"style":77},[76],[25,4985,239],{"className":4986},[81],[25,4988,52],{"className":4989},[87],[25,4991,55],{"className":4992},[81,82],[25,4994],{"className":4995,"style":902},[441],[25,4997,907,4999],{"className":4998},[906],[25,5000,911],{"style":910},[25,5002],{"className":5003,"style":902},[441],[25,5005,55],{"className":5006},[81,82],[25,5008,58],{"className":5009},[94]," holds? This is indeed a good idea. So, as we proceed, as long as we can prove that ",[25,5012,5014,5051],{"className":5013,"translate":29},[28],[25,5015,5017],{"className":5016},[33],[35,5018,5019],{"xmlns":37},[39,5020,5021,5048],{},[42,5022,5023,5025,5027,5029,5031,5033,5036,5038,5040,5042,5044,5046],{},[45,5024,47],{},[49,5026,52],{"stretchy":51},[45,5028,55],{},[49,5030,58],{"stretchy":51},[49,5032,301],{},[45,5034,5035],{"mathvariant":142},"Ω",[49,5037,52],{"stretchy":51},[45,5039,55],{},[45,5041,842],{},[49,5043,845],{},[45,5045,55],{},[49,5047,58],{"stretchy":51},[60,5049,5050],{"encoding":62},"T(n) = \\Omega(n \\log n)",[25,5052,5054,5081],{"className":5053,"ariaHidden":68},[67],[25,5055,5057,5060,5063,5066,5069,5072,5075,5078],{"className":5056},[72],[25,5058],{"className":5059,"style":77},[76],[25,5061,47],{"className":5062,"style":83},[81,82],[25,5064,52],{"className":5065},[87],[25,5067,55],{"className":5068},[81,82],[25,5070,58],{"className":5071},[94],[25,5073],{"className":5074,"style":442},[441],[25,5076,301],{"className":5077},[446],[25,5079],{"className":5080,"style":442},[441],[25,5082,5084,5087,5090,5093,5096,5099,5104,5107,5110],{"className":5083},[72],[25,5085],{"className":5086,"style":77},[76],[25,5088,5035],{"className":5089},[81],[25,5091,52],{"className":5092},[87],[25,5094,55],{"className":5095},[81,82],[25,5097],{"className":5098,"style":902},[441],[25,5100,907,5102],{"className":5101},[906],[25,5103,911],{"style":910},[25,5105],{"className":5106,"style":902},[441],[25,5108,55],{"className":5109},[81,82],[25,5111,58],{"className":5112},[94]," holds, we can obtain the tight bound mentioned above. Removing the asymptotic notation, this equation is equivalent to ",[25,5115,5117,5153],{"className":5116,"translate":29},[28],[25,5118,5120],{"className":5119},[33],[35,5121,5122],{"xmlns":37},[39,5123,5124,5150],{},[42,5125,5126,5128,5130,5132,5134,5136,5142,5144,5146,5148],{},[45,5127,47],{},[49,5129,52],{"stretchy":51},[45,5131,55],{},[49,5133,58],{"stretchy":51},[49,5135,2447],{},[946,5137,5138,5140],{},[45,5139,950],{},[145,5141,147],{},[45,5143,55],{},[45,5145,842],{},[49,5147,845],{},[45,5149,55],{},[60,5151,5152],{"encoding":62},"T(n) \\ge c_2 n \\log n",[25,5154,5156,5183],{"className":5155,"ariaHidden":68},[67],[25,5157,5159,5162,5165,5168,5171,5174,5177,5180],{"className":5158},[72],[25,5160],{"className":5161,"style":77},[76],[25,5163,47],{"className":5164,"style":83},[81,82],[25,5166,52],{"className":5167},[87],[25,5169,55],{"className":5170},[81,82],[25,5172,58],{"className":5173},[94],[25,5175],{"className":5176,"style":442},[441],[25,5178,2447],{"className":5179},[446],[25,5181],{"className":5182,"style":442},[441],[25,5184,5186,5189,5229,5232,5235,5240,5243],{"className":5185},[72],[25,5187],{"className":5188,"style":1000},[76],[25,5190,5192,5195],{"className":5191},[81],[25,5193,950],{"className":5194},[81,82],[25,5196,5198],{"className":5197},[1010],[25,5199,5201,5221],{"className":5200},[484,485],[25,5202,5204,5218],{"className":5203},[489],[25,5205,5207],{"className":5206,"style":1020},[493],[25,5208,5209,5212],{"style":1023},[25,5210],{"className":5211,"style":1027},[501],[25,5213,5215],{"className":5214},[1031,1032,1033,1034],[25,5216,147],{"className":5217},[81,1034],[25,5219,519],{"className":5220},[518],[25,5222,5224],{"className":5223},[489],[25,5225,5227],{"className":5226,"style":1047},[493],[25,5228],{},[25,5230,55],{"className":5231},[81,82],[25,5233],{"className":5234,"style":902},[441],[25,5236,907,5238],{"className":5237},[906],[25,5239,911],{"style":910},[25,5241],{"className":5242,"style":902},[441],[25,5244,55],{"className":5245},[81,82],". Using mathematical induction again, confirming that this holds in the initial stage, and then assuming that the solution holds at ",[25,5248,5250,5267],{"className":5249,"translate":29},[28],[25,5251,5253],{"className":5252},[33],[35,5254,5255],{"xmlns":37},[39,5256,5257,5265],{},[42,5258,5259,5261,5263],{},[45,5260,55],{},[45,5262,143],{"mathvariant":142},[145,5264,147],{},[60,5266,150],{"encoding":62},[25,5268,5270],{"className":5269,"ariaHidden":68},[67],[25,5271,5273,5276,5279],{"className":5272},[72],[25,5274],{"className":5275,"style":77},[76],[25,5277,55],{"className":5278},[81,82],[25,5280,166],{"className":5281},[81],", we can derive that the solution holds at ",[25,5284,5286,5299],{"className":5285,"translate":29},[28],[25,5287,5289],{"className":5288},[33],[35,5290,5291],{"xmlns":37},[39,5292,5293,5297],{},[42,5294,5295],{},[45,5296,55],{},[60,5298,55],{"encoding":62},[25,5300,5302],{"className":5301,"ariaHidden":68},[67],[25,5303,5305,5308],{"className":5304},[72],[25,5306],{"className":5307,"style":121},[76],[25,5309,55],{"className":5310},[81,82],". Therefore, the proof is complete. Since the proof steps are almost identical to those just mentioned, we will not elaborate on them, and you can try it yourself.",[10,5313,5314],{},"The substitution method relies on accurate guesses, which can be impractical. Therefore, in practice, a more common approach is to find a guess using other methods and then use the substitution method to verify the guess. The recursive tree method we will introduce next provides this possibility.",[14,5316,5318],{"id":5317},"recursive-tree","Recursive Tree",[10,5320,5321,5322,5489],{},"Expanding the recurrence relation layer by layer directly yields a tree structure. For example, let's take ",[25,5323,5325,5378],{"className":5324,"translate":29},[28],[25,5326,5328],{"className":5327},[33],[35,5329,5330],{"xmlns":37},[39,5331,5332,5375],{},[42,5333,5334,5336,5338,5340,5342,5344,5347,5349,5351,5353,5355,5358,5360,5362,5364,5366,5373],{},[45,5335,47],{},[49,5337,52],{"stretchy":51},[45,5339,55],{},[49,5341,58],{"stretchy":51},[49,5343,301],{},[145,5345,5346],{},"3",[45,5348,47],{},[49,5350,52],{"stretchy":51},[45,5352,55],{},[45,5354,143],{"mathvariant":142},[145,5356,5357],{},"4",[49,5359,58],{"stretchy":51},[49,5361,390],{},[45,5363,239],{"mathvariant":142},[49,5365,52],{"stretchy":51},[5367,5368,5369,5371],"msup",{},[45,5370,55],{},[145,5372,147],{},[49,5374,58],{"stretchy":51},[60,5376,5377],{"encoding":62},"T(n) = 3 T(n\u002F4) + \\Theta(n^2)",[25,5379,5381,5408,5442],{"className":5380,"ariaHidden":68},[67],[25,5382,5384,5387,5390,5393,5396,5399,5402,5405],{"className":5383},[72],[25,5385],{"className":5386,"style":77},[76],[25,5388,47],{"className":5389,"style":83},[81,82],[25,5391,52],{"className":5392},[87],[25,5394,55],{"className":5395},[81,82],[25,5397,58],{"className":5398},[94],[25,5400],{"className":5401,"style":442},[441],[25,5403,301],{"className":5404},[446],[25,5406],{"className":5407,"style":442},[441],[25,5409,5411,5414,5417,5420,5423,5426,5430,5433,5436,5439],{"className":5410},[72],[25,5412],{"className":5413,"style":77},[76],[25,5415,5346],{"className":5416},[81],[25,5418,47],{"className":5419,"style":83},[81,82],[25,5421,52],{"className":5422},[87],[25,5424,55],{"className":5425},[81,82],[25,5427,5429],{"className":5428},[81],"\u002F4",[25,5431,58],{"className":5432},[94],[25,5434],{"className":5435,"style":600},[441],[25,5437,390],{"className":5438},[604],[25,5440],{"className":5441,"style":600},[441],[25,5443,5445,5449,5452,5455,5486],{"className":5444},[72],[25,5446],{"className":5447,"style":5448},[76],"height:1.0641em;vertical-align:-0.25em;",[25,5450,239],{"className":5451},[81],[25,5453,52],{"className":5454},[87],[25,5456,5458,5461],{"className":5457},[81],[25,5459,55],{"className":5460},[81,82],[25,5462,5464],{"className":5463},[1010],[25,5465,5467],{"className":5466},[484],[25,5468,5470],{"className":5469},[489],[25,5471,5474],{"className":5472,"style":5473},[493],"height:0.8141em;",[25,5475,5477,5480],{"style":5476},"top:-3.063em;margin-right:0.05em;",[25,5478],{"className":5479,"style":1027},[501],[25,5481,5483],{"className":5482},[1031,1032,1033,1034],[25,5484,147],{"className":5485},[81,1034],[25,5487,58],{"className":5488},[94]," as an example and see how to draw its recursive tree.",[10,5491,5492],{},"Expanding the first layer, we draw three child nodes under the root node.",[10,5494,5495,5496,5560,5561,5635],{},"Here, ",[25,5497,5499,5519],{"className":5498,"translate":29},[28],[25,5500,5502],{"className":5501},[33],[35,5503,5504],{"xmlns":37},[39,5505,5506,5516],{},[42,5507,5508,5510],{},[45,5509,950],{},[5367,5511,5512,5514],{},[45,5513,55],{},[145,5515,147],{},[60,5517,5518],{"encoding":62},"cn^2",[25,5520,5522],{"className":5521,"ariaHidden":68},[67],[25,5523,5525,5528,5531],{"className":5524},[72],[25,5526],{"className":5527,"style":5473},[76],[25,5529,950],{"className":5530},[81,82],[25,5532,5534,5537],{"className":5533},[81],[25,5535,55],{"className":5536},[81,82],[25,5538,5540],{"className":5539},[1010],[25,5541,5543],{"className":5542},[484],[25,5544,5546],{"className":5545},[489],[25,5547,5549],{"className":5548,"style":5473},[493],[25,5550,5551,5554],{"style":5476},[25,5552],{"className":5553,"style":1027},[501],[25,5555,5557],{"className":5556},[1031,1032,1033,1034],[25,5558,147],{"className":5559},[81,1034]," represents the cost of decomposition and merging at this layer, which corresponds to the part represented by ",[25,5562,5564,5588],{"className":5563,"translate":29},[28],[25,5565,5567],{"className":5566},[33],[35,5568,5569],{"xmlns":37},[39,5570,5571,5585],{},[42,5572,5573,5575,5577,5583],{},[45,5574,239],{"mathvariant":142},[49,5576,52],{"stretchy":51},[5367,5578,5579,5581],{},[45,5580,55],{},[145,5582,147],{},[49,5584,58],{"stretchy":51},[60,5586,5587],{"encoding":62},"\\Theta(n^2)",[25,5589,5591],{"className":5590,"ariaHidden":68},[67],[25,5592,5594,5597,5600,5603,5632],{"className":5593},[72],[25,5595],{"className":5596,"style":5448},[76],[25,5598,239],{"className":5599},[81],[25,5601,52],{"className":5602},[87],[25,5604,5606,5609],{"className":5605},[81],[25,5607,55],{"className":5608},[81,82],[25,5610,5612],{"className":5611},[1010],[25,5613,5615],{"className":5614},[484],[25,5616,5618],{"className":5617},[489],[25,5619,5621],{"className":5620,"style":5473},[493],[25,5622,5623,5626],{"style":5476},[25,5624],{"className":5625,"style":1027},[501],[25,5627,5629],{"className":5628},[1031,1032,1033,1034],[25,5630,147],{"className":5631},[81,1034],[25,5633,58],{"className":5634},[94]," in the recurrence relation.",[10,5637,5638],{},"Next, further expand to the second layer.",[10,5640,5641],{},[5642,5643],"img",{"alt":5318,"src":5644},"https:\u002F\u002Fimage-assets.dreams.plus\u002F20231218224727.webp",[10,5646,5647,5648,5792,5793,5866,5867,5895,5896,270],{},"Here, the cost of each node in the first layer is still ",[25,5649,5651,5679],{"className":5650,"translate":29},[28],[25,5652,5654],{"className":5653},[33],[35,5655,5656],{"xmlns":37},[39,5657,5658,5676],{},[42,5659,5660,5662,5664,5670],{},[45,5661,950],{},[49,5663,52],{"stretchy":51},[2916,5665,5666,5668],{},[45,5667,55],{},[145,5669,5357],{},[5367,5671,5672,5674],{},[49,5673,58],{"stretchy":51},[145,5675,147],{},[60,5677,5678],{"encoding":62},"c(\\frac{n}{4})^2",[25,5680,5682],{"className":5681,"ariaHidden":68},[67],[25,5683,5685,5689,5692,5695,5763],{"className":5684},[72],[25,5686],{"className":5687,"style":5688},[76],"height:1.1591em;vertical-align:-0.345em;",[25,5690,950],{"className":5691},[81,82],[25,5693,52],{"className":5694},[87],[25,5696,5698,5701,5760],{"className":5697},[81],[25,5699],{"className":5700},[87,721],[25,5702,5704],{"className":5703},[2916],[25,5705,5707,5752],{"className":5706},[484,485],[25,5708,5710,5749],{"className":5709},[489],[25,5711,5713,5727,5735],{"className":5712,"style":3064},[493],[25,5714,5715,5718],{"style":3067},[25,5716],{"className":5717,"style":548},[501],[25,5719,5721],{"className":5720},[1031,1032,1033,1034],[25,5722,5724],{"className":5723},[81,1034],[25,5725,5357],{"className":5726},[81,1034],[25,5728,5729,5732],{"style":3082},[25,5730],{"className":5731,"style":548},[501],[25,5733],{"className":5734,"style":3090},[3089],[25,5736,5737,5740],{"style":3093},[25,5738],{"className":5739,"style":548},[501],[25,5741,5743],{"className":5742},[1031,1032,1033,1034],[25,5744,5746],{"className":5745},[81,1034],[25,5747,55],{"className":5748},[81,82,1034],[25,5750,519],{"className":5751},[518],[25,5753,5755],{"className":5754},[489],[25,5756,5758],{"className":5757,"style":3115},[493],[25,5759],{},[25,5761],{"className":5762},[94,721],[25,5764,5766,5769],{"className":5765},[94],[25,5767,58],{"className":5768},[94],[25,5770,5772],{"className":5771},[1010],[25,5773,5775],{"className":5774},[484],[25,5776,5778],{"className":5777},[489],[25,5779,5781],{"className":5780,"style":5473},[493],[25,5782,5783,5786],{"style":5476},[25,5784],{"className":5785,"style":1027},[501],[25,5787,5789],{"className":5788},[1031,1032,1033,1034],[25,5790,147],{"className":5791},[81,1034],", still corresponding to the part represented by ",[25,5794,5796,5819],{"className":5795,"translate":29},[28],[25,5797,5799],{"className":5798},[33],[35,5800,5801],{"xmlns":37},[39,5802,5803,5817],{},[42,5804,5805,5807,5809,5815],{},[45,5806,239],{"mathvariant":142},[49,5808,52],{"stretchy":51},[5367,5810,5811,5813],{},[45,5812,55],{},[145,5814,147],{},[49,5816,58],{"stretchy":51},[60,5818,5587],{"encoding":62},[25,5820,5822],{"className":5821,"ariaHidden":68},[67],[25,5823,5825,5828,5831,5834,5863],{"className":5824},[72],[25,5826],{"className":5827,"style":5448},[76],[25,5829,239],{"className":5830},[81],[25,5832,52],{"className":5833},[87],[25,5835,5837,5840],{"className":5836},[81],[25,5838,55],{"className":5839},[81,82],[25,5841,5843],{"className":5842},[1010],[25,5844,5846],{"className":5845},[484],[25,5847,5849],{"className":5848},[489],[25,5850,5852],{"className":5851,"style":5473},[493],[25,5853,5854,5857],{"style":5476},[25,5855],{"className":5856,"style":1027},[501],[25,5858,5860],{"className":5859},[1031,1032,1033,1034],[25,5861,147],{"className":5862},[81,1034],[25,5864,58],{"className":5865},[94]," in the recurrence relation, but the data size has changed from ",[25,5868,5870,5883],{"className":5869,"translate":29},[28],[25,5871,5873],{"className":5872},[33],[35,5874,5875],{"xmlns":37},[39,5876,5877,5881],{},[42,5878,5879],{},[45,5880,55],{},[60,5882,55],{"encoding":62},[25,5884,5886],{"className":5885,"ariaHidden":68},[67],[25,5887,5889,5892],{"className":5888},[72],[25,5890],{"className":5891,"style":121},[76],[25,5893,55],{"className":5894},[81,82]," to ",[25,5897,5899,5917],{"className":5898,"translate":29},[28],[25,5900,5902],{"className":5901},[33],[35,5903,5904],{"xmlns":37},[39,5905,5906,5914],{},[42,5907,5908,5910,5912],{},[45,5909,55],{},[45,5911,143],{"mathvariant":142},[145,5913,5357],{},[60,5915,5916],{"encoding":62},"n\u002F4",[25,5918,5920],{"className":5919,"ariaHidden":68},[67],[25,5921,5923,5926,5929],{"className":5922},[72],[25,5924],{"className":5925,"style":77},[76],[25,5927,55],{"className":5928},[81,82],[25,5930,5429],{"className":5931},[81],[10,5933,5934],{},"Continue expanding in this way until the leaf nodes.",[10,5936,5937],{},[5642,5938],{"alt":5318,"src":5939},"https:\u002F\u002Fimage-assets.dreams.plus\u002F20231218224626.webp",[10,5941,5942,5943,6031,6032,6287,6288,270],{},"This completes a complete recursive tree. The data size of each layer is reduced to one-fourth of the previous one, until the data size of the leaf node is 1, so the total number of layers is ",[25,5944,5946,5970],{"className":5945,"translate":29},[28],[25,5947,5949],{"className":5948},[33],[35,5950,5951],{"xmlns":37},[39,5952,5953,5967],{},[42,5954,5955,5965],{},[946,5956,5957,5963],{},[42,5958,5959,5961],{},[45,5960,842],{},[49,5962,845],{},[145,5964,5357],{},[45,5966,55],{},[60,5968,5969],{"encoding":62},"\\log_4 n",[25,5971,5973],{"className":5972,"ariaHidden":68},[67],[25,5974,5976,5980,6025,6028],{"className":5975},[72],[25,5977],{"className":5978,"style":5979},[76],"height:0.9386em;vertical-align:-0.2441em;",[25,5981,5983,5988],{"className":5982},[906],[25,5984,907,5986],{"className":5985},[906],[25,5987,911],{"style":910},[25,5989,5991],{"className":5990},[1010],[25,5992,5994,6016],{"className":5993},[484,485],[25,5995,5997,6013],{"className":5996},[489],[25,5998,6001],{"className":5999,"style":6000},[493],"height:0.207em;",[25,6002,6004,6007],{"style":6003},"top:-2.4559em;margin-right:0.05em;",[25,6005],{"className":6006,"style":1027},[501],[25,6008,6010],{"className":6009},[1031,1032,1033,1034],[25,6011,5357],{"className":6012},[81,1034],[25,6014,519],{"className":6015},[518],[25,6017,6019],{"className":6018},[489],[25,6020,6023],{"className":6021,"style":6022},[493],"height:0.2441em;",[25,6024],{},[25,6026],{"className":6027,"style":902},[441],[25,6029,55],{"className":6030},[81,82],". The rightmost side of the figure calculates the total cost of all nodes in each layer, and the calculation is simple: multiply the cost of a single node by the number of nodes in that layer. In the last layer, the cost of a single node is a constant, and the number of nodes is ",[25,6033,6035,6085],{"className":6034,"translate":29},[28],[25,6036,6038],{"className":6037},[33],[35,6039,6040],{"xmlns":37},[39,6041,6042,6082],{},[42,6043,6044,6062,6064],{},[5367,6045,6046,6048],{},[145,6047,5346],{},[42,6049,6050,6060],{},[946,6051,6052,6058],{},[42,6053,6054,6056],{},[45,6055,842],{},[49,6057,845],{},[145,6059,5357],{},[45,6061,55],{},[49,6063,301],{},[5367,6065,6066,6068],{},[45,6067,55],{},[42,6069,6070,6080],{},[946,6071,6072,6078],{},[42,6073,6074,6076],{},[45,6075,842],{},[49,6077,845],{},[145,6079,5357],{},[145,6081,5346],{},[60,6083,6084],{"encoding":62},"3^{\\log_4 n} = n^{\\log_4 3}",[25,6086,6088,6197],{"className":6087,"ariaHidden":68},[67],[25,6089,6091,6095,6188,6191,6194],{"className":6090},[72],[25,6092],{"className":6093,"style":6094},[76],"height:0.8491em;",[25,6096,6098,6101],{"className":6097},[81],[25,6099,5346],{"className":6100},[81],[25,6102,6104],{"className":6103},[1010],[25,6105,6107],{"className":6106},[484],[25,6108,6110],{"className":6109},[489],[25,6111,6113],{"className":6112,"style":6094},[493],[25,6114,6115,6118],{"style":5476},[25,6116],{"className":6117,"style":1027},[501],[25,6119,6121],{"className":6120},[1031,1032,1033,1034],[25,6122,6124,6181,6185],{"className":6123},[81,1034],[25,6125,6127,6141],{"className":6126},[906,1034],[25,6128,6130,6134,6138],{"className":6129},[906,1034],[25,6131,6133],{"className":6132},[1034],"l",[25,6135,6137],{"className":6136},[1034],"o",[25,6139,911],{"className":6140,"style":910},[1034],[25,6142,6144],{"className":6143},[1010],[25,6145,6147,6172],{"className":6146},[484,485],[25,6148,6150,6169],{"className":6149},[489],[25,6151,6154],{"className":6152,"style":6153},[493],"height:0.1944em;",[25,6155,6157,6161],{"style":6156},"top:-2.2341em;margin-right:0.0714em;",[25,6158],{"className":6159,"style":6160},[501],"height:2.5em;",[25,6162,6166],{"className":6163},[1031,6164,6165,1034],"reset-size3","size1",[25,6167,5357],{"className":6168},[81,1034],[25,6170,519],{"className":6171},[518],[25,6173,6175],{"className":6174},[489],[25,6176,6179],{"className":6177,"style":6178},[493],"height:0.2659em;",[25,6180],{},[25,6182],{"className":6183,"style":6184},[441,1034],"margin-right:0.1952em;",[25,6186,55],{"className":6187},[81,82,1034],[25,6189],{"className":6190,"style":442},[441],[25,6192,301],{"className":6193},[446],[25,6195],{"className":6196,"style":442},[441],[25,6198,6200,6203],{"className":6199},[72],[25,6201],{"className":6202,"style":6094},[76],[25,6204,6206,6209],{"className":6205},[81],[25,6207,55],{"className":6208},[81,82],[25,6210,6212],{"className":6211},[1010],[25,6213,6215],{"className":6214},[484],[25,6216,6218],{"className":6217},[489],[25,6219,6221],{"className":6220,"style":6094},[493],[25,6222,6223,6226],{"style":5476},[25,6224],{"className":6225,"style":1027},[501],[25,6227,6229],{"className":6228},[1031,1032,1033,1034],[25,6230,6232,6281,6284],{"className":6231},[81,1034],[25,6233,6235,6247],{"className":6234},[906,1034],[25,6236,6238,6241,6244],{"className":6237},[906,1034],[25,6239,6133],{"className":6240},[1034],[25,6242,6137],{"className":6243},[1034],[25,6245,911],{"className":6246,"style":910},[1034],[25,6248,6250],{"className":6249},[1010],[25,6251,6253,6273],{"className":6252},[484,485],[25,6254,6256,6270],{"className":6255},[489],[25,6257,6259],{"className":6258,"style":6153},[493],[25,6260,6261,6264],{"style":6156},[25,6262],{"className":6263,"style":6160},[501],[25,6265,6267],{"className":6266},[1031,6164,6165,1034],[25,6268,5357],{"className":6269},[81,1034],[25,6271,519],{"className":6272},[518],[25,6274,6276],{"className":6275},[489],[25,6277,6279],{"className":6278,"style":6178},[493],[25,6280],{},[25,6282],{"className":6283,"style":6184},[441,1034],[25,6285,5346],{"className":6286},[81,1034],", and we denote the total cost as ",[25,6289,6291,6327],{"className":6290,"translate":29},[28],[25,6292,6294],{"className":6293},[33],[35,6295,6296],{"xmlns":37},[39,6297,6298,6324],{},[42,6299,6300,6302,6304,6322],{},[45,6301,239],{"mathvariant":142},[49,6303,52],{"stretchy":51},[5367,6305,6306,6308],{},[45,6307,55],{},[42,6309,6310,6320],{},[946,6311,6312,6318],{},[42,6313,6314,6316],{},[45,6315,842],{},[49,6317,845],{},[145,6319,5357],{},[145,6321,5346],{},[49,6323,58],{"stretchy":51},[60,6325,6326],{"encoding":62},"\\Theta(n^{\\log_4 3})",[25,6328,6330],{"className":6329,"ariaHidden":68},[67],[25,6331,6333,6337,6340,6343,6427],{"className":6332},[72],[25,6334],{"className":6335,"style":6336},[76],"height:1.0991em;vertical-align:-0.25em;",[25,6338,239],{"className":6339},[81],[25,6341,52],{"className":6342},[87],[25,6344,6346,6349],{"className":6345},[81],[25,6347,55],{"className":6348},[81,82],[25,6350,6352],{"className":6351},[1010],[25,6353,6355],{"className":6354},[484],[25,6356,6358],{"className":6357},[489],[25,6359,6361],{"className":6360,"style":6094},[493],[25,6362,6363,6366],{"style":5476},[25,6364],{"className":6365,"style":1027},[501],[25,6367,6369],{"className":6368},[1031,1032,1033,1034],[25,6370,6372,6421,6424],{"className":6371},[81,1034],[25,6373,6375,6387],{"className":6374},[906,1034],[25,6376,6378,6381,6384],{"className":6377},[906,1034],[25,6379,6133],{"className":6380},[1034],[25,6382,6137],{"className":6383},[1034],[25,6385,911],{"className":6386,"style":910},[1034],[25,6388,6390],{"className":6389},[1010],[25,6391,6393,6413],{"className":6392},[484,485],[25,6394,6396,6410],{"className":6395},[489],[25,6397,6399],{"className":6398,"style":6153},[493],[25,6400,6401,6404],{"style":6156},[25,6402],{"className":6403,"style":6160},[501],[25,6405,6407],{"className":6406},[1031,6164,6165,1034],[25,6408,5357],{"className":6409},[81,1034],[25,6411,519],{"className":6412},[518],[25,6414,6416],{"className":6415},[489],[25,6417,6419],{"className":6418,"style":6178},[493],[25,6420],{},[25,6422],{"className":6423,"style":6184},[441,1034],[25,6425,5346],{"className":6426},[81,1034],[25,6428,58],{"className":6429},[94],[10,6431,6432],{},"Next, we just need to add up the costs of all nodes to obtain the solution to the recurrence relation.",[25,6434,6436],{"className":6435,"translate":29},[277],[25,6437,6439,6946],{"className":6438,"translate":29},[28],[25,6440,6442],{"className":6441},[33],[35,6443,6444],{"xmlns":37,"display":286},[39,6445,6446,6943],{},[308,6447,6448,6598,6701,6786,6854,6913],{"rowspacing":310,"columnalign":3580,"columnspacing":3581},[314,6449,6450,6464],{},[317,6451,6452],{},[320,6453,6454],{"scriptlevel":322,"displaystyle":68},[42,6455,6456,6458,6460,6462],{},[45,6457,47],{},[49,6459,52],{"stretchy":51},[45,6461,55],{},[49,6463,58],{"stretchy":51},[317,6465,6466],{},[320,6467,6468],{"scriptlevel":322,"displaystyle":68},[42,6469,6470,6472,6474,6476,6482,6484,6491,6493,6499,6501,6517,6519,6525,6527,6530,6532,6564,6566,6572,6574,6576,6578,6596],{},[42,6471],{},[49,6473,301],{},[45,6475,950],{},[5367,6477,6478,6480],{},[45,6479,55],{},[145,6481,147],{},[49,6483,390],{},[2916,6485,6486,6488],{},[145,6487,5346],{},[145,6489,6490],{},"16",[45,6492,950],{},[5367,6494,6495,6497],{},[45,6496,55],{},[145,6498,147],{},[49,6500,390],{},[5367,6502,6503,6515],{},[42,6504,6505,6507,6513],{},[49,6506,52],{"fence":68},[2916,6508,6509,6511],{},[145,6510,5346],{},[145,6512,6490],{},[49,6514,58],{"fence":68},[145,6516,147],{},[45,6518,950],{},[5367,6520,6521,6523],{},[45,6522,55],{},[145,6524,147],{},[49,6526,390],{},[49,6528,6529],{},"⋯",[49,6531,390],{},[5367,6533,6534,6546],{},[42,6535,6536,6538,6544],{},[49,6537,52],{"fence":68},[2916,6539,6540,6542],{},[145,6541,5346],{},[145,6543,6490],{},[49,6545,58],{"fence":68},[42,6547,6548,6558,6560,6562],{},[946,6549,6550,6556],{},[42,6551,6552,6554],{},[45,6553,842],{},[49,6555,845],{},[145,6557,5357],{},[45,6559,55],{},[49,6561,2958],{},[145,6563,339],{},[45,6565,950],{},[5367,6567,6568,6570],{},[45,6569,55],{},[145,6571,147],{},[49,6573,390],{},[45,6575,239],{"mathvariant":142},[49,6577,52],{"stretchy":51},[5367,6579,6580,6582],{},[45,6581,55],{},[42,6583,6584,6594],{},[946,6585,6586,6592],{},[42,6587,6588,6590],{},[45,6589,842],{},[49,6591,845],{},[145,6593,5357],{},[145,6595,5346],{},[49,6597,58],{"stretchy":51},[314,6599,6600,6606],{},[317,6601,6602],{},[320,6603,6604],{"scriptlevel":322,"displaystyle":68},[42,6605],{},[317,6607,6608],{},[320,6609,6610],{"scriptlevel":322,"displaystyle":68},[42,6611,6612,6614,6616,6649,6675,6677,6679,6681,6699],{},[42,6613],{},[49,6615,301],{},[6617,6618,6619,6622,6631],"munderover",{},[49,6620,6621],{},"∑",[42,6623,6624,6627,6629],{},[45,6625,6626],{},"i",[49,6628,301],{},[145,6630,322],{},[42,6632,6633,6643,6645,6647],{},[946,6634,6635,6641],{},[42,6636,6637,6639],{},[45,6638,842],{},[49,6640,845],{},[145,6642,5357],{},[45,6644,55],{},[49,6646,2958],{},[145,6648,339],{},[42,6650,6651,6667,6669],{},[5367,6652,6653,6665],{},[42,6654,6655,6657,6663],{},[49,6656,52],{"fence":68},[2916,6658,6659,6661],{},[145,6660,5346],{},[145,6662,6490],{},[49,6664,58],{"fence":68},[45,6666,6626],{},[45,6668,950],{},[5367,6670,6671,6673],{},[45,6672,55],{},[145,6674,147],{},[49,6676,390],{},[45,6678,239],{"mathvariant":142},[49,6680,52],{"stretchy":51},[5367,6682,6683,6685],{},[45,6684,55],{},[42,6686,6687,6697],{},[946,6688,6689,6695],{},[42,6690,6691,6693],{},[45,6692,842],{},[49,6694,845],{},[145,6696,5357],{},[145,6698,5346],{},[49,6700,58],{"stretchy":51},[314,6702,6703,6709],{},[317,6704,6705],{},[320,6706,6707],{"scriptlevel":322,"displaystyle":68},[42,6708],{},[317,6710,6711],{},[320,6712,6713],{"scriptlevel":322,"displaystyle":68},[42,6714,6715,6717,6719,6734,6760,6762,6764,6766,6784],{},[42,6716],{},[49,6718,4605],{},[6617,6720,6721,6723,6731],{},[49,6722,6621],{},[42,6724,6725,6727,6729],{},[45,6726,6626],{},[49,6728,301],{},[145,6730,322],{},[45,6732,6733],{"mathvariant":142},"∞",[42,6735,6736,6752,6754],{},[5367,6737,6738,6750],{},[42,6739,6740,6742,6748],{},[49,6741,52],{"fence":68},[2916,6743,6744,6746],{},[145,6745,5346],{},[145,6747,6490],{},[49,6749,58],{"fence":68},[45,6751,6626],{},[45,6753,950],{},[5367,6755,6756,6758],{},[45,6757,55],{},[145,6759,147],{},[49,6761,390],{},[45,6763,239],{"mathvariant":142},[49,6765,52],{"stretchy":51},[5367,6767,6768,6770],{},[45,6769,55],{},[42,6771,6772,6782],{},[946,6773,6774,6780],{},[42,6775,6776,6778],{},[45,6777,842],{},[49,6779,845],{},[145,6781,5357],{},[145,6783,5346],{},[49,6785,58],{"stretchy":51},[314,6787,6788,6794],{},[317,6789,6790],{},[320,6791,6792],{"scriptlevel":322,"displaystyle":68},[42,6793],{},[317,6795,6796],{},[320,6797,6798],{"scriptlevel":322,"displaystyle":68},[42,6799,6800,6802,6804,6820,6822,6828,6830,6832,6834,6852],{},[42,6801],{},[49,6803,301],{},[2916,6805,6806,6808],{},[145,6807,339],{},[42,6809,6810,6812,6814,6816,6818],{},[145,6811,339],{},[49,6813,2958],{},[145,6815,5346],{},[45,6817,143],{"mathvariant":142},[145,6819,6490],{},[45,6821,950],{},[5367,6823,6824,6826],{},[45,6825,55],{},[145,6827,147],{},[49,6829,390],{},[45,6831,239],{"mathvariant":142},[49,6833,52],{"stretchy":51},[5367,6835,6836,6838],{},[45,6837,55],{},[42,6839,6840,6850],{},[946,6841,6842,6848],{},[42,6843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T(n) &= cn^2 + \\frac{3}{16} cn^2 + \\left(\\frac{3}{16}\\right)^2 cn^2 + \\cdots + \\left( \\frac{3}{16} \\right)^{\\log_4 n-1} cn^2 + \\Theta(n^{\\log_4 3}) \\\\ &=\\sum_{i=0}^{\\log_4 n-1}{\\left(\\frac{3}{16}\\right)^i cn^2} + \\Theta(n^{\\log_4 3})\\\\ &\\lt \\sum_{i=0}^{\\infty}{\\left(\\frac{3}{16}\\right)^i cn^2} + \\Theta(n^{\\log_4 3}) \\\\ &= \\frac{1}{1 - 3\u002F16}cn^2 + \\Theta(n^{\\log_4 3}) \\\\ &= \\frac{16}{13} cn^2 + \\Theta(n^{\\log_4 3}) \\\\ &= O(n^2) 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in the third line, we made a simplification by replacing a finite geometric series with an infinite geometric series. Although the total size has increased, this infinite decreasing geometric series has a limit, so this replacement can simplify the derivation process. The second term in the fifth line is a lower-order term relative to the first term, so it is discarded, resulting in the final result.",[10,8951,8952,8953,9016,9017,9128,9129,270],{},"Similar to before, we have only proven the upper bound of the recurrence relation. Naturally, we want to try whether it can also be proven as a lower bound. In the above equation, each term in the summation contains ",[25,8954,8956,8975],{"className":8955,"translate":29},[28],[25,8957,8959],{"className":8958},[33],[35,8960,8961],{"xmlns":37},[39,8962,8963,8973],{},[42,8964,8965,8967],{},[45,8966,950],{},[5367,8968,8969,8971],{},[45,8970,55],{},[145,8972,147],{},[60,8974,5518],{"encoding":62},[25,8976,8978],{"className":8977,"ariaHidden":68},[67],[25,8979,8981,8984,8987],{"className":8980},[72],[25,8982],{"className":8983,"style":5473},[76],[25,8985,950],{"className":8986},[81,82],[25,8988,8990,8993],{"className":8989},[81],[25,8991,55],{"className":8992},[81,82],[25,8994,8996],{"className":8995},[1010],[25,8997,8999],{"className":8998},[484],[25,9000,9002],{"className":9001},[489],[25,9003,9005],{"className":9004,"style":5473},[493],[25,9006,9007,9010],{"style":5476},[25,9008],{"className":9009,"style":1027},[501],[25,9011,9013],{"className":9012},[1031,1032,1033,1034],[25,9014,147],{"className":9015},[81,1034],", and each term is positive, so ",[25,9018,9020,9054],{"className":9019,"translate":29},[28],[25,9021,9023],{"className":9022},[33],[35,9024,9025],{"xmlns":37},[39,9026,9027,9051],{},[42,9028,9029,9031,9033,9035,9037,9039,9041,9043,9049],{},[45,9030,47],{},[49,9032,52],{"stretchy":51},[45,9034,55],{},[49,9036,58],{"stretchy":51},[49,9038,301],{},[45,9040,5035],{"mathvariant":142},[49,9042,52],{"stretchy":51},[5367,9044,9045,9047],{},[45,9046,55],{},[145,9048,147],{},[49,9050,58],{"stretchy":51},[60,9052,9053],{"encoding":62},"T(n) = \\Omega(n^2)",[25,9055,9057,9084],{"className":9056,"ariaHidden":68},[67],[25,9058,9060,9063,9066,9069,9072,9075,9078,9081],{"className":9059},[72],[25,9061],{"className":9062,"style":77},[76],[25,9064,47],{"className":9065,"style":83},[81,82],[25,9067,52],{"className":9068},[87],[25,9070,55],{"className":9071},[81,82],[25,9073,58],{"className":9074},[94],[25,9076],{"className":9077,"style":442},[441],[25,9079,301],{"className":9080},[446],[25,9082],{"className":9083,"style":442},[441],[25,9085,9087,9090,9093,9096,9125],{"className":9086},[72],[25,9088],{"className":9089,"style":5448},[76],[25,9091,5035],{"className":9092},[81],[25,9094,52],{"className":9095},[87],[25,9097,9099,9102],{"className":9098},[81],[25,9100,55],{"className":9101},[81,82],[25,9103,9105],{"className":9104},[1010],[25,9106,9108],{"className":9107},[484],[25,9109,9111],{"className":9110},[489],[25,9112,9114],{"className":9113,"style":5473},[493],[25,9115,9116,9119],{"style":5476},[25,9117],{"className":9118,"style":1027},[501],[25,9120,9122],{"className":9121},[1031,1032,1033,1034],[25,9123,147],{"className":9124},[81,1034],[25,9126,58],{"className":9127},[94]," obviously holds. Therefore, the solution to the recurrence relation is ",[25,9130,9132,9166],{"className":9131,"translate":29},[28],[25,9133,9135],{"className":9134},[33],[35,9136,9137],{"xmlns":37},[39,9138,9139,9163],{},[42,9140,9141,9143,9145,9147,9149,9151,9153,9155,9161],{},[45,9142,47],{},[49,9144,52],{"stretchy":51},[45,9146,55],{},[49,9148,58],{"stretchy":51},[49,9150,301],{},[45,9152,239],{"mathvariant":142},[49,9154,52],{"stretchy":51},[5367,9156,9157,9159],{},[45,9158,55],{},[145,9160,147],{},[49,9162,58],{"stretchy":51},[60,9164,9165],{"encoding":62},"T(n) = \\Theta(n^2)",[25,9167,9169,9196],{"className":9168,"ariaHidden":68},[67],[25,9170,9172,9175,9178,9181,9184,9187,9190,9193],{"className":9171},[72],[25,9173],{"className":9174,"style":77},[76],[25,9176,47],{"className":9177,"style":83},[81,82],[25,9179,52],{"className":9180},[87],[25,9182,55],{"className":9183},[81,82],[25,9185,58],{"className":9186},[94],[25,9188],{"className":9189,"style":442},[441],[25,9191,301],{"className":9192},[446],[25,9194],{"className":9195,"style":442},[441],[25,9197,9199,9202,9205,9208,9237],{"className":9198},[72],[25,9200],{"className":9201,"style":5448},[76],[25,9203,239],{"className":9204},[81],[25,9206,52],{"className":9207},[87],[25,9209,9211,9214],{"className":9210},[81],[25,9212,55],{"className":9213},[81,82],[25,9215,9217],{"className":9216},[1010],[25,9218,9220],{"className":9219},[484],[25,9221,9223],{"className":9222},[489],[25,9224,9226],{"className":9225,"style":5473},[493],[25,9227,9228,9231],{"style":5476},[25,9229],{"className":9230,"style":1027},[501],[25,9232,9234],{"className":9233},[1031,1032,1033,1034],[25,9235,147],{"className":9236},[81,1034],[25,9238,58],{"className":9239},[94],[10,9241,9242],{},"In this example, we directly calculated the solution to the recurrence relation using the recursive tree method. Of course, we can use the substitution method to verify whether this solution is correct. In some cases, it may be slightly difficult to calculate the exact solution using the recursive tree method, but it does not prevent us from making a reasonable guess and then further verifying this guess using the substitution method.",[14,9244,9246],{"id":9245},"master-theorem","Master Theorem",[10,9248,9249],{},"Both the substitution method and the recursion tree method have their drawbacks. The former requires an accurate guess, while the latter involves drawing a recursion tree and deriving results. If you derive too much, you may find that the forms of most recursive solutions seem to be the same. From the derivation process above, it can be observed that the final solution depends entirely on the sum of costs in the middle layers and has nothing to do with the leaf layer because the latter is a lower-order term. In other recursion trees, the final solution may depend entirely on the cost of the leaf layer, unrelated to the middle layers. Therefore, we may be able to classify recursive formulas and directly write out the final solution according to some rules. This is the idea behind the master theorem.",[10,9251,9252],{},"When the divide-and-conquer method evenly splits the data each time, the recursive formula generally has the following form.",[25,9254,9256],{"className":9255,"translate":29},[277],[25,9257,9259,9327],{"className":9258,"translate":29},[28],[25,9260,9262],{"className":9261},[33],[35,9263,9264],{"xmlns":37,"display":286},[39,9265,9266,9324],{},[308,9267,9269],{"width":9268},"100%",[314,9270,9271,9274,9317,9319],{},[317,9272],{"width":9273},"50%",[317,9275,9276],{},[42,9277,9278,9280,9282,9284,9286,9288,9291,9293,9306,9308,9311,9313,9315],{},[45,9279,47],{},[49,9281,52],{"stretchy":51},[45,9283,55],{},[49,9285,58],{"stretchy":51},[49,9287,301],{},[45,9289,9290],{},"a",[45,9292,47],{},[42,9294,9295,9297,9304],{},[49,9296,52],{"fence":68},[2916,9298,9299,9301],{},[45,9300,55],{},[45,9302,9303],{},"b",[49,9305,58],{"fence":68},[49,9307,390],{},[45,9309,9310],{},"f",[49,9312,52],{"stretchy":51},[45,9314,55],{},[49,9316,58],{"stretchy":51},[317,9318],{"width":9273},[317,9320,9321],{},[349,9322,9323],{},"(2)",[60,9325,9326],{"encoding":62},"T(n) = aT\\left(\\frac{n}{b}\\right) + f(n) \\tag{2}",[25,9328,9330,9357,9459,9478],{"className":9329,"ariaHidden":68},[67],[25,9331,9333,9336,9339,9342,9345,9348,9351,9354],{"className":9332},[72],[25,9334],{"className":9335,"style":77},[76],[25,9337,47],{"className":9338,"style":83},[81,82],[25,9340,52],{"className":9341},[87],[25,9343,55],{"className":9344},[81,82],[25,9346,58],{"className":9347},[94],[25,9349],{"className":9350,"style":442},[441],[25,9352,301],{"className":9353},[446],[25,9355],{"className":9356,"style":442},[441],[25,9358,9360,9364,9367,9370,9373,9450,9453,9456],{"className":9359},[72],[25,9361],{"className":9362,"style":9363},[76],"height:1.836em;vertical-align:-0.686em;",[25,9365,9290],{"className":9366},[81,82],[25,9368,47],{"className":9369,"style":83},[81,82],[25,9371],{"className":9372,"style":902},[441],[25,9374,9376,9382,9444],{"className":9375},[460],[25,9377,9379],{"className":9378,"style":465},[87,464],[25,9380,52],{"className":9381},[469,3954],[25,9383,9385,9388,9441],{"className":9384},[81],[25,9386],{"className":9387},[87,721],[25,9389,9391],{"className":9390},[2916],[25,9392,9394,9433],{"className":9393},[484,485],[25,9395,9397,9430],{"className":9396},[489],[25,9398,9400,9411,9419],{"className":9399,"style":3973},[493],[25,9401,9402,9405],{"style":3976},[25,9403],{"className":9404,"style":548},[501],[25,9406,9408],{"className":9407},[81],[25,9409,9303],{"className":9410},[81,82],[25,9412,9413,9416],{"style":3082},[25,9414],{"className":9415,"style":548},[501],[25,9417],{"className":9418,"style":3090},[3089],[25,9420,9421,9424],{"style":3996},[25,9422],{"className":9423,"style":548},[501],[25,9425,9427],{"className":9426},[81],[25,9428,55],{"className":9429},[81,82],[25,9431,519],{"className":9432},[518],[25,9434,9436],{"className":9435},[489],[25,9437,9439],{"className":9438,"style":4015},[493],[25,9440],{},[25,9442],{"className":9443},[94,721],[25,9445,9447],{"className":9446,"style":465},[94,464],[25,9448,58],{"className":9449},[469,3954],[25,9451],{"className":9452,"style":600},[441],[25,9454,390],{"className":9455},[604],[25,9457],{"className":9458,"style":600},[441],[25,9460,9462,9465,9469,9472,9475],{"className":9461},[72],[25,9463],{"className":9464,"style":77},[76],[25,9466,9310],{"className":9467,"style":9468},[81,82],"margin-right:0.10764em;",[25,9470,52],{"className":9471},[87],[25,9473,55],{"className":9474},[81,82],[25,9476,58],{"className":9477},[94],[25,9479,9481,9484],{"className":9480},[4544],[25,9482],{"className":9483,"style":9363},[76],[25,9485,9487,9490,9496],{"className":9486},[81,659],[25,9488,52],{"className":9489},[81],[25,9491,9493],{"className":9492},[81],[25,9494,147],{"className":9495},[81],[25,9497,58],{"className":9498},[81],[10,9500,9501,9502,9530,9531,9560,9561,9605],{},"where ",[25,9503,9505,9518],{"className":9504,"translate":29},[28],[25,9506,9508],{"className":9507},[33],[35,9509,9510],{"xmlns":37},[39,9511,9512,9516],{},[42,9513,9514],{},[45,9515,9290],{},[60,9517,9290],{"encoding":62},[25,9519,9521],{"className":9520,"ariaHidden":68},[67],[25,9522,9524,9527],{"className":9523},[72],[25,9525],{"className":9526,"style":121},[76],[25,9528,9290],{"className":9529},[81,82]," is a positive integer representing the number of subproblems when dividing each time, ",[25,9532,9534,9547],{"className":9533,"translate":29},[28],[25,9535,9537],{"className":9536},[33],[35,9538,9539],{"xmlns":37},[39,9540,9541,9545],{},[42,9542,9543],{},[45,9544,9303],{},[60,9546,9303],{"encoding":62},[25,9548,9550],{"className":9549,"ariaHidden":68},[67],[25,9551,9553,9557],{"className":9552},[72],[25,9554],{"className":9555,"style":9556},[76],"height:0.6944em;",[25,9558,9303],{"className":9559},[81,82]," is an integer greater than 1 representing the reduction factor in problem size each time, and ",[25,9562,9564,9584],{"className":9563,"translate":29},[28],[25,9565,9567],{"className":9566},[33],[35,9568,9569],{"xmlns":37},[39,9570,9571,9581],{},[42,9572,9573,9575,9577,9579],{},[45,9574,9310],{},[49,9576,52],{"stretchy":51},[45,9578,55],{},[49,9580,58],{"stretchy":51},[60,9582,9583],{"encoding":62},"f(n)",[25,9585,9587],{"className":9586,"ariaHidden":68},[67],[25,9588,9590,9593,9596,9599,9602],{"className":9589},[72],[25,9591],{"className":9592,"style":77},[76],[25,9594,9310],{"className":9595,"style":9468},[81,82],[25,9597,52],{"className":9598},[87],[25,9600,55],{"className":9601},[81,82],[25,9603,58],{"className":9604},[94]," is an asymptotically positive function representing the cost of dividing and merging.",[10,9607,9608,9609,9637,9638,9666,9667,9710],{},"To find a unified solution applicable to any ",[25,9610,9612,9625],{"className":9611,"translate":29},[28],[25,9613,9615],{"className":9614},[33],[35,9616,9617],{"xmlns":37},[39,9618,9619,9623],{},[42,9620,9621],{},[45,9622,9290],{},[60,9624,9290],{"encoding":62},[25,9626,9628],{"className":9627,"ariaHidden":68},[67],[25,9629,9631,9634],{"className":9630},[72],[25,9632],{"className":9633,"style":121},[76],[25,9635,9290],{"className":9636},[81,82],", ",[25,9639,9641,9654],{"className":9640,"translate":29},[28],[25,9642,9644],{"className":9643},[33],[35,9645,9646],{"xmlns":37},[39,9647,9648,9652],{},[42,9649,9650],{},[45,9651,9303],{},[60,9653,9303],{"encoding":62},[25,9655,9657],{"className":9656,"ariaHidden":68},[67],[25,9658,9660,9663],{"className":9659},[72],[25,9661],{"className":9662,"style":9556},[76],[25,9664,9303],{"className":9665},[81,82],", and ",[25,9668,9670,9689],{"className":9669,"translate":29},[28],[25,9671,9673],{"className":9672},[33],[35,9674,9675],{"xmlns":37},[39,9676,9677,9687],{},[42,9678,9679,9681,9683,9685],{},[45,9680,9310],{},[49,9682,52],{"stretchy":51},[45,9684,55],{},[49,9686,58],{"stretchy":51},[60,9688,9583],{"encoding":62},[25,9690,9692],{"className":9691,"ariaHidden":68},[67],[25,9693,9695,9698,9701,9704,9707],{"className":9694},[72],[25,9696],{"className":9697,"style":77},[76],[25,9699,9310],{"className":9700,"style":9468},[81,82],[25,9702,52],{"className":9703},[87],[25,9705,55],{"className":9706},[81,82],[25,9708,58],{"className":9709},[94],", we still need to use the recursion tree to calculate the costs of all intermediate and leaf nodes and try to derive a simple result.",[10,9712,9713],{},[5642,9714],{"alt":9715,"src":9716},"","https:\u002F\u002Fimage-assets.dreams.plus\u002F20231218224657.webp",[10,9718,9719,9720,10181],{},"From the graph, we can see that the total cost of all nodes is ",[25,9721,9723,9827],{"className":9722,"translate":29},[28],[25,9724,9726],{"className":9725},[33],[35,9727,9728],{"xmlns":37},[39,9729,9730,9824],{},[42,9731,9732,9734,9736,9738,9740,9742,9744,9746,9764,9766,9768,9800],{},[45,9733,47],{},[49,9735,52],{"stretchy":51},[45,9737,55],{},[49,9739,58],{"stretchy":51},[49,9741,301],{},[45,9743,239],{"mathvariant":142},[49,9745,52],{"stretchy":51},[5367,9747,9748,9750],{},[45,9749,55],{},[42,9751,9752,9762],{},[946,9753,9754,9760],{},[42,9755,9756,9758],{},[45,9757,842],{},[49,9759,845],{},[45,9761,9303],{},[45,9763,9290],{},[49,9765,58],{"stretchy":51},[49,9767,390],{},[9769,9770,9771,9773,9782],"msubsup",{},[49,9772,6621],{},[42,9774,9775,9778,9780],{},[45,9776,9777],{},"j",[49,9779,301],{},[145,9781,322],{},[42,9783,9784,9794,9796,9798],{},[946,9785,9786,9792],{},[42,9787,9788,9790],{},[45,9789,842],{},[49,9791,845],{},[45,9793,9303],{},[45,9795,55],{},[49,9797,2958],{},[145,9799,339],{},[42,9801,9802,9808,9810,9812,9814,9816,9822],{},[5367,9803,9804,9806],{},[45,9805,9290],{},[45,9807,9777],{},[45,9809,9310],{},[49,9811,52],{"stretchy":51},[45,9813,55],{},[45,9815,143],{"mathvariant":142},[5367,9817,9818,9820],{},[45,9819,9303],{},[45,9821,9777],{},[49,9823,58],{"stretchy":51},[60,9825,9826],{"encoding":62},"T(n) = \\Theta(n^{\\log_b a}) + \\sum_{j=0}^{\\log_b n-1}{a^j f(n\u002Fb^j)}",[25,9828,9830,9857,9966],{"className":9829,"ariaHidden":68},[67],[25,9831,9833,9836,9839,9842,9845,9848,9851,9854],{"className":9832},[72],[25,9834],{"className":9835,"style":77},[76],[25,9837,47],{"className":9838,"style":83},[81,82],[25,9840,52],{"className":9841},[87],[25,9843,55],{"className":9844},[81,82],[25,9846,58],{"className":9847},[94],[25,9849],{"className":9850,"style":442},[441],[25,9852,301],{"className":9853},[446],[25,9855],{"className":9856,"style":442},[441],[25,9858,9860,9863,9866,9869,9954,9957,9960,9963],{"className":9859},[72],[25,9861],{"className":9862,"style":6336},[76],[25,9864,239],{"className":9865},[81],[25,9867,52],{"className":9868},[87],[25,9870,9872,9875],{"className":9871},[81],[25,9873,55],{"className":9874},[81,82],[25,9876,9878],{"className":9877},[1010],[25,9879,9881],{"className":9880},[484],[25,9882,9884],{"className":9883},[489],[25,9885,9887],{"className":9886,"style":6094},[493],[25,9888,9889,9892],{"style":5476},[25,9890],{"className":9891,"style":1027},[501],[25,9893,9895],{"className":9894},[1031,1032,1033,1034],[25,9896,9898,9948,9951],{"className":9897},[81,1034],[25,9899,9901,9913],{"className":9900},[906,1034],[25,9902,9904,9907,9910],{"className":9903},[906,1034],[25,9905,6133],{"className":9906},[1034],[25,9908,6137],{"className":9909},[1034],[25,9911,911],{"className":9912,"style":910},[1034],[25,9914,9916],{"className":9915},[1010],[25,9917,9919,9940],{"className":9918},[484,485],[25,9920,9922,9937],{"className":9921},[489],[25,9923,9926],{"className":9924,"style":9925},[493],"height:0.2302em;",[25,9927,9928,9931],{"style":6156},[25,9929],{"className":9930,"style":6160},[501],[25,9932,9934],{"className":9933},[1031,6164,6165,1034],[25,9935,9303],{"className":9936},[81,82,1034],[25,9938,519],{"className":9939},[518],[25,9941,9943],{"className":9942},[489],[25,9944,9946],{"className":9945,"style":6178},[493],[25,9947],{},[25,9949],{"className":9950,"style":6184},[441,1034],[25,9952,9290],{"className":9953},[81,82,1034],[25,9955,58],{"className":9956},[94],[25,9958],{"className":9959,"style":600},[441],[25,9961,390],{"className":9962},[604],[25,9964],{"className":9965,"style":600},[441],[25,9967,9969,9973,10101,10104],{"className":9968},[72],[25,9970],{"className":9971,"style":9972},[76],"height:1.43em;vertical-align:-0.413em;",[25,9974,9976,9981],{"className":9975},[906],[25,9977,6621],{"className":9978,"style":9980},[906,7763,9979],"small-op","position:relative;top:0em;",[25,9982,9984],{"className":9983},[1010],[25,9985,9987,10092],{"className":9986},[484,485],[25,9988,9990,10089],{"className":9989},[489],[25,9991,9994,10016],{"className":9992,"style":9993},[493],"height:1.017em;",[25,9995,9997,10000],{"style":9996},"top:-2.4231em;margin-left:0em;margin-right:0.05em;",[25,9998],{"className":9999,"style":1027},[501],[25,10001,10003],{"className":10002},[1031,1032,1033,1034],[25,10004,10006,10010,10013],{"className":10005},[81,1034],[25,10007,9777],{"className":10008,"style":10009},[81,82,1034],"margin-right:0.05724em;",[25,10011,301],{"className":10012},[446,1034],[25,10014,322],{"className":10015},[81,1034],[25,10017,10019,10022],{"style":10018},"top:-3.2309em;margin-right:0.05em;",[25,10020],{"className":10021,"style":1027},[501],[25,10023,10025],{"className":10024},[1031,1032,1033,1034],[25,10026,10028,10077,10080,10083,10086],{"className":10027},[81,1034],[25,10029,10031,10043],{"className":10030},[906,1034],[25,10032,10034,10037,10040],{"className":10033},[906,1034],[25,10035,6133],{"className":10036},[1034],[25,10038,6137],{"className":10039},[1034],[25,10041,911],{"className":10042,"style":910},[1034],[25,10044,10046],{"className":10045},[1010],[25,10047,10049,10069],{"className":10048},[484,485],[25,10050,10052,10066],{"className":10051},[489],[25,10053,10055],{"className":10054,"style":9925},[493],[25,10056,10057,10060],{"style":6156},[25,10058],{"className":10059,"style":6160},[501],[25,10061,10063],{"className":10062},[1031,6164,6165,1034],[25,10064,9303],{"className":10065},[81,82,1034],[25,10067,519],{"className":10068},[518],[25,10070,10072],{"className":10071},[489],[25,10073,10075],{"className":10074,"style":6178},[493],[25,10076],{},[25,10078],{"className":10079,"style":6184},[441,1034],[25,10081,55],{"className":10082},[81,82,1034],[25,10084,2958],{"className":10085},[604,1034],[25,10087,339],{"className":10088},[81,1034],[25,10090,519],{"className":10091},[518],[25,10093,10095],{"className":10094},[489],[25,10096,10099],{"className":10097,"style":10098},[493],"height:0.413em;",[25,10100],{},[25,10102],{"className":10103,"style":902},[441],[25,10105,10107,10137,10140,10143,10146,10149,10178],{"className":10106},[81],[25,10108,10110,10113],{"className":10109},[81],[25,10111,9290],{"className":10112},[81,82],[25,10114,10116],{"className":10115},[1010],[25,10117,10119],{"className":10118},[484],[25,10120,10122],{"className":10121},[489],[25,10123,10126],{"className":10124,"style":10125},[493],"height:0.8247em;",[25,10127,101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Further derivation requires a breakthrough idea, and I don't know how the original researchers discovered the following pattern, but it is indeed magical.",[10,10183,10184,10185,2702,10228,10369,10370,10648],{},"Researchers found that the relationship between 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Define",[25,10650,10652],{"className":10651,"translate":29},[277],[25,10653,10655,10746],{"className":10654,"translate":29},[28],[25,10656,10658],{"className":10657},[33],[35,10659,10660],{"xmlns":37,"display":286},[39,10661,10662,10743],{},[308,10663,10664],{"width":9268},[314,10665,10666,10668,10736,10738],{},[317,10667],{"width":9273},[317,10669,10670],{},[42,10671,10672,10674,10676,10678,10680,10682,10712],{},[45,10673,911],{},[49,10675,52],{"stretchy":51},[45,10677,55],{},[49,10679,58],{"stretchy":51},[49,10681,301],{},[6617,10683,10684,10686,10694],{},[49,10685,6621],{},[42,10687,10688,10690,10692],{},[45,10689,9777],{},[49,10691,301],{},[145,10693,322],{},[42,10695,10696,10706,10708,10710],{},[946,10697,10698,10704],{},[42,10699,10700,10702],{},[45,10701,842],{},[49,10703,845],{},[45,10705,9303],{},[45,10707,55],{},[49,10709,2958],{},[145,10711,339],{},[42,10713,10714,10720,10722,10724,10726,10728,10734],{},[5367,10715,10716,10718],{},[45,10717,9290],{},[45,10719,9777],{},[45,10721,9310],{},[49,10723,52],{"stretchy":51},[45,10725,55],{},[45,10727,143],{"mathvariant":142},[5367,10729,10730,10732],{},[45,10731,9303],{},[45,10733,9777],{},[49,10735,58],{"stretchy":51},[317,10737],{"width":9273},[317,10739,10740],{},[349,10741,10742],{},"(3)",[60,10744,10745],{"encoding":62},"g(n) = \\sum_{j=0}^{\\log_b n-1}{a^j f(n\u002Fb^j)} \\tag{3}",[25,10747,10749,10777,10990],{"className":10748,"ariaHidden":68},[67],[25,10750,10752,10755,10759,10762,10765,10768,10771,10774],{"className":10751},[72],[25,10753],{"className":10754,"style":77},[76],[25,10756,911],{"className":10757,"style":10758},[81,82],"margin-right:0.03588em;",[25,10760,52],{"className":10761},[87],[25,10763,55],{"className":10764},[81,82],[25,10766,58],{"className":10767},[94],[25,10769],{"className":10770,"style":442},[441],[25,10772,301],{"className":10773},[446],[25,10775],{"className":10776,"style":442},[441],[25,10778,10780,10784,10910,10913],{"className":10779},[72],[25,10781],{"className":10782,"style":10783},[76],"height:3.347em;vertical-align:-1.4138em;",[25,10785,10787],{"className":10786},[906,7719],[25,10788,10790,10901],{"className":10789},[484,485],[25,10791,10793,10898],{"className":10792},[489],[25,10794,10796,10816,10826],{"className":10795,"style":7729},[493],[25,10797,10798,10801],{"style":7732},[25,10799],{"className":10800,"style":7736},[501],[25,10802,10804],{"className":10803},[1031,1032,1033,1034],[25,10805,10807,10810,10813],{"className":10806},[81,1034],[25,10808,9777],{"className":10809,"style":10009},[81,82,1034],[25,10811,301],{"className":10812},[446,1034],[25,10814,322],{"className":10815},[81,1034],[25,10817,10818,10821],{"style":7754},[25,10819],{"className":10820,"style":7736},[501],[25,10822,10823],{},[25,10824,6621],{"className":10825},[906,7763,7764],[25,10827,10828,10831],{"style":7767},[25,10829],{"className":10830,"style":7736},[501],[25,10832,10834],{"className":10833},[1031,1032,1033,1034],[25,10835,10837,10886,10889,10892,10895],{"className":10836},[81,1034],[25,10838,10840,10852],{"className":10839},[906,1034],[25,10841,10843,10846,10849],{"className":10842},[906,1034],[25,10844,6133],{"className":10845},[1034],[25,10847,6137],{"className":10848},[1034],[25,10850,911],{"className":10851,"style":910},[1034],[25,10853,10855],{"className":10854},[1010],[25,10856,10858,10878],{"className":10857},[484,485],[25,10859,10861,10875],{"className":10860},[489],[25,10862,10864],{"className":10863,"style":9925},[493],[25,10865,10866,10869],{"style":6156},[25,10867],{"className":10868,"style":6160},[501],[25,10870,10872],{"className":10871},[1031,6164,6165,1034],[25,10873,9303],{"className":10874},[81,82,1034],[25,10876,519],{"className":10877},[518],[25,10879,10881],{"className":10880},[489],[25,10882,10884],{"className":10883,"style":6178},[493],[25,10885],{},[25,10887],{"className":10888,"style":6184},[441,1034],[25,10890,55],{"className":10891},[81,82,1034],[25,10893,2958],{"className":10894},[604,1034],[25,10896,339],{"className":10897},[81,1034],[25,10899,519],{"className":10900},[518],[25,10902,10904],{"className":10903},[489],[25,10905,10908],{"className":10906,"style":10907},[493],"height:1.4138em;",[25,10909],{},[25,10911],{"className":10912,"style":902},[441],[25,10914,10916,10946,10949,10952,10955,10958,10987],{"className":10915},[81],[25,10917,10919,10922],{"className":10918},[81],[25,10920,9290],{"className":10921},[81,82],[25,10923,10925],{"className":10924},[1010],[25,10926,10928],{"className":10927},[484],[25,10929,10931],{"className":10930},[489],[25,10932,10935],{"className":10933,"style":10934},[493],"height:0.8747em;",[25,10936,10937,10940],{"style":7110},[25,10938],{"className":10939,"style":1027},[501],[25,10941,10943],{"className":10942},[1031,1032,1033,1034],[25,10944,9777],{"className":10945,"style":10009},[81,82,1034],[25,10947,9310],{"className":10948,"style":9468},[81,82],[25,10950,52],{"className":10951},[87],[25,10953,55],{"className":10954},[81,82],[25,10956,143],{"className":10957},[81],[25,10959,10961,10964],{"className":10960},[81],[25,10962,9303],{"className":10963},[81,82],[25,10965,10967],{"className":10966},[1010],[25,10968,10970],{"className":10969},[484],[25,10971,10973],{"className":10972},[489],[25,10974,10976],{"className":10975,"style":10934},[493],[25,10977,10978,10981],{"style":7110},[25,10979],{"className":10980,"style":1027},[501],[25,10982,10984],{"className":10983},[1031,1032,1033,1034],[25,10985,9777],{"className":10986,"style":10009},[81,82,1034],[25,10988,58],{"className":10989},[94],[25,10991,10993,10996],{"className":10992},[4544],[25,10994],{"className":10995,"style":10783},[76],[25,10997,10999,11002,11008],{"className":10998},[81,659],[25,11000,52],{"className":11001},[81],[25,11003,11005],{"className":11004},[81],[25,11006,5346],{"className":11007},[81],[25,11009,58],{"className":11010},[81],[10,11012,11013],{},"Now let's analyze three cases.",[11015,11016,11017,11582,11783],"ol",{},[11018,11019,11020,11021,11161,11162,11214,11215,11403,11404,270],"li",{},"If ",[25,11022,11024,11059],{"className":11023,"translate":29},[28],[25,11025,11027],{"className":11026},[33],[35,11028,11029],{"xmlns":37},[39,11030,11031,11057],{},[42,11032,11033,11035,11037,11055],{},[45,11034,239],{"mathvariant":142},[49,11036,52],{"stretchy":51},[5367,11038,11039,11041],{},[45,11040,55],{},[42,11042,11043,11053],{},[946,11044,11045,11051],{},[42,11046,11047,11049],{},[45,11048,842],{},[49,11050,845],{},[45,11052,9303],{},[45,11054,9290],{},[49,11056,58],{"stretchy":51},[60,11058,10266],{"encoding":62},[25,11060,11062],{"className":11061,"ariaHidden":68},[67],[25,11063,11065,11068,11071,11074,11158],{"className":11064},[72],[25,11066],{"className":11067,"style":6336},[76],[25,11069,239],{"className":11070},[81],[25,11072,52],{"className":11073},[87],[25,11075,11077,11080],{"className":11076},[81],[25,11078,55],{"className":11079},[81,82],[25,11081,11083],{"className":11082},[1010],[25,11084,11086],{"className":11085},[484],[25,11087,11089],{"className":11088},[489],[25,11090,11092],{"className":11091,"style":6094},[493],[25,11093,11094,11097],{"style":5476},[25,11095],{"className":11096,"style":1027},[501],[25,11098,11100],{"className":11099},[1031,1032,1033,1034],[25,11101,11103,11152,11155],{"className":11102},[81,1034],[25,11104,11106,11118],{"className":11105},[906,1034],[25,11107,11109,11112,11115],{"className":11108},[906,1034],[25,11110,6133],{"className":11111},[1034],[25,11113,6137],{"className":11114},[1034],[25,11116,911],{"className":11117,"style":910},[1034],[25,11119,11121],{"className":11120},[1010],[25,11122,11124,11144],{"className":11123},[484,485],[25,11125,11127,11141],{"className":11126},[489],[25,11128,11130],{"className":11129,"style":9925},[493],[25,11131,11132,11135],{"style":6156},[25,11133],{"className":11134,"style":6160},[501],[25,11136,11138],{"className":11137},[1031,6164,6165,1034],[25,11139,9303],{"className":11140},[81,82,1034],[25,11142,519],{"className":11143},[518],[25,11145,11147],{"className":11146},[489],[25,11148,11150],{"className":11149,"style":6178},[493],[25,11151],{},[25,11153],{"className":11154,"style":6184},[441,1034],[25,11156,9290],{"className":11157},[81,82,1034],[25,11159,58],{"className":11160},[94]," is larger, we can assume that there exists a constant ",[25,11163,11165,11184],{"className":11164,"translate":29},[28],[25,11166,11168],{"className":11167},[33],[35,11169,11170],{"xmlns":37},[39,11171,11172,11181],{},[42,11173,11174,11177,11179],{},[45,11175,11176],{},"ϵ",[49,11178,411],{},[145,11180,322],{},[60,11182,11183],{"encoding":62},"\\epsilon > 0",[25,11185,11187,11205],{"className":11186,"ariaHidden":68},[67],[25,11188,11190,11193,11196,11199,11202],{"className":11189},[72],[25,11191],{"className":11192,"style":1867},[76],[25,11194,11176],{"className":11195},[81,82],[25,11197],{"className":11198,"style":442},[441],[25,11200,411],{"className":11201},[446],[25,11203],{"className":11204,"style":442},[441],[25,11206,11208,11211],{"className":11207},[72],[25,11209],{"className":11210,"style":1220},[76],[25,11212,322],{"className":11213},[81]," such that ",[25,11216,11218,11268],{"className":11217,"translate":29},[28],[25,11219,11221],{"className":11220},[33],[35,11222,11223],{"xmlns":37},[39,11224,11225,11265],{},[42,11226,11227,11229,11231,11233,11235,11237,11239,11241,11263],{},[45,11228,9310],{},[49,11230,52],{"stretchy":51},[45,11232,55],{},[49,11234,58],{"stretchy":51},[49,11236,301],{},[45,11238,835],{},[49,11240,52],{"stretchy":51},[5367,11242,11243,11245],{},[45,11244,55],{},[42,11246,11247,11257,11259,11261],{},[946,11248,11249,11255],{},[42,11250,11251,11253],{},[45,11252,842],{},[49,11254,845],{},[45,11256,9303],{},[45,11258,9290],{},[49,11260,2958],{},[45,11262,11176],{},[49,11264,58],{"stretchy":51},[60,11266,11267],{"encoding":62},"f(n) = O(n^{\\log_b a-\\epsilon})",[25,11269,11271,11298],{"className":11270,"ariaHidden":68},[67],[25,11272,11274,11277,11280,11283,11286,11289,11292,11295],{"className":11273},[72],[25,11275],{"className":11276,"style":77},[76],[25,11278,9310],{"className":11279,"style":9468},[81,82],[25,11281,52],{"className":11282},[87],[25,11284,55],{"className":11285},[81,82],[25,11287,58],{"className":11288},[94],[25,11290],{"className":11291,"style":442},[441],[25,11293,301],{"className":11294},[446],[25,11296],{"className":11297,"style":442},[441],[25,11299,11301,11304,11307,11310,11400],{"className":11300},[72],[25,11302],{"className":11303,"style":6336},[76],[25,11305,835],{"className":11306,"style":892},[81,82],[25,11308,52],{"className":11309},[87],[25,11311,11313,11316],{"className":11312},[81],[25,11314,55],{"className":11315},[81,82],[25,11317,11319],{"className":11318},[1010],[25,11320,11322],{"className":11321},[484],[25,11323,11325],{"className":11324},[489],[25,11326,11328],{"className":11327,"style":6094},[493],[25,11329,11330,11333],{"style":5476},[25,11331],{"className":11332,"style":1027},[501],[25,11334,11336],{"className":11335},[1031,1032,1033,1034],[25,11337,11339,11388,11391,11394,11397],{"className":11338},[81,1034],[25,11340,11342,11354],{"className":11341},[906,1034],[25,11343,11345,11348,11351],{"className":11344},[906,1034],[25,11346,6133],{"className":11347},[1034],[25,11349,6137],{"className":11350},[1034],[25,11352,911],{"className":11353,"style":910},[1034],[25,11355,11357],{"className":11356},[1010],[25,11358,11360,11380],{"className":11359},[484,485],[25,11361,11363,11377],{"className":11362},[489],[25,11364,11366],{"className":11365,"style":9925},[493],[25,11367,11368,11371],{"style":6156},[25,11369],{"className":11370,"style":6160},[501],[25,11372,11374],{"className":11373},[1031,6164,6165,1034],[25,11375,9303],{"className":11376},[81,82,1034],[25,11378,519],{"className":11379},[518],[25,11381,11383],{"className":11382},[489],[25,11384,11386],{"className":11385,"style":6178},[493],[25,11387],{},[25,11389],{"className":11390,"style":6184},[441,1034],[25,11392,9290],{"className":11393},[81,82,1034],[25,11395,2958],{"className":11396},[604,1034],[25,11398,11176],{"className":11399},[81,82,1034],[25,11401,58],{"className":11402},[94],". Substituting this into the summation formula, it can be simplified to ",[25,11405,11407,11453],{"className":11406,"translate":29},[28],[25,11408,11410],{"className":11409},[33],[35,11411,11412],{"xmlns":37},[39,11413,11414,11450],{},[42,11415,11416,11418,11420,11422,11424,11426,11428,11430,11448],{},[45,11417,911],{},[49,11419,52],{"stretchy":51},[45,11421,55],{},[49,11423,58],{"stretchy":51},[49,11425,301],{},[45,11427,835],{},[49,11429,52],{"stretchy":51},[5367,11431,11432,11434],{},[45,11433,55],{},[42,11435,11436,11446],{},[946,11437,11438,11444],{},[42,11439,11440,11442],{},[45,11441,842],{},[49,11443,845],{},[45,11445,9303],{},[45,11447,9290],{},[49,11449,58],{"stretchy":51},[60,11451,11452],{"encoding":62},"g(n) = O(n^{\\log_b a})",[25,11454,11456,11483],{"className":11455,"ariaHidden":68},[67],[25,11457,11459,11462,11465,11468,11471,11474,11477,11480],{"className":11458},[72],[25,11460],{"className":11461,"style":77},[76],[25,11463,911],{"className":11464,"style":10758},[81,82],[25,11466,52],{"className":11467},[87],[25,11469,55],{"className":11470},[81,82],[25,11472,58],{"className":11473},[94],[25,11475],{"className":11476,"style":442},[441],[25,11478,301],{"className":11479},[446],[25,11481],{"className":11482,"style":442},[441],[25,11484,11486,11489,11492,11495,11579],{"className":11485},[72],[25,11487],{"className":11488,"style":6336},[76],[25,11490,835],{"className":11491,"style":892},[81,82],[25,11493,52],{"className":11494},[87],[25,11496,11498,11501],{"className":11497},[81],[25,11499,55],{"className":11500},[81,82],[25,11502,11504],{"className":11503},[1010],[25,11505,11507],{"className":11506},[484],[25,11508,11510],{"className":11509},[489],[25,11511,11513],{"className":11512,"style":6094},[493],[25,11514,11515,11518],{"style":5476},[25,11516],{"className":11517,"style":1027},[501],[25,11519,11521],{"className":11520},[1031,1032,1033,1034],[25,11522,11524,11573,11576],{"className":11523},[81,1034],[25,11525,11527,11539],{"className":11526},[906,1034],[25,11528,11530,11533,11536],{"className":11529},[906,1034],[25,11531,6133],{"className":11532},[1034],[25,11534,6137],{"className":11535},[1034],[25,11537,911],{"className":11538,"style":910},[1034],[25,11540,11542],{"className":11541},[1010],[25,11543,11545,11565],{"className":11544},[484,485],[25,11546,11548,11562],{"className":11547},[489],[25,11549,11551],{"className":11550,"style":9925},[493],[25,11552,11553,11556],{"style":6156},[25,11554],{"className":11555,"style":6160},[501],[25,11557,11559],{"className":11558},[1031,6164,6165,1034],[25,11560,9303],{"className":11561},[81,82,1034],[25,11563,519],{"className":11564},[518],[25,11566,11568],{"className":11567},[489],[25,11569,11571],{"className":11570,"style":6178},[493],[25,11572],{},[25,11574],{"className":11575,"style":6184},[441,1034],[25,11577,9290],{"className":11578},[81,82,1034],[25,11580,58],{"className":11581},[94],[11018,11583,11584,11585,270],{},"If both are equally large, directly substitute into equation (3) and simplify to get ",[25,11586,11588,11640],{"className":11587,"translate":29},[28],[25,11589,11591],{"className":11590},[33],[35,11592,11593],{"xmlns":37},[39,11594,11595,11637],{},[42,11596,11597,11599,11601,11603,11605,11607,11609,11611,11629,11631,11633,11635],{},[45,11598,911],{},[49,11600,52],{"stretchy":51},[45,11602,55],{},[49,11604,58],{"stretchy":51},[49,11606,301],{},[45,11608,239],{"mathvariant":142},[49,11610,52],{"stretchy":51},[5367,11612,11613,11615],{},[45,11614,55],{},[42,11616,11617,11627],{},[946,11618,11619,11625],{},[42,11620,11621,11623],{},[45,11622,842],{},[49,11624,845],{},[45,11626,9303],{},[45,11628,9290],{},[45,11630,842],{},[49,11632,845],{},[45,11634,55],{},[49,11636,58],{"stretchy":51},[60,11638,11639],{"encoding":62},"g(n) = \\Theta(n^{\\log_b a} \\log n)",[25,11641,11643,11670],{"className":11642,"ariaHidden":68},[67],[25,11644,11646,11649,11652,11655,11658,11661,11664,11667],{"className":11645},[72],[25,11647],{"className":11648,"style":77},[76],[25,11650,911],{"className":11651,"style":10758},[81,82],[25,11653,52],{"className":11654},[87],[25,11656,55],{"className":11657},[81,82],[25,11659,58],{"className":11660},[94],[25,11662],{"className":11663,"style":442},[441],[25,11665,301],{"className":11666},[446],[25,11668],{"className":11669,"style":442},[441],[25,11671,11673,11676,11679,11682,11766,11769,11774,11777,11780],{"className":11672},[72],[25,11674],{"className":11675,"style":6336},[76],[25,11677,239],{"className":11678},[81],[25,11680,52],{"className":11681},[87],[25,11683,11685,11688],{"className":11684},[81],[25,11686,55],{"className":11687},[81,82],[25,11689,11691],{"className":11690},[1010],[25,11692,11694],{"className":11693},[484],[25,11695,11697],{"className":11696},[489],[25,11698,11700],{"className":11699,"style":6094},[493],[25,11701,11702,11705],{"style":5476},[25,11703],{"className":11704,"style":1027},[501],[25,11706,11708],{"className":11707},[1031,1032,1033,1034],[25,11709,11711,11760,11763],{"className":11710},[81,1034],[25,11712,11714,11726],{"className":11713},[906,1034],[25,11715,11717,11720,11723],{"className":11716},[906,1034],[25,11718,6133],{"className":11719},[1034],[25,11721,6137],{"className":11722},[1034],[25,11724,911],{"className":11725,"style":910},[1034],[25,11727,11729],{"className":11728},[1010],[25,11730,11732,11752],{"className":11731},[484,485],[25,11733,11735,11749],{"className":11734},[489],[25,11736,11738],{"className":11737,"style":9925},[493],[25,11739,11740,11743],{"style":6156},[25,11741],{"className":11742,"style":6160},[501],[25,11744,11746],{"className":11745},[1031,6164,6165,1034],[25,11747,9303],{"className":11748},[81,82,1034],[25,11750,519],{"className":11751},[518],[25,11753,11755],{"className":11754},[489],[25,11756,11758],{"className":11757,"style":6178},[493],[25,11759],{},[25,11761],{"className":11762,"style":6184},[441,1034],[25,11764,9290],{"className":11765},[81,82,1034],[25,11767],{"className":11768,"style":902},[441],[25,11770,907,11772],{"className":11771},[906],[25,11773,911],{"style":910},[25,11775],{"className":11776,"style":902},[441],[25,11778,55],{"className":11779},[81,82],[25,11781,58],{"className":11782},[94],[11018,11784,11020,11785,11828,11829,11880,11881,9637,11909,12010,12011,270],{},[25,11786,11788,11807],{"className":11787,"translate":29},[28],[25,11789,11791],{"className":11790},[33],[35,11792,11793],{"xmlns":37},[39,11794,11795,11805],{},[42,11796,11797,11799,11801,11803],{},[45,11798,9310],{},[49,11800,52],{"stretchy":51},[45,11802,55],{},[49,11804,58],{"stretchy":51},[60,11806,9583],{"encoding":62},[25,11808,11810],{"className":11809,"ariaHidden":68},[67],[25,11811,11813,11816,11819,11822,11825],{"className":11812},[72],[25,11814],{"className":11815,"style":77},[76],[25,11817,9310],{"className":11818,"style":9468},[81,82],[25,11820,52],{"className":11821},[87],[25,11823,55],{"className":11824},[81,82],[25,11826,58],{"className":11827},[94]," is larger and, for some constant ",[25,11830,11832,11850],{"className":11831,"translate":29},[28],[25,11833,11835],{"className":11834},[33],[35,11836,11837],{"xmlns":37},[39,11838,11839,11847],{},[42,11840,11841,11843,11845],{},[45,11842,950],{},[49,11844,4605],{},[145,11846,339],{},[60,11848,11849],{"encoding":62},"c \u003C 1",[25,11851,11853,11871],{"className":11852,"ariaHidden":68},[67],[25,11854,11856,11859,11862,11865,11868],{"className":11855},[72],[25,11857],{"className":11858,"style":1867},[76],[25,11860,950],{"className":11861},[81,82],[25,11863],{"className":11864,"style":442},[441],[25,11866,4605],{"className":11867},[446],[25,11869],{"className":11870,"style":442},[441],[25,11872,11874,11877],{"className":11873},[72],[25,11875],{"className":11876,"style":1220},[76],[25,11878,339],{"className":11879},[81],", for all sufficiently large ",[25,11882,11884,11897],{"className":11883,"translate":29},[28],[25,11885,11887],{"className":11886},[33],[35,11888,11889],{"xmlns":37},[39,11890,11891,11895],{},[42,11892,11893],{},[45,11894,55],{},[60,11896,55],{"encoding":62},[25,11898,11900],{"className":11899,"ariaHidden":68},[67],[25,11901,11903,11906],{"className":11902},[72],[25,11904],{"className":11905,"style":121},[76],[25,11907,55],{"className":11908},[81,82],[25,11910,11912,11950],{"className":11911,"translate":29},[28],[25,11913,11915],{"className":11914},[33],[35,11916,11917],{"xmlns":37},[39,11918,11919,11947],{},[42,11920,11921,11923,11925,11927,11929,11931,11933,11935,11937,11939,11941,11943,11945],{},[45,11922,9290],{},[45,11924,9310],{},[49,11926,52],{"stretchy":51},[45,11928,55],{},[45,11930,143],{"mathvariant":142},[45,11932,9303],{},[49,11934,58],{"stretchy":51},[49,11936,944],{},[45,11938,950],{},[45,11940,9310],{},[49,11942,52],{"stretchy":51},[45,11944,55],{},[49,11946,58],{"stretchy":51},[60,11948,11949],{"encoding":62},"a f(n\u002Fb) \\le c f(n)",[25,11951,11953,11989],{"className":11952,"ariaHidden":68},[67],[25,11954,11956,11959,11962,11965,11968,11971,11974,11977,11980,11983,11986],{"className":11955},[72],[25,11957],{"className":11958,"style":77},[76],[25,11960,9290],{"className":11961},[81,82],[25,11963,9310],{"className":11964,"style":9468},[81,82],[25,11966,52],{"className":11967},[87],[25,11969,55],{"className":11970},[81,82],[25,11972,143],{"className":11973},[81],[25,11975,9303],{"className":11976},[81,82],[25,11978,58],{"className":11979},[94],[25,11981],{"className":11982,"style":442},[441],[25,11984,944],{"className":11985},[446],[25,11987],{"className":11988,"style":442},[441],[25,11990,11992,11995,11998,12001,12004,12007],{"className":11991},[72],[25,11993],{"className":11994,"style":77},[76],[25,11996,950],{"className":11997},[81,82],[25,11999,9310],{"className":12000,"style":9468},[81,82],[25,12002,52],{"className":12003},[87],[25,12005,55],{"className":12006},[81,82],[25,12008,58],{"className":12009},[94]," holds, then it can be deduced that ",[25,12012,12014,12050],{"className":12013,"translate":29},[28],[25,12015,12017],{"className":12016},[33],[35,12018,12019],{"xmlns":37},[39,12020,12021,12047],{},[42,12022,12023,12025,12027,12029,12031,12033,12035,12037,12039,12041,12043,12045],{},[45,12024,911],{},[49,12026,52],{"stretchy":51},[45,12028,55],{},[49,12030,58],{"stretchy":51},[49,12032,301],{},[45,12034,239],{"mathvariant":142},[49,12036,52],{"stretchy":51},[45,12038,9310],{},[49,12040,52],{"stretchy":51},[45,12042,55],{},[49,12044,58],{"stretchy":51},[49,12046,58],{"stretchy":51},[60,12048,12049],{"encoding":62},"g(n) = \\Theta(f(n))",[25,12051,12053,12080],{"className":12052,"ariaHidden":68},[67],[25,12054,12056,12059,12062,12065,12068,12071,12074,12077],{"className":12055},[72],[25,12057],{"className":12058,"style":77},[76],[25,12060,911],{"className":12061,"style":10758},[81,82],[25,12063,52],{"className":12064},[87],[25,12066,55],{"className":12067},[81,82],[25,12069,58],{"className":12070},[94],[25,12072],{"className":12073,"style":442},[441],[25,12075,301],{"className":12076},[446],[25,12078],{"className":12079,"style":442},[441],[25,12081,12083,12086,12089,12092,12095,12098,12101],{"className":12082},[72],[25,12084],{"className":12085,"style":77},[76],[25,12087,239],{"className":12088},[81],[25,12090,52],{"className":12091},[87],[25,12093,9310],{"className":12094,"style":9468},[81,82],[25,12096,52],{"className":12097},[87],[25,12099,55],{"className":12100},[81,82],[25,12102,12104],{"className":12103},[94],"))",[10,12106,12107],{},"Here, the complete proof process is not provided, partly because I don't want to make it too long, and secondly, I feel that these proofs are somewhat cumbersome and not very meaningful. Interested students can refer to the original book.",[10,12109,12110,12111,12155],{},"Substituting the functions ",[25,12112,12114,12134],{"className":12113,"translate":29},[28],[25,12115,12117],{"className":12116},[33],[35,12118,12119],{"xmlns":37},[39,12120,12121,12131],{},[42,12122,12123,12125,12127,12129],{},[45,12124,911],{},[49,12126,52],{"stretchy":51},[45,12128,55],{},[49,12130,58],{"stretchy":51},[60,12132,12133],{"encoding":62},"g(n)",[25,12135,12137],{"className":12136,"ariaHidden":68},[67],[25,12138,12140,12143,12146,12149,12152],{"className":12139},[72],[25,12141],{"className":12142,"style":77},[76],[25,12144,911],{"className":12145,"style":10758},[81,82],[25,12147,52],{"className":12148},[87],[25,12150,55],{"className":12151},[81,82],[25,12153,58],{"className":12154},[94]," for these three cases into the complete solution, we get:",[12157,12158,12159,12608,13097],"ul",{},[11018,12160,12161,12162],{},"For case 1: ",[25,12163,12165,12263],{"className":12164,"translate":29},[28],[25,12166,12168],{"className":12167},[33],[35,12169,12170],{"xmlns":37},[39,12171,12172,12260],{},[42,12173,12174,12176,12178,12180,12182,12184,12186,12188,12206,12208,12210,12212,12214,12232,12234,12236,12238,12240,12258],{},[45,12175,47],{},[49,12177,52],{"stretchy":51},[45,12179,55],{},[49,12181,58],{"stretchy":51},[49,12183,301],{},[45,12185,239],{"mathvariant":142},[49,12187,52],{"stretchy":51},[5367,12189,12190,12192],{},[45,12191,55],{},[42,12193,12194,12204],{},[946,12195,12196,12202],{},[42,12197,12198,12200],{},[45,12199,842],{},[49,12201,845],{},[45,12203,9303],{},[45,12205,9290],{},[49,12207,58],{"stretchy":51},[49,12209,390],{},[45,12211,835],{},[49,12213,52],{"stretchy":51},[5367,12215,12216,12218],{},[45,12217,55],{},[42,12219,12220,12230],{},[946,12221,12222,12228],{},[42,12223,12224,12226],{},[45,12225,842],{},[49,12227,845],{},[45,12229,9303],{},[45,12231,9290],{},[49,12233,58],{"stretchy":51},[49,12235,301],{},[45,12237,239],{"mathvariant":142},[49,12239,52],{"stretchy":51},[5367,12241,12242,12244],{},[45,12243,55],{},[42,12245,12246,12256],{},[946,12247,12248,12254],{},[42,12249,12250,12252],{},[45,12251,842],{},[49,12253,845],{},[45,12255,9303],{},[45,12257,9290],{},[49,12259,58],{"stretchy":51},[60,12261,12262],{"encoding":62},"T(n) = \\Theta(n^{\\log_b a}) + O(n^{\\log_b a}) = \\Theta(n^{\\log_b a})",[25,12264,12266,12293,12401,12509],{"className":12265,"ariaHidden":68},[67],[25,12267,12269,12272,12275,12278,12281,12284,12287,12290],{"className":12268},[72],[25,12270],{"className":12271,"style":77},[76],[25,12273,47],{"className":12274,"style":83},[81,82],[25,12276,52],{"className":12277},[87],[25,12279,55],{"className":12280},[81,82],[25,12282,58],{"className":12283},[94],[25,12285],{"className":12286,"style":442},[441],[25,12288,301],{"className":12289},[446],[25,12291],{"className":12292,"style":442},[441],[25,12294,12296,12299,12302,12305,12389,12392,12395,12398],{"className":12295},[72],[25,12297],{"className":12298,"style":6336},[76],[25,12300,239],{"className":12301},[81],[25,12303,52],{"className":12304},[87],[25,12306,12308,12311],{"className":12307},[81],[25,12309,55],{"className":12310},[81,82],[25,12312,12314],{"className":12313},[1010],[25,12315,12317],{"className":12316},[484],[25,12318,12320],{"className":12319},[489],[25,12321,12323],{"className":12322,"style":6094},[493],[25,12324,12325,12328],{"style":5476},[25,12326],{"className":12327,"style":1027},[501],[25,12329,12331],{"className":12330},[1031,1032,1033,1034],[25,12332,12334,12383,12386],{"className":12333},[81,1034],[25,12335,12337,12349],{"className":12336},[906,1034],[25,12338,12340,12343,12346],{"className":12339},[906,1034],[25,12341,6133],{"className":12342},[1034],[25,12344,6137],{"className":12345},[1034],[25,12347,911],{"className":12348,"style":910},[1034],[25,12350,12352],{"className":12351},[1010],[25,12353,12355,12375],{"className":12354},[484,485],[25,12356,12358,12372],{"className":12357},[489],[25,12359,12361],{"className":12360,"style":9925},[493],[25,12362,12363,12366],{"style":6156},[25,12364],{"className":12365,"style":6160},[501],[25,12367,12369],{"className":12368},[1031,6164,6165,1034],[25,12370,9303],{"className":12371},[81,82,1034],[25,12373,519],{"className":12374},[518],[25,12376,12378],{"className":12377},[489],[25,12379,12381],{"className":12380,"style":6178},[493],[25,12382],{},[25,12384],{"className":12385,"style":6184},[441,1034],[25,12387,9290],{"className":12388},[81,82,1034],[25,12390,58],{"className":12391},[94],[25,12393],{"className":12394,"style":600},[441],[25,12396,390],{"className":12397},[604],[25,12399],{"className":12400,"style":600},[441],[25,12402,12404,12407,12410,12413,12497,12500,12503,12506],{"className":12403},[72],[25,12405],{"className":12406,"style":6336},[76],[25,12408,835],{"className":12409,"style":892},[81,82],[25,12411,52],{"className":12412},[87],[25,12414,12416,12419],{"className":12415},[81],[25,12417,55],{"className":12418},[81,82],[25,12420,12422],{"className":12421},[1010],[25,12423,12425],{"className":12424},[484],[25,12426,12428],{"className":12427},[489],[25,12429,12431],{"className":12430,"style":6094},[493],[25,12432,12433,12436],{"style":5476},[25,12434],{"className":12435,"style":1027},[501],[25,12437,12439],{"className":12438},[1031,1032,1033,1034],[25,12440,12442,12491,12494],{"className":12441},[81,1034],[25,12443,12445,12457],{"className":12444},[906,1034],[25,12446,12448,12451,12454],{"className":12447},[906,1034],[25,12449,6133],{"className":12450},[1034],[25,12452,6137],{"className":12453},[1034],[25,12455,911],{"className":12456,"style":910},[1034],[25,12458,12460],{"className":12459},[1010],[25,12461,12463,12483],{"className":12462},[484,485],[25,12464,12466,12480],{"className":12465},[489],[25,12467,12469],{"className":12468,"style":9925},[493],[25,12470,12471,12474],{"style":6156},[25,12472],{"className":12473,"style":6160},[501],[25,12475,12477],{"className":12476},[1031,6164,6165,1034],[25,12478,9303],{"className":12479},[81,82,1034],[25,12481,519],{"className":12482},[518],[25,12484,12486],{"className":12485},[489],[25,12487,12489],{"className":12488,"style":6178},[493],[25,12490],{},[25,12492],{"className":12493,"style":6184},[441,1034],[25,12495,9290],{"className":12496},[81,82,1034],[25,12498,58],{"className":12499},[94],[25,12501],{"className":12502,"style":442},[441],[25,12504,301],{"className":12505},[446],[25,12507],{"className":12508,"style":442},[441],[25,12510,12512,12515,12518,12521,12605],{"className":12511},[72],[25,12513],{"className":12514,"style":6336},[76],[25,12516,239],{"className":12517},[81],[25,12519,52],{"className":12520},[87],[25,12522,12524,12527],{"className":12523},[81],[25,12525,55],{"className":12526},[81,82],[25,12528,12530],{"className":12529},[1010],[25,12531,12533],{"className":12532},[484],[25,12534,12536],{"className":12535},[489],[25,12537,12539],{"className":12538,"style":6094},[493],[25,12540,12541,12544],{"style":5476},[25,12542],{"className":12543,"style":1027},[501],[25,12545,12547],{"className":12546},[1031,1032,1033,1034],[25,12548,12550,12599,12602],{"className":12549},[81,1034],[25,12551,12553,12565],{"className":12552},[906,1034],[25,12554,12556,12559,12562],{"className":12555},[906,1034],[25,12557,6133],{"className":12558},[1034],[25,12560,6137],{"className":12561},[1034],[25,12563,911],{"className":12564,"style":910},[1034],[25,12566,12568],{"className":12567},[1010],[25,12569,12571,12591],{"className":12570},[484,485],[25,12572,12574,12588],{"className":12573},[489],[25,12575,12577],{"className":12576,"style":9925},[493],[25,12578,12579,12582],{"style":6156},[25,12580],{"className":12581,"style":6160},[501],[25,12583,12585],{"className":12584},[1031,6164,6165,1034],[25,12586,9303],{"className":12587},[81,82,1034],[25,12589,519],{"className":12590},[518],[25,12592,12594],{"className":12593},[489],[25,12595,12597],{"className":12596,"style":6178},[493],[25,12598],{},[25,12600],{"className":12601,"style":6184},[441,1034],[25,12603,9290],{"className":12604},[81,82,1034],[25,12606,58],{"className":12607},[94],[11018,12609,12610,12611],{},"For case 2: ",[25,12612,12614,12724],{"className":12613,"translate":29},[28],[25,12615,12617],{"className":12616},[33],[35,12618,12619],{"xmlns":37},[39,12620,12621,12721],{},[42,12622,12623,12625,12627,12629,12631,12633,12635,12637,12655,12657,12659,12661,12663,12681,12683,12685,12687,12689,12691,12693,12695,12713,12715,12717,12719],{},[45,12624,47],{},[49,12626,52],{"stretchy":51},[45,12628,55],{},[49,12630,58],{"stretchy":51},[49,12632,301],{},[45,12634,239],{"mathvariant":142},[49,12636,52],{"stretchy":51},[5367,12638,12639,12641],{},[45,12640,55],{},[42,12642,12643,12653],{},[946,12644,12645,12651],{},[42,12646,12647,12649],{},[45,12648,842],{},[49,12650,845],{},[45,12652,9303],{},[45,12654,9290],{},[49,12656,58],{"stretchy":51},[49,12658,390],{},[45,12660,239],{"mathvariant":142},[49,12662,52],{"stretchy":51},[5367,12664,12665,12667],{},[45,12666,55],{},[42,12668,12669,12679],{},[946,12670,12671,12677],{},[42,12672,12673,12675],{},[45,12674,842],{},[49,12676,845],{},[45,12678,9303],{},[45,12680,9290],{},[45,12682,842],{},[49,12684,845],{},[45,12686,55],{},[49,12688,58],{"stretchy":51},[49,12690,301],{},[45,12692,239],{"mathvariant":142},[49,12694,52],{"stretchy":51},[5367,12696,12697,12699],{},[45,12698,55],{},[42,12700,12701,12711],{},[946,12702,12703,12709],{},[42,12704,12705,12707],{},[45,12706,842],{},[49,12708,845],{},[45,12710,9303],{},[45,12712,9290],{},[45,12714,842],{},[49,12716,845],{},[45,12718,55],{},[49,12720,58],{"stretchy":51},[60,12722,12723],{"encoding":62},"T(n) = \\Theta(n^{\\log_b a}) + \\Theta(n^{\\log_b a} \\log n) = \\Theta(n^{\\log_b a} \\log n)",[25,12725,12727,12754,12862,12984],{"className":12726,"ariaHidden":68},[67],[25,12728,12730,12733,12736,12739,12742,12745,12748,12751],{"className":12729},[72],[25,12731],{"className":12732,"style":77},[76],[25,12734,47],{"className":12735,"style":83},[81,82],[25,12737,52],{"className":12738},[87],[25,12740,55],{"className":12741},[81,82],[25,12743,58],{"className":12744},[94],[25,12746],{"className":12747,"style":442},[441],[25,12749,301],{"className":12750},[446],[25,12752],{"className":12753,"style":442},[441],[25,12755,12757,12760,12763,12766,12850,12853,12856,12859],{"className":12756},[72],[25,12758],{"className":12759,"style":6336},[76],[25,12761,239],{"className":12762},[81],[25,12764,52],{"className":12765},[87],[25,12767,12769,12772],{"className":12768},[81],[25,12770,55],{"className":12771},[81,82],[25,12773,12775],{"className":12774},[1010],[25,12776,12778],{"className":12777},[484],[25,12779,12781],{"className":12780},[489],[25,12782,12784],{"className":12783,"style":6094},[493],[25,12785,12786,12789],{"style":5476},[25,12787],{"className":12788,"style":1027},[501],[25,12790,12792],{"className":12791},[1031,1032,1033,1034],[25,12793,12795,12844,12847],{"className":12794},[81,1034],[25,12796,12798,12810],{"className":12797},[906,1034],[25,12799,12801,12804,12807],{"className":12800},[906,1034],[25,12802,6133],{"className":12803},[1034],[25,12805,6137],{"className":12806},[1034],[25,12808,911],{"className":12809,"style":910},[1034],[25,12811,12813],{"className":12812},[1010],[25,12814,12816,12836],{"className":12815},[484,485],[25,12817,12819,12833],{"className":12818},[489],[25,12820,12822],{"className":12821,"style":9925},[493],[25,12823,12824,12827],{"style":6156},[25,12825],{"className":12826,"style":6160},[501],[25,12828,12830],{"className":12829},[1031,6164,6165,1034],[25,12831,9303],{"className":12832},[81,82,1034],[25,12834,519],{"className":12835},[518],[25,12837,12839],{"className":12838},[489],[25,12840,12842],{"className":12841,"style":6178},[493],[25,12843],{},[25,12845],{"className":12846,"style":6184},[441,1034],[25,12848,9290],{"className":12849},[81,82,1034],[25,12851,58],{"className":12852},[94],[25,12854],{"className":12855,"style":600},[441],[25,12857,390],{"className":12858},[604],[25,12860],{"className":12861,"style":600},[441],[25,12863,12865,12868,12871,12874,12958,12961,12966,12969,12972,12975,12978,12981],{"className":12864},[72],[25,12866],{"className":12867,"style":6336},[76],[25,12869,239],{"className":12870},[81],[25,12872,52],{"className":12873},[87],[25,12875,12877,12880],{"className":12876},[81],[25,12878,55],{"className":12879},[81,82],[25,12881,12883],{"className":12882},[1010],[25,12884,12886],{"className":12885},[484],[25,12887,12889],{"className":12888},[489],[25,12890,12892],{"className":12891,"style":6094},[493],[25,12893,12894,12897],{"style":5476},[25,12895],{"className":12896,"style":1027},[501],[25,12898,12900],{"className":12899},[1031,1032,1033,1034],[25,12901,12903,12952,12955],{"className":12902},[81,1034],[25,12904,12906,12918],{"className":12905},[906,1034],[25,12907,12909,12912,12915],{"className":12908},[906,1034],[25,12910,6133],{"className":12911},[1034],[25,12913,6137],{"className":12914},[1034],[25,12916,911],{"className":12917,"style":910},[1034],[25,12919,12921],{"className":12920},[1010],[25,12922,12924,12944],{"className":12923},[484,485],[25,12925,12927,12941],{"className":12926},[489],[25,12928,12930],{"className":12929,"style":9925},[493],[25,12931,12932,12935],{"style":6156},[25,12933],{"className":12934,"style":6160},[501],[25,12936,12938],{"className":12937},[1031,6164,6165,1034],[25,12939,9303],{"className":12940},[81,82,1034],[25,12942,519],{"className":12943},[518],[25,12945,12947],{"className":12946},[489],[25,12948,12950],{"className":12949,"style":6178},[493],[25,12951],{},[25,12953],{"className":12954,"style":6184},[441,1034],[25,12956,9290],{"className":12957},[81,82,1034],[25,12959],{"className":12960,"style":902},[441],[25,12962,907,12964],{"className":12963},[906],[25,12965,911],{"style":910},[25,12967],{"className":12968,"style":902},[441],[25,12970,55],{"className":12971},[81,82],[25,12973,58],{"className":12974},[94],[25,12976],{"className":12977,"style":442},[441],[25,12979,301],{"className":12980},[446],[25,12982],{"className":12983,"style":442},[441],[25,12985,12987,12990,12993,12996,13080,13083,13088,13091,13094],{"className":12986},[72],[25,12988],{"className":12989,"style":6336},[76],[25,12991,239],{"className":12992},[81],[25,12994,52],{"className":12995},[87],[25,12997,12999,13002],{"className":12998},[81],[25,13000,55],{"className":13001},[81,82],[25,13003,13005],{"className":13004},[1010],[25,13006,13008],{"className":13007},[484],[25,13009,13011],{"className":13010},[489],[25,13012,13014],{"className":13013,"style":6094},[493],[25,13015,13016,13019],{"style":5476},[25,13017],{"className":13018,"style":1027},[501],[25,13020,13022],{"className":13021},[1031,1032,1033,1034],[25,13023,13025,13074,13077],{"className":13024},[81,1034],[25,13026,13028,13040],{"className":13027},[906,1034],[25,13029,13031,13034,13037],{"className":13030},[906,1034],[25,13032,6133],{"className":13033},[1034],[25,13035,6137],{"className":13036},[1034],[25,13038,911],{"className":13039,"style":910},[1034],[25,13041,13043],{"className":13042},[1010],[25,13044,13046,13066],{"className":13045},[484,485],[25,13047,13049,13063],{"className":13048},[489],[25,13050,13052],{"className":13051,"style":9925},[493],[25,13053,13054,13057],{"style":6156},[25,13055],{"className":13056,"style":6160},[501],[25,13058,13060],{"className":13059},[1031,6164,6165,1034],[25,13061,9303],{"className":13062},[81,82,1034],[25,13064,519],{"className":13065},[518],[25,13067,13069],{"className":13068},[489],[25,13070,13072],{"className":13071,"style":6178},[493],[25,13073],{},[25,13075],{"className":13076,"style":6184},[441,1034],[25,13078,9290],{"className":13079},[81,82,1034],[25,13081],{"className":13082,"style":902},[441],[25,13084,907,13086],{"className":13085},[906],[25,13087,911],{"style":910},[25,13089],{"className":13090,"style":902},[441],[25,13092,55],{"className":13093},[81,82],[25,13095,58],{"className":13096},[94],[11018,13098,13099,13100],{},"For case 3: ",[25,13101,13103,13181],{"className":13102,"translate":29},[28],[25,13104,13106],{"className":13105},[33],[35,13107,13108],{"xmlns":37},[39,13109,13110,13178],{},[42,13111,13112,13114,13116,13118,13120,13122,13124,13126,13144,13146,13148,13150,13152,13154,13156,13158,13160,13162,13164,13166,13168,13170,13172,13174,13176],{},[45,13113,47],{},[49,13115,52],{"stretchy":51},[45,13117,55],{},[49,13119,58],{"stretchy":51},[49,13121,301],{},[45,13123,239],{"mathvariant":142},[49,13125,52],{"stretchy":51},[5367,13127,13128,13130],{},[45,13129,55],{},[42,13131,13132,13142],{},[946,13133,13134,13140],{},[42,13135,13136,13138],{},[45,13137,842],{},[49,13139,845],{},[45,13141,9303],{},[45,13143,9290],{},[49,13145,58],{"stretchy":51},[49,13147,390],{},[45,13149,239],{"mathvariant":142},[49,13151,52],{"stretchy":51},[45,13153,9310],{},[49,13155,52],{"stretchy":51},[45,13157,55],{},[49,13159,58],{"stretchy":51},[49,13161,58],{"stretchy":51},[49,13163,301],{},[45,13165,239],{"mathvariant":142},[49,13167,52],{"stretchy":51},[45,13169,9310],{},[49,13171,52],{"stretchy":51},[45,13173,55],{},[49,13175,58],{"stretchy":51},[49,13177,58],{"stretchy":51},[60,13179,13180],{"encoding":62},"T(n) = \\Theta(n^{\\log_b a}) + \\Theta(f(n)) = \\Theta(f(n))",[25,13182,13184,13211,13319,13352],{"className":13183,"ariaHidden":68},[67],[25,13185,13187,13190,13193,13196,13199,13202,13205,13208],{"className":13186},[72],[25,13188],{"className":13189,"style":77},[76],[25,13191,47],{"className":13192,"style":83},[81,82],[25,13194,52],{"className":13195},[87],[25,13197,55],{"className":13198},[81,82],[25,13200,58],{"className":13201},[94],[25,13203],{"className":13204,"style":442},[441],[25,13206,301],{"className":13207},[446],[25,13209],{"className":13210,"style":442},[441],[25,13212,13214,13217,13220,13223,13307,13310,13313,13316],{"className":13213},[72],[25,13215],{"className":13216,"style":6336},[76],[25,13218,239],{"className":13219},[81],[25,13221,52],{"className":13222},[87],[25,13224,13226,13229],{"className":13225},[81],[25,13227,55],{"className":13228},[81,82],[25,13230,13232],{"className":13231},[1010],[25,13233,13235],{"className":13234},[484],[25,13236,13238],{"className":13237},[489],[25,13239,13241],{"className":13240,"style":6094},[493],[25,13242,13243,13246],{"style":5476},[25,13244],{"className":13245,"style":1027},[501],[25,13247,13249],{"className":13248},[1031,1032,1033,1034],[25,13250,13252,13301,13304],{"className":13251},[81,1034],[25,13253,13255,13267],{"className":13254},[906,1034],[25,13256,13258,13261,13264],{"className":13257},[906,1034],[25,13259,6133],{"className":13260},[1034],[25,13262,6137],{"className":13263},[1034],[25,13265,911],{"className":13266,"style":910},[1034],[25,13268,13270],{"className":13269},[1010],[25,13271,13273,13293],{"className":13272},[484,485],[25,13274,13276,13290],{"className":13275},[489],[25,13277,13279],{"className":13278,"style":9925},[493],[25,13280,13281,13284],{"style":6156},[25,13282],{"className":13283,"style":6160},[501],[25,13285,13287],{"className":13286},[1031,6164,6165,1034],[25,13288,9303],{"className":13289},[81,82,1034],[25,13291,519],{"className":13292},[518],[25,13294,13296],{"className":13295},[489],[25,13297,13299],{"className":13298,"style":6178},[493],[25,13300],{},[25,13302],{"className":13303,"style":6184},[441,1034],[25,13305,9290],{"className":13306},[81,82,1034],[25,13308,58],{"className":13309},[94],[25,13311],{"className":13312,"style":600},[441],[25,13314,390],{"className":13315},[604],[25,13317],{"className":13318,"style":600},[441],[25,13320,13322,13325,13328,13331,13334,13337,13340,13343,13346,13349],{"className":13321},[72],[25,13323],{"className":13324,"style":77},[76],[25,13326,239],{"className":13327},[81],[25,13329,52],{"className":13330},[87],[25,13332,9310],{"className":13333,"style":9468},[81,82],[25,13335,52],{"className":13336},[87],[25,13338,55],{"className":13339},[81,82],[25,13341,12104],{"className":13342},[94],[25,13344],{"className":13345,"style":442},[441],[25,13347,301],{"className":13348},[446],[25,13350],{"className":13351,"style":442},[441],[25,13353,13355,13358,13361,13364,13367,13370,13373],{"className":13354},[72],[25,13356],{"className":13357,"style":77},[76],[25,13359,239],{"className":13360},[81],[25,13362,52],{"className":13363},[87],[25,13365,9310],{"className":13366,"style":9468},[81,82],[25,13368,52],{"className":13369},[87],[25,13371,55],{"className":13372},[81,82],[25,13374,12104],{"className":13375},[94],[10,13377,13378],{},"Now, for any recursive formula, we only need to determine which case it belongs to and directly substitute into the corresponding formula. To facilitate understanding, let's go through a few examples.",[10,13380,13381],{},[13382,13383,13384],"strong",{},"Example 1:",[10,13386,13387],{},[25,13388,13390,13431],{"className":13389,"translate":29},[28],[25,13391,13393],{"className":13392},[33],[35,13394,13395],{"xmlns":37},[39,13396,13397,13428],{},[42,13398,13399,13401,13403,13405,13407,13409,13412,13414,13416,13418,13420,13422,13424,13426],{},[45,13400,47],{},[49,13402,52],{"stretchy":51},[45,13404,55],{},[49,13406,58],{"stretchy":51},[49,13408,301],{},[145,13410,13411],{},"9",[45,13413,47],{},[49,13415,52],{"stretchy":51},[45,13417,55],{},[45,13419,143],{"mathvariant":142},[145,13421,5346],{},[49,13423,58],{"stretchy":51},[49,13425,390],{},[45,13427,55],{},[60,13429,13430],{"encoding":62},"T(n) = 9 T(n\u002F3) + n",[25,13432,13434,13461,13495],{"className":13433,"ariaHidden":68},[67],[25,13435,13437,13440,13443,13446,13449,13452,13455,13458],{"className":13436},[72],[25,13438],{"className":13439,"style":77},[76],[25,13441,47],{"className":13442,"style":83},[81,82],[25,13444,52],{"className":13445},[87],[25,13447,55],{"className":13448},[81,82],[25,13450,58],{"className":13451},[94],[25,13453],{"className":13454,"style":442},[441],[25,13456,301],{"className":13457},[446],[25,13459],{"className":13460,"style":442},[441],[25,13462,13464,13467,13470,13473,13476,13479,13483,13486,13489,13492],{"className":13463},[72],[25,13465],{"className":13466,"style":77},[76],[25,13468,13411],{"className":13469},[81],[25,13471,47],{"className":13472,"style":83},[81,82],[25,13474,52],{"className":13475},[87],[25,13477,55],{"className":13478},[81,82],[25,13480,13482],{"className":13481},[81],"\u002F3",[25,13484,58],{"className":13485},[94],[25,13487],{"className":13488,"style":600},[441],[25,13490,390],{"className":13491},[604],[25,13493],{"className":13494,"style":600},[441],[25,13496,13498,13501],{"className":13497},[72],[25,13499],{"className":13500,"style":121},[76],[25,13502,55],{"className":13503},[81,82],[10,13505,13506,13507,9637,13558,9637,13609,2632,13675,13740,13741,13814,13815,270],{},"For this recursive formula, we have ",[25,13508,13510,13528],{"className":13509,"translate":29},[28],[25,13511,13513],{"className":13512},[33],[35,13514,13515],{"xmlns":37},[39,13516,13517,13525],{},[42,13518,13519,13521,13523],{},[45,13520,9290],{},[49,13522,301],{},[145,13524,13411],{},[60,13526,13527],{"encoding":62},"a = 9",[25,13529,13531,13549],{"className":13530,"ariaHidden":68},[67],[25,13532,13534,13537,13540,13543,13546],{"className":13533},[72],[25,13535],{"className":13536,"style":121},[76],[25,13538,9290],{"className":13539},[81,82],[25,13541],{"className":13542,"style":442},[441],[25,13544,301],{"className":13545},[446],[25,13547],{"className":13548,"style":442},[441],[25,13550,13552,13555],{"className":13551},[72],[25,13553],{"className":13554,"style":1220},[76],[25,13556,13411],{"className":13557},[81],[25,13559,13561,13579],{"className":13560,"translate":29},[28],[25,13562,13564],{"className":13563},[33],[35,13565,13566],{"xmlns":37},[39,13567,13568,13576],{},[42,13569,13570,13572,13574],{},[45,13571,9303],{},[49,13573,301],{},[145,13575,5346],{},[60,13577,13578],{"encoding":62},"b = 3",[25,13580,13582,13600],{"className":13581,"ariaHidden":68},[67],[25,13583,13585,13588,13591,13594,13597],{"className":13584},[72],[25,13586],{"className":13587,"style":9556},[76],[25,13589,9303],{"className":13590},[81,82],[25,13592],{"className":13593,"style":442},[441],[25,13595,301],{"className":13596},[446],[25,13598],{"className":13599,"style":442},[441],[25,13601,13603,13606],{"className":13602},[72],[25,13604],{"className":13605,"style":1220},[76],[25,13607,5346],{"className":13608},[81],[25,13610,13612,13636],{"className":13611,"translate":29},[28],[25,13613,13615],{"className":13614},[33],[35,13616,13617],{"xmlns":37},[39,13618,13619,13633],{},[42,13620,13621,13623,13625,13627,13629,13631],{},[45,13622,9310],{},[49,13624,52],{"stretchy":51},[45,13626,55],{},[49,13628,58],{"stretchy":51},[49,13630,301],{},[45,13632,55],{},[60,13634,13635],{"encoding":62},"f(n) = n",[25,13637,13639,13666],{"className":13638,"ariaHidden":68},[67],[25,13640,13642,13645,13648,13651,13654,13657,13660,13663],{"className":13641},[72],[25,13643],{"className":13644,"style":77},[76],[25,13646,9310],{"className":13647,"style":9468},[81,82],[25,13649,52],{"className":13650},[87],[25,13652,55],{"className":13653},[81,82],[25,13655,58],{"className":13656},[94],[25,13658],{"className":13659,"style":442},[441],[25,13661,301],{"className":13662},[446],[25,13664],{"className":13665,"style":442},[441],[25,13667,13669,13672],{"className":13668},[72],[25,13670],{"className":13671,"style":121},[76],[25,13673,55],{"className":13674},[81,82],[25,13676,13678,13701],{"className":13677,"translate":29},[28],[25,13679,13681],{"className":13680},[33],[35,13682,13683],{"xmlns":37},[39,13684,13685,13699],{},[42,13686,13687,13689,13691,13693,13695,13697],{},[45,13688,9310],{},[49,13690,52],{"stretchy":51},[45,13692,55],{},[49,13694,58],{"stretchy":51},[49,13696,301],{},[45,13698,55],{},[60,13700,13635],{"encoding":62},[25,13702,13704,13731],{"className":13703,"ariaHidden":68},[67],[25,13705,13707,13710,13713,13716,13719,13722,13725,13728],{"className":13706},[72],[25,13708],{"className":13709,"style":77},[76],[25,13711,9310],{"className":13712,"style":9468},[81,82],[25,13714,52],{"className":13715},[87],[25,13717,55],{"className":13718},[81,82],[25,13720,58],{"className":13721},[94],[25,13723],{"className":13724,"style":442},[441],[25,13726,301],{"className":13727},[446],[25,13729],{"className":13730,"style":442},[441],[25,13732,13734,13737],{"className":13733},[72],[25,13735],{"className":13736,"style":121},[76],[25,13738,55],{"className":13739},[81,82]," is asymptotically smaller than ",[25,13742,13744,13767],{"className":13743,"translate":29},[28],[25,13745,13747],{"className":13746},[33],[35,13748,13749],{"xmlns":37},[39,13750,13751,13765],{},[42,13752,13753,13755,13757,13763],{},[45,13754,239],{"mathvariant":142},[49,13756,52],{"stretchy":51},[5367,13758,13759,13761],{},[45,13760,55],{},[145,13762,147],{},[49,13764,58],{"stretchy":51},[60,13766,5587],{"encoding":62},[25,13768,13770],{"className":13769,"ariaHidden":68},[67],[25,13771,13773,13776,13779,13782,13811],{"className":13772},[72],[25,13774],{"className":13775,"style":5448},[76],[25,13777,239],{"className":13778},[81],[25,13780,52],{"className":13781},[87],[25,13783,13785,13788],{"className":13784},[81],[25,13786,55],{"className":13787},[81,82],[25,13789,13791],{"className":13790},[1010],[25,13792,13794],{"className":13793},[484],[25,13795,13797],{"className":13796},[489],[25,13798,13800],{"className":13799,"style":5473},[493],[25,13801,13802,13805],{"style":5476},[25,13803],{"className":13804,"style":1027},[501],[25,13806,13808],{"className":13807},[1031,1032,1033,1034],[25,13809,147],{"className":13810},[81,1034],[25,13812,58],{"className":13813},[94],", it can be applied to the first case of the master theorem, and the solution is ",[25,13816,13818,13851],{"className":13817,"translate":29},[28],[25,13819,13821],{"className":13820},[33],[35,13822,13823],{"xmlns":37},[39,13824,13825,13849],{},[42,13826,13827,13829,13831,13833,13835,13837,13839,13841,13847],{},[45,13828,47],{},[49,13830,52],{"stretchy":51},[45,13832,55],{},[49,13834,58],{"stretchy":51},[49,13836,301],{},[45,13838,239],{"mathvariant":142},[49,13840,52],{"stretchy":51},[5367,13842,13843,13845],{},[45,13844,55],{},[145,13846,147],{},[49,13848,58],{"stretchy":51},[60,13850,9165],{"encoding":62},[25,13852,13854,13881],{"className":13853,"ariaHidden":68},[67],[25,13855,13857,13860,13863,13866,13869,13872,13875,13878],{"className":13856},[72],[25,13858],{"className":13859,"style":77},[76],[25,13861,47],{"className":13862,"style":83},[81,82],[25,13864,52],{"className":13865},[87],[25,13867,55],{"className":13868},[81,82],[25,13870,58],{"className":13871},[94],[25,13873],{"className":13874,"style":442},[441],[25,13876,301],{"className":13877},[446],[25,13879],{"className":13880,"style":442},[441],[25,13882,13884,13887,13890,13893,13922],{"className":13883},[72],[25,13885],{"className":13886,"style":5448},[76],[25,13888,239],{"className":13889},[81],[25,13891,52],{"className":13892},[87],[25,13894,13896,13899],{"className":13895},[81],[25,13897,55],{"className":13898},[81,82],[25,13900,13902],{"className":13901},[1010],[25,13903,13905],{"className":13904},[484],[25,13906,13908],{"className":13907},[489],[25,13909,13911],{"className":13910,"style":5473},[493],[25,13912,13913,13916],{"style":5476},[25,13914],{"className":13915,"style":1027},[501],[25,13917,13919],{"className":13918},[1031,1032,1033,1034],[25,13920,147],{"className":13921},[81,1034],[25,13923,58],{"className":13924},[94],[10,13926,13927],{},[13382,13928,13929],{},"Example 2:",[10,13931,13932],{},[25,13933,13935,13975],{"className":13934,"translate":29},[28],[25,13936,13938],{"className":13937},[33],[35,13939,13940],{"xmlns":37},[39,13941,13942,13972],{},[42,13943,13944,13946,13948,13950,13952,13954,13956,13958,13960,13962,13964,13966,13968,13970],{},[45,13945,47],{},[49,13947,52],{"stretchy":51},[45,13949,55],{},[49,13951,58],{"stretchy":51},[49,13953,301],{},[45,13955,47],{},[49,13957,52],{"stretchy":51},[145,13959,147],{},[45,13961,55],{},[45,13963,143],{"mathvariant":142},[145,13965,5346],{},[49,13967,58],{"stretchy":51},[49,13969,390],{},[145,13971,339],{},[60,13973,13974],{"encoding":62},"T(n) = T(2n\u002F3) + 1",[25,13976,13978,14005,14038],{"className":13977,"ariaHidden":68},[67],[25,13979,13981,13984,13987,13990,13993,13996,13999,14002],{"className":13980},[72],[25,13982],{"className":13983,"style":77},[76],[25,13985,47],{"className":13986,"style":83},[81,82],[25,13988,52],{"className":13989},[87],[25,13991,55],{"className":13992},[81,82],[25,13994,58],{"className":13995},[94],[25,13997],{"className":13998,"style":442},[441],[25,14000,301],{"className":14001},[446],[25,14003],{"className":14004,"style":442},[441],[25,14006,14008,14011,14014,14017,14020,14023,14026,14029,14032,14035],{"className":14007},[72],[25,14009],{"className":14010,"style":77},[76],[25,14012,47],{"className":14013,"style":83},[81,82],[25,14015,52],{"className":14016},[87],[25,14018,147],{"className":14019},[81],[25,14021,55],{"className":14022},[81,82],[25,14024,13482],{"className":14025},[81],[25,14027,58],{"className":14028},[94],[25,14030],{"className":14031,"style":600},[441],[25,14033,390],{"className":14034},[604],[25,14036],{"className":14037,"style":600},[441],[25,14039,14041,14044],{"className":14040},[72],[25,14042],{"className":14043,"style":1220},[76],[25,14045,339],{"className":14046},[81],[10,14048,14049,14050,9637,14101,9637,14157,2632,14223,14557,14558,14601,14602,270],{},"For this formula, ",[25,14051,14053,14071],{"className":14052,"translate":29},[28],[25,14054,14056],{"className":14055},[33],[35,14057,14058],{"xmlns":37},[39,14059,14060,14068],{},[42,14061,14062,14064,14066],{},[45,14063,9290],{},[49,14065,301],{},[145,14067,339],{},[60,14069,14070],{"encoding":62},"a = 1",[25,14072,14074,14092],{"className":14073,"ariaHidden":68},[67],[25,14075,14077,14080,14083,14086,14089],{"className":14076},[72],[25,14078],{"className":14079,"style":121},[76],[25,14081,9290],{"className":14082},[81,82],[25,14084],{"className":14085,"style":442},[441],[25,14087,301],{"className":14088},[446],[25,14090],{"className":14091,"style":442},[441],[25,14093,14095,14098],{"className":14094},[72],[25,14096],{"className":14097,"style":1220},[76],[25,14099,339],{"className":14100},[81],[25,14102,14104,14126],{"className":14103,"translate":29},[28],[25,14105,14107],{"className":14106},[33],[35,14108,14109],{"xmlns":37},[39,14110,14111,14123],{},[42,14112,14113,14115,14117,14119,14121],{},[45,14114,9303],{},[49,14116,301],{},[145,14118,5346],{},[45,14120,143],{"mathvariant":142},[145,14122,147],{},[60,14124,14125],{"encoding":62},"b = 3\u002F2",[25,14127,14129,14147],{"className":14128,"ariaHidden":68},[67],[25,14130,14132,14135,14138,14141,14144],{"className":14131},[72],[25,14133],{"className":14134,"style":9556},[76],[25,14136,9303],{"className":14137},[81,82],[25,14139],{"className":14140,"style":442},[441],[25,14142,301],{"className":14143},[446],[25,14145],{"className":14146,"style":442},[441],[25,14148,14150,14153],{"className":14149},[72],[25,14151],{"className":14152,"style":77},[76],[25,14154,14156],{"className":14155},[81],"3\u002F2",[25,14158,14160,14184],{"className":14159,"translate":29},[28],[25,14161,14163],{"className":14162},[33],[35,14164,14165],{"xmlns":37},[39,14166,14167,14181],{},[42,14168,14169,14171,14173,14175,14177,14179],{},[45,14170,9310],{},[49,14172,52],{"stretchy":51},[45,14174,55],{},[49,14176,58],{"stretchy":51},[49,14178,301],{},[145,14180,339],{},[60,14182,14183],{"encoding":62},"f(n) = 1",[25,14185,14187,14214],{"className":14186,"ariaHidden":68},[67],[25,14188,14190,14193,14196,14199,14202,14205,14208,14211],{"className":14189},[72],[25,14191],{"className":14192,"style":77},[76],[25,14194,9310],{"className":14195,"style":9468},[81,82],[25,14197,52],{"className":14198},[87],[25,14200,55],{"className":14201},[81,82],[25,14203,58],{"className":14204},[94],[25,14206],{"className":14207,"style":442},[441],[25,14209,301],{"className":14210},[446],[25,14212],{"className":14213,"style":442},[441],[25,14215,14217,14220],{"className":14216},[72],[25,14218],{"className":14219,"style":1220},[76],[25,14221,339],{"className":14222},[81],[25,14224,14226,14294],{"className":14225,"translate":29},[28],[25,14227,14229],{"className":14228},[33],[35,14230,14231],{"xmlns":37},[39,14232,14233,14291],{},[42,14234,14235,14253,14255,14279,14281,14287,14289],{},[5367,14236,14237,14239],{},[45,14238,55],{},[42,14240,14241,14251],{},[946,14242,14243,14249],{},[42,14244,14245,14247],{},[45,14246,842],{},[49,14248,845],{},[45,14250,9303],{},[45,14252,9290],{},[49,14254,301],{},[5367,14256,14257,14259],{},[45,14258,55],{},[42,14260,14261,14277],{},[946,14262,14263,14269],{},[42,14264,14265,14267],{},[45,14266,842],{},[49,14268,845],{},[42,14270,14271,14273,14275],{},[145,14272,5346],{},[45,14274,143],{"mathvariant":142},[145,14276,147],{},[145,14278,339],{},[49,14280,301],{},[5367,14282,14283,14285],{},[45,14284,55],{},[145,14286,322],{},[49,14288,301],{},[145,14290,339],{},[60,14292,14293],{"encoding":62},"n^{\\log_b a} = n^{\\log_{3\u002F2} 1} = n^0 = 1",[25,14295,14297,14396,14504,14548],{"className":14296,"ariaHidden":68},[67],[25,14298,14300,14303,14387,14390,14393],{"className":14299},[72],[25,14301],{"className":14302,"style":6094},[76],[25,14304,14306,14309],{"className":14305},[81],[25,14307,55],{"className":14308},[81,82],[25,14310,14312],{"className":14311},[1010],[25,14313,14315],{"className":14314},[484],[25,14316,14318],{"className":14317},[489],[25,14319,14321],{"className":14320,"style":6094},[493],[25,14322,14323,14326],{"style":5476},[25,14324],{"className":14325,"style":1027},[501],[25,14327,14329],{"className":14328},[1031,1032,1033,1034],[25,14330,14332,14381,14384],{"className":14331},[81,1034],[25,14333,14335,14347],{"className":14334},[906,1034],[25,14336,14338,14341,14344],{"className":14337},[906,1034],[25,14339,6133],{"className":14340},[1034],[25,14342,6137],{"className":14343},[1034],[25,14345,911],{"className":14346,"style":910},[1034],[25,14348,14350],{"className":14349},[1010],[25,14351,14353,14373],{"className":14352},[484,485],[25,14354,14356,14370],{"className":14355},[489],[25,14357,14359],{"className":14358,"style":9925},[493],[25,14360,14361,14364],{"style":6156},[25,14362],{"className":14363,"style":6160},[501],[25,14365,14367],{"className":14366},[1031,6164,6165,1034],[25,14368,9303],{"className":14369},[81,82,1034],[25,14371,519],{"className":14372},[518],[25,14374,14376],{"className":14375},[489],[25,14377,14379],{"className":14378,"style":6178},[493],[25,14380],{},[25,14382],{"className":14383,"style":6184},[441,1034],[25,14385,9290],{"className":14386},[81,82,1034],[25,14388],{"className":14389,"style":442},[441],[25,14391,301],{"className":14392},[446],[25,14394],{"className":14395,"style":442},[441],[25,14397,14399,14403,14495,14498,14501],{"className":14398},[72],[25,14400],{"className":14401,"style":14402},[76],"height:0.905em;",[25,14404,14406,14409],{"className":14405},[81],[25,14407,55],{"className":14408},[81,82],[25,14410,14412],{"className":14411},[1010],[25,14413,14415],{"className":14414},[484],[25,14416,14418],{"className":14417},[489],[25,14419,14421],{"className":14420,"style":14402},[493],[25,14422,14424,14427],{"style":14423},"top:-3.1189em;margin-right:0.05em;",[25,14425],{"className":14426,"style":1027},[501],[25,14428,14430],{"className":14429},[1031,1032,1033,1034],[25,14431,14433,14489,14492],{"className":14432},[81,1034],[25,14434,14436,14448],{"className":14435},[906,1034],[25,14437,14439,14442,14445],{"className":14438},[906,1034],[25,14440,6133],{"className":14441},[1034],[25,14443,6137],{"className":14444},[1034],[25,14446,911],{"className":14447,"style":910},[1034],[25,14449,14451],{"className":14450},[1010],[25,14452,14454,14480],{"className":14453},[484,485],[25,14455,14457,14477],{"className":14456},[489],[25,14458,14461],{"className":14459,"style":14460},[493],"height:0.2698em;",[25,14462,14464,14468],{"style":14463},"top:-2.2698em;margin-right:0.0714em;",[25,14465],{"className":14466,"style":14467},[501],"height:2.5357em;",[25,14469,14471],{"className":14470},[1031,6164,6165,1034],[25,14472,14474],{"className":14473},[81,1034],[25,14475,14156],{"className":14476},[81,1034],[25,14478,519],{"className":14479},[518],[25,14481,14483],{"className":14482},[489],[25,14484,14487],{"className":14485,"style":14486},[493],"height:0.4444em;",[25,14488],{},[25,14490],{"className":14491,"style":6184},[441,1034],[25,14493,339],{"className":14494},[81,1034],[25,14496],{"className":14497,"style":442},[441],[25,14499,301],{"className":14500},[446],[25,14502],{"className":14503,"style":442},[441],[25,14505,14507,14510,14539,14542,14545],{"className":14506},[72],[25,14508],{"className":14509,"style":5473},[76],[25,14511,14513,14516],{"className":14512},[81],[25,14514,55],{"className":14515},[81,82],[25,14517,14519],{"className":14518},[1010],[25,14520,14522],{"className":14521},[484],[25,14523,14525],{"className":14524},[489],[25,14526,14528],{"className":14527,"style":5473},[493],[25,14529,14530,14533],{"style":5476},[25,14531],{"className":14532,"style":1027},[501],[25,14534,14536],{"className":14535},[1031,1032,1033,1034],[25,14537,322],{"className":14538},[81,1034],[25,14540],{"className":14541,"style":442},[441],[25,14543,301],{"className":14544},[446],[25,14546],{"className":14547,"style":442},[441],[25,14549,14551,14554],{"className":14550},[72],[25,14552],{"className":14553,"style":1220},[76],[25,14555,339],{"className":14556},[81],", which is exactly equal to ",[25,14559,14561,14580],{"className":14560,"translate":29},[28],[25,14562,14564],{"className":14563},[33],[35,14565,14566],{"xmlns":37},[39,14567,14568,14578],{},[42,14569,14570,14572,14574,14576],{},[45,14571,9310],{},[49,14573,52],{"stretchy":51},[45,14575,55],{},[49,14577,58],{"stretchy":51},[60,14579,9583],{"encoding":62},[25,14581,14583],{"className":14582,"ariaHidden":68},[67],[25,14584,14586,14589,14592,14595,14598],{"className":14585},[72],[25,14587],{"className":14588,"style":77},[76],[25,14590,9310],{"className":14591,"style":9468},[81,82],[25,14593,52],{"className":14594},[87],[25,14596,55],{"className":14597},[81,82],[25,14599,58],{"className":14600},[94],", it can be applied to the second case of the master theorem, and the solution is ",[25,14603,14605,14639],{"className":14604,"translate":29},[28],[25,14606,14608],{"className":14607},[33],[35,14609,14610],{"xmlns":37},[39,14611,14612,14636],{},[42,14613,14614,14616,14618,14620,14622,14624,14626,14628,14630,14632,14634],{},[45,14615,47],{},[49,14617,52],{"stretchy":51},[45,14619,55],{},[49,14621,58],{"stretchy":51},[49,14623,301],{},[45,14625,239],{"mathvariant":142},[49,14627,52],{"stretchy":51},[45,14629,842],{},[49,14631,845],{},[45,14633,55],{},[49,14635,58],{"stretchy":51},[60,14637,14638],{"encoding":62},"T(n) = \\Theta(\\log n)",[25,14640,14642,14669],{"className":14641,"ariaHidden":68},[67],[25,14643,14645,14648,14651,14654,14657,14660,14663,14666],{"className":14644},[72],[25,14646],{"className":14647,"style":77},[76],[25,14649,47],{"className":14650,"style":83},[81,82],[25,14652,52],{"className":14653},[87],[25,14655,55],{"className":14656},[81,82],[25,14658,58],{"className":14659},[94],[25,14661],{"className":14662,"style":442},[441],[25,14664,301],{"className":14665},[446],[25,14667],{"className":14668,"style":442},[441],[25,14670,14672,14675,14678,14681,14686,14689,14692],{"className":14671},[72],[25,14673],{"className":14674,"style":77},[76],[25,14676,239],{"className":14677},[81],[25,14679,52],{"className":14680},[87],[25,14682,907,14684],{"className":14683},[906],[25,14685,911],{"style":910},[25,14687],{"className":14688,"style":902},[441],[25,14690,55],{"className":14691},[81,82],[25,14693,58],{"className":14694},[94],[10,14696,14697],{},[13382,14698,14699],{},"Example 3:",[10,14701,14702],{},[25,14703,14705,14751],{"className":14704,"translate":29},[28],[25,14706,14708],{"className":14707},[33],[35,14709,14710],{"xmlns":37},[39,14711,14712,14748],{},[42,14713,14714,14716,14718,14720,14722,14724,14726,14728,14730,14732,14734,14736,14738,14740,14742,14744,14746],{},[45,14715,47],{},[49,14717,52],{"stretchy":51},[45,14719,55],{},[49,14721,58],{"stretchy":51},[49,14723,301],{},[145,14725,5346],{},[45,14727,47],{},[49,14729,52],{"stretchy":51},[45,14731,55],{},[45,14733,143],{"mathvariant":142},[145,14735,5357],{},[49,14737,58],{"stretchy":51},[49,14739,390],{},[45,14741,55],{},[45,14743,842],{},[49,14745,845],{},[45,14747,55],{},[60,14749,14750],{"encoding":62},"T(n) = 3T(n\u002F4) + n\\log n",[25,14752,14754,14781,14814],{"className":14753,"ariaHidden":68},[67],[25,14755,14757,14760,14763,14766,14769,14772,14775,14778],{"className":14756},[72],[25,14758],{"className":14759,"style":77},[76],[25,14761,47],{"className":14762,"style":83},[81,82],[25,14764,52],{"className":14765},[87],[25,14767,55],{"className":14768},[81,82],[25,14770,58],{"className":14771},[94],[25,14773],{"className":14774,"style":442},[441],[25,14776,301],{"className":14777},[446],[25,14779],{"className":14780,"style":442},[441],[25,14782,14784,14787,14790,14793,14796,14799,14802,14805,14808,14811],{"className":14783},[72],[25,14785],{"className":14786,"style":77},[76],[25,14788,5346],{"className":14789},[81],[25,14791,47],{"className":14792,"style":83},[81,82],[25,14794,52],{"className":14795},[87],[25,14797,55],{"className":14798},[81,82],[25,14800,5429],{"className":14801},[81],[25,14803,58],{"className":14804},[94],[25,14806],{"className":14807,"style":600},[441],[25,14809,390],{"className":14810},[604],[25,14812],{"className":14813,"style":600},[441],[25,14815,14817,14820,14823,14826,14831,14834],{"className":14816},[72],[25,14818],{"className":14819,"style":1000},[76],[25,14821,55],{"className":14822},[81,82],[25,14824],{"className":14825,"style":902},[441],[25,14827,907,14829],{"className":14828},[906],[25,14830,911],{"style":910},[25,14832],{"className":14833,"style":902},[441],[25,14835,55],{"className":14836},[81,82],[10,14838,14049,14839,9637,14890,9637,14941,2632,15027,9666,15328,15413,15414,15476,15477,15527,15528,9637,15556,15656,15657,9637,15685,9666,15713,15756],{},[25,14840,14842,14860],{"className":14841,"translate":29},[28],[25,14843,14845],{"className":14844},[33],[35,14846,14847],{"xmlns":37},[39,14848,14849,14857],{},[42,14850,14851,14853,14855],{},[45,14852,9290],{},[49,14854,301],{},[145,14856,5346],{},[60,14858,14859],{"encoding":62},"a = 3",[25,14861,14863,14881],{"className":14862,"ariaHidden":68},[67],[25,14864,14866,14869,14872,14875,14878],{"className":14865},[72],[25,14867],{"className":14868,"style":121},[76],[25,14870,9290],{"className":14871},[81,82],[25,14873],{"className":14874,"style":442},[441],[25,14876,301],{"className":14877},[446],[25,14879],{"className":14880,"style":442},[441],[25,14882,14884,14887],{"className":14883},[72],[25,14885],{"className":14886,"style":1220},[76],[25,14888,5346],{"className":14889},[81],[25,14891,14893,14911],{"className":14892,"translate":29},[28],[25,14894,14896],{"className":14895},[33],[35,14897,14898],{"xmlns":37},[39,14899,14900,14908],{},[42,14901,14902,14904,14906],{},[45,14903,9303],{},[49,14905,301],{},[145,14907,5357],{},[60,14909,14910],{"encoding":62},"b = 4",[25,14912,14914,14932],{"className":14913,"ariaHidden":68},[67],[25,14915,14917,14920,14923,14926,14929],{"className":14916},[72],[25,14918],{"className":14919,"style":9556},[76],[25,14921,9303],{"className":14922},[81,82],[25,14924],{"className":14925,"style":442},[441],[25,14927,301],{"className":14928},[446],[25,14930],{"className":14931,"style":442},[441],[25,14933,14935,14938],{"className":14934},[72],[25,14936],{"className":14937,"style":1220},[76],[25,14939,5357],{"className":14940},[81],[25,14942,14944,14974],{"className":14943,"translate":29},[28],[25,14945,14947],{"className":14946},[33],[35,14948,14949],{"xmlns":37},[39,14950,14951,14971],{},[42,14952,14953,14955,14957,14959,14961,14963,14965,14967,14969],{},[45,14954,9310],{},[49,14956,52],{"stretchy":51},[45,14958,55],{},[49,14960,58],{"stretchy":51},[49,14962,301],{},[45,14964,55],{},[45,14966,842],{},[49,14968,845],{},[45,14970,55],{},[60,14972,14973],{"encoding":62},"f(n) = n\\log n",[25,14975,14977,15004],{"className":14976,"ariaHidden":68},[67],[25,14978,14980,14983,14986,14989,14992,14995,14998,15001],{"className":14979},[72],[25,14981],{"className":14982,"style":77},[76],[25,14984,9310],{"className":14985,"style":9468},[81,82],[25,14987,52],{"className":14988},[87],[25,14990,55],{"className":14991},[81,82],[25,14993,58],{"className":14994},[94],[25,14996],{"className":14997,"style":442},[441],[25,14999,301],{"className":15000},[446],[25,15002],{"className":15003,"style":442},[441],[25,15005,15007,15010,15013,15016,15021,15024],{"className":15006},[72],[25,15008],{"className":15009,"style":1000},[76],[25,15011,55],{"className":15012},[81,82],[25,15014],{"className":15015,"style":902},[441],[25,15017,907,15019],{"className":15018},[906],[25,15020,911],{"style":910},[25,15022],{"className":15023,"style":902},[441],[25,15025,55],{"className":15026},[81,82],[25,15028,15030,15089],{"className":15029,"translate":29},[28],[25,15031,15033],{"className":15032},[33],[35,15034,15035],{"xmlns":37},[39,15036,15037,15086],{},[42,15038,15039,15057,15059,15077,15079],{},[5367,15040,15041,15043],{},[45,15042,55],{},[42,15044,15045,15055],{},[946,15046,15047,15053],{},[42,15048,15049,15051],{},[45,15050,842],{},[49,15052,845],{},[45,15054,9303],{},[45,15056,9290],{},[49,15058,301],{},[5367,15060,15061,15063],{},[45,15062,55],{},[42,15064,15065,15075],{},[946,15066,15067,15073],{},[42,15068,15069,15071],{},[45,15070,842],{},[49,15072,845],{},[145,15074,5357],{},[145,15076,5346],{},[49,15078,301],{},[5367,15080,15081,15083],{},[45,15082,55],{},[145,15084,15085],{},"0.793",[60,15087,15088],{"encoding":62},"n^{\\log_b a} = n^{\\log_4 3} = n^{0.793}",[25,15090,15092,15191,15290],{"className":15091,"ariaHidden":68},[67],[25,15093,15095,15098,15182,15185,15188],{"className":15094},[72],[25,15096],{"className":15097,"style":6094},[76],[25,15099,15101,15104],{"className":15100},[81],[25,15102,55],{"className":15103},[81,82],[25,15105,15107],{"className":15106},[1010],[25,15108,15110],{"className":15109},[484],[25,15111,15113],{"className":15112},[489],[25,15114,15116],{"className":15115,"style":6094},[493],[25,15117,15118,15121],{"style":5476},[25,15119],{"className":15120,"style":1027},[501],[25,15122,15124],{"className":15123},[1031,1032,1033,1034],[25,15125,15127,15176,15179],{"className":15126},[81,1034],[25,15128,15130,15142],{"className":15129},[906,1034],[25,15131,15133,15136,15139],{"className":15132},[906,1034],[25,15134,6133],{"className":15135},[1034],[25,15137,6137],{"className":15138},[1034],[25,15140,911],{"className":15141,"style":910},[1034],[25,15143,15145],{"className":15144},[1010],[25,15146,15148,15168],{"className":15147},[484,485],[25,15149,15151,15165],{"className":15150},[489],[25,15152,15154],{"className":15153,"style":9925},[493],[25,15155,15156,15159],{"style":6156},[25,15157],{"className":15158,"style":6160},[501],[25,15160,15162],{"className":15161},[1031,6164,6165,1034],[25,15163,9303],{"className":15164},[81,82,1034],[25,15166,519],{"className":15167},[518],[25,15169,15171],{"className":15170},[489],[25,15172,15174],{"className":15173,"style":6178},[493],[25,15175],{},[25,15177],{"className":15178,"style":6184},[441,1034],[25,15180,9290],{"className":15181},[81,82,1034],[25,15183],{"className":15184,"style":442},[441],[25,15186,301],{"className":15187},[446],[25,15189],{"className":15190,"style":442},[441],[25,15192,15194,15197,15281,15284,15287],{"className":15193},[72],[25,15195],{"className":15196,"style":6094},[76],[25,15198,15200,15203],{"className":15199},[81],[25,15201,55],{"className":15202},[81,82],[25,15204,15206],{"className":15205},[1010],[25,15207,15209],{"className":15208},[484],[25,15210,15212],{"className":15211},[489],[25,15213,15215],{"className":15214,"style":6094},[493],[25,15216,15217,15220],{"style":5476},[25,15218],{"className":15219,"style":1027},[501],[25,15221,15223],{"className":15222},[1031,1032,1033,1034],[25,15224,15226,15275,15278],{"className":15225},[81,1034],[25,15227,15229,15241],{"className":15228},[906,1034],[25,15230,15232,15235,15238],{"className":15231},[906,1034],[25,15233,6133],{"className":15234},[1034],[25,15236,6137],{"className":15237},[1034],[25,15239,911],{"className":15240,"style":910},[1034],[25,15242,15244],{"className":15243},[1010],[25,15245,15247,15267],{"className":15246},[484,485],[25,15248,15250,15264],{"className":15249},[489],[25,15251,15253],{"className":15252,"style":6153},[493],[25,15254,15255,15258],{"style":6156},[25,15256],{"className":15257,"style":6160},[501],[25,15259,15261],{"className":15260},[1031,6164,6165,1034],[25,15262,5357],{"className":15263},[81,1034],[25,15265,519],{"className":15266},[518],[25,15268,15270],{"className":15269},[489],[25,15271,15273],{"className":15272,"style":6178},[493],[25,15274],{},[25,15276],{"className":15277,"style":6184},[441,1034],[25,15279,5346],{"className":15280},[81,1034],[25,15282],{"className":15283,"style":442},[441],[25,15285,301],{"className":15286},[446],[25,15288],{"className":15289,"style":442},[441],[25,15291,15293,15296],{"className":15292},[72],[25,15294],{"className":15295,"style":5473},[76],[25,15297,15299,15302],{"className":15298},[81],[25,15300,55],{"className":15301},[81,82],[25,15303,15305],{"className":15304},[1010],[25,15306,15308],{"className":15307},[484],[25,15309,15311],{"className":15310},[489],[25,15312,15314],{"className":15313,"style":5473},[493],[25,15315,15316,15319],{"style":5476},[25,15317],{"className":15318,"style":1027},[501],[25,15320,15322],{"className":15321},[1031,1032,1033,1034],[25,15323,15325],{"className":15324},[81,1034],[25,15326,15085],{"className":15327},[81,1034],[25,15329,15331,15360],{"className":15330,"translate":29},[28],[25,15332,15334],{"className":15333},[33],[35,15335,15336],{"xmlns":37},[39,15337,15338,15358],{},[42,15339,15340,15342,15344,15346,15348,15350,15352,15354,15356],{},[45,15341,9310],{},[49,15343,52],{"stretchy":51},[45,15345,55],{},[49,15347,58],{"stretchy":51},[49,15349,301],{},[45,15351,55],{},[45,15353,842],{},[49,15355,845],{},[45,15357,55],{},[60,15359,14973],{"encoding":62},[25,15361,15363,15390],{"className":15362,"ariaHidden":68},[67],[25,15364,15366,15369,15372,15375,15378,15381,15384,15387],{"className":15365},[72],[25,15367],{"className":15368,"style":77},[76],[25,15370,9310],{"className":15371,"style":9468},[81,82],[25,15373,52],{"className":15374},[87],[25,15376,55],{"className":15377},[81,82],[25,15379,58],{"className":15380},[94],[25,15382],{"className":15383,"style":442},[441],[25,15385,301],{"className":15386},[446],[25,15388],{"className":15389,"style":442},[441],[25,15391,15393,15396,15399,15402,15407,15410],{"className":15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is asymptotically larger than ",[25,15415,15417,15435],{"className":15416,"translate":29},[28],[25,15418,15420],{"className":15419},[33],[35,15421,15422],{"xmlns":37},[39,15423,15424,15432],{},[42,15425,15426],{},[5367,15427,15428,15430],{},[45,15429,55],{},[145,15431,15085],{},[60,15433,15434],{"encoding":62},"n^{0.793}",[25,15436,15438],{"className":15437,"ariaHidden":68},[67],[25,15439,15441,15444],{"className":15440},[72],[25,15442],{"className":15443,"style":5473},[76],[25,15445,15447,15450],{"className":15446},[81],[25,15448,55],{"className":15449},[81,82],[25,15451,15453],{"className":15452},[1010],[25,15454,15456],{"className":15455},[484],[25,15457,15459],{"className":15458},[489],[25,15460,15462],{"className":15461,"style":5473},[493],[25,15463,15464,15467],{"style":5476},[25,15465],{"className":15466,"style":1027},[501],[25,15468,15470],{"className":15469},[1031,1032,1033,1034],[25,15471,15473],{"className":15472},[81,1034],[25,15474,15085],{"className":15475},[81,1034],", it should consider the third case of the master theorem. However, we need to check if the additional condition holds: for some constant ",[25,15478,15480,15497],{"className":15479,"translate":29},[28],[25,15481,15483],{"className":15482},[33],[35,15484,15485],{"xmlns":37},[39,15486,15487,15495],{},[42,15488,15489,15491,15493],{},[45,15490,950],{},[49,15492,4605],{},[145,15494,339],{},[60,15496,11849],{"encoding":62},[25,15498,15500,15518],{"className":15499,"ariaHidden":68},[67],[25,15501,15503,15506,15509,15512,15515],{"className":15502},[72],[25,15504],{"className":15505,"style":1867},[76],[25,15507,950],{"className":15508},[81,82],[25,15510],{"className":15511,"style":442},[441],[25,15513,4605],{"className":15514},[446],[25,15516],{"className":15517,"style":442},[441],[25,15519,15521,15524],{"className":15520},[72],[25,15522],{"className":15523,"style":1220},[76],[25,15525,339],{"className":15526},[81]," and all sufficiently large ",[25,15529,15531,15544],{"className":15530,"translate":29},[28],[25,15532,15534],{"className":15533},[33],[35,15535,15536],{"xmlns":37},[39,15537,15538,15542],{},[42,15539,15540],{},[45,15541,55],{},[60,15543,55],{"encoding":62},[25,15545,15547],{"className":15546,"ariaHidden":68},[67],[25,15548,15550,15553],{"className":15549},[72],[25,15551],{"className":15552,"style":121},[76],[25,15554,55],{"className":15555},[81,82],[25,15557,15559,15596],{"className":15558,"translate":29},[28],[25,15560,15562],{"className":15561},[33],[35,15563,15564],{"xmlns":37},[39,15565,15566,15594],{},[42,15567,15568,15570,15572,15574,15576,15578,15580,15582,15584,15586,15588,15590,15592],{},[45,15569,9290],{},[45,15571,9310],{},[49,15573,52],{"stretchy":51},[45,15575,55],{},[45,15577,143],{"mathvariant":142},[45,15579,9303],{},[49,15581,58],{"stretchy":51},[49,15583,944],{},[45,15585,950],{},[45,15587,9310],{},[49,15589,52],{"stretchy":51},[45,15591,55],{},[49,15593,58],{"stretchy":51},[60,15595,11949],{"encoding":62},[25,15597,15599,15635],{"className":15598,"ariaHidden":68},[67],[25,15600,15602,15605,15608,15611,15614,15617,15620,15623,15626,15629,15632],{"className":15601},[72],[25,15603],{"className":15604,"style":77},[76],[25,15606,9290],{"className":15607},[81,82],[25,15609,9310],{"className":15610,"style":9468},[81,82],[25,15612,52],{"className":15613},[87],[25,15615,55],{"className":15616},[81,82],[25,15618,143],{"className":15619},[81],[25,15621,9303],{"className":15622},[81,82],[25,15624,58],{"className":15625},[94],[25,15627],{"className":15628,"style":442},[441],[25,15630,944],{"className":15631},[446],[25,15633],{"className":15634,"style":442},[441],[25,15636,15638,15641,15644,15647,15650,15653],{"className":15637},[72],[25,15639],{"className":15640,"style":77},[76],[25,15642,950],{"className":15643},[81,82],[25,15645,9310],{"className":15646,"style":9468},[81,82],[25,15648,52],{"className":15649},[87],[25,15651,55],{"className":15652},[81,82],[25,15654,58],{"className":15655},[94]," holds. Substituting ",[25,15658,15660,15673],{"className":15659,"translate":29},[28],[25,15661,15663],{"className":15662},[33],[35,15664,15665],{"xmlns":37},[39,15666,15667,15671],{},[42,15668,15669],{},[45,15670,9290],{},[60,15672,9290],{"encoding":62},[25,15674,15676],{"className":15675,"ariaHidden":68},[67],[25,15677,15679,15682],{"className":15678},[72],[25,15680],{"className":15681,"style":121},[76],[25,15683,9290],{"className":15684},[81,82],[25,15686,15688,15701],{"className":15687,"translate":29},[28],[25,15689,15691],{"className":15690},[33],[35,15692,15693],{"xmlns":37},[39,15694,15695,15699],{},[42,15696,15697],{},[45,15698,9303],{},[60,15700,9303],{"encoding":62},[25,15702,15704],{"className":15703,"ariaHidden":68},[67],[25,15705,15707,15710],{"className":15706},[72],[25,15708],{"className":15709,"style":9556},[76],[25,15711,9303],{"className":15712},[81,82],[25,15714,15716,15735],{"className":15715,"translate":29},[28],[25,15717,15719],{"className":15718},[33],[35,15720,15721],{"xmlns":37},[39,15722,15723,15733],{},[42,15724,15725,15727,15729,15731],{},[45,15726,9310],{},[49,15728,52],{"stretchy":51},[45,15730,55],{},[49,15732,58],{"stretchy":51},[60,15734,9583],{"encoding":62},[25,15736,15738],{"className":15737,"ariaHidden":68},[67],[25,15739,15741,15744,15747,15750,15753],{"className":15740},[72],[25,15742],{"className":15743,"style":77},[76],[25,15745,9310],{"className":15746,"style":9468},[81,82],[25,15748,52],{"className":15749},[87],[25,15751,55],{"className":15752},[81,82],[25,15754,58],{"className":15755},[94]," into the inequality, we get",[25,15758,15760],{"className":15759,"translate":29},[277],[25,15761,15763,15879],{"className":15762,"translate":29},[28],[25,15764,15766],{"className":15765},[33],[35,15767,15768],{"xmlns":37,"display":286},[39,15769,15770,15876],{},[42,15771,15772,15774,15780,15782,15784,15790,15792,15794,15796,15798,15800,15802,15805,15808,15810,15816,15818,15820,15822,15824,15826,15828,15830,15832,15834,15836,15838,15840,15842,15844,15846,15862,15864,15866,15868,15870],{},[145,15773,5346],{},[2916,15775,15776,15778],{},[45,15777,55],{},[145,15779,5357],{},[45,15781,842],{},[49,15783,845],{},[2916,15785,15786,15788],{},[45,15787,55],{},[145,15789,5357],{},[49,15791,944],{},[45,15793,950],{},[45,15795,55],{},[45,15797,842],{},[49,15799,845],{},[45,15801,55],{},[349,15803,15804],{},"  ",[49,15806,15807],{},"⟹",[349,15809,15804],{},[2916,15811,15812,15814],{},[145,15813,5346],{},[145,15815,5357],{},[49,15817,52],{"stretchy":51},[45,15819,842],{},[49,15821,845],{},[45,15823,55],{},[49,15825,2958],{},[145,15827,147],{},[49,15829,58],{"stretchy":51},[49,15831,944],{},[45,15833,950],{},[45,15835,842],{},[49,15837,845],{},[45,15839,55],{},[349,15841,15804],{},[49,15843,15807],{},[349,15845,15804],{},[42,15847,15848,15850,15856,15858,15860],{},[49,15849,52],{"fence":68},[2916,15851,15852,15854],{},[145,15853,5346],{},[145,15855,5357],{},[49,15857,2958],{},[45,15859,950],{},[49,15861,58],{"fence":68},[45,15863,842],{},[49,15865,845],{},[45,15867,55],{},[49,15869,944],{},[2916,15871,15872,15874],{},[145,15873,5346],{},[145,15875,147],{},[60,15877,15878],{"encoding":62},"3\\frac{n}{4}\\log\\frac{n}{4} \\le cn\\log n \\implies \\frac{3}{4}( \\log n - 2) \\le c\\log n \\implies \\left(\\frac{3}{4} - c\\right) \\log n \\le \\frac{3}{2}",[25,15880,15882,16036,16077,16169,16190,16228,16347],{"className":15881,"ariaHidden":68},[67],[25,15883,15885,15889,15892,15954,15957,15962,15965,16027,16030,16033],{"className":15884},[72],[25,15886],{"className":15887,"style":15888},[76],"height:1.7936em;vertical-align:-0.686em;",[25,15890,5346],{"className":15891},[81],[25,15893,15895,15898,15951],{"className":15894},[81],[25,15896],{"className":15897},[87,721],[25,15899,15901],{"className":15900},[2916],[25,15902,15904,15943],{"className":15903},[484,485],[25,15905,15907,15940],{"className":15906},[489],[25,15908,15910,15921,15929],{"className":15909,"style":3973},[493],[25,15911,15912,15915],{"style":3976},[25,15913],{"className":15914,"style":548},[501],[25,15916,15918],{"className":15917},[81],[25,15919,5357],{"className":15920},[81],[25,15922,15923,15926],{"style":3082},[25,15924],{"className":15925,"style":548},[501],[25,15927],{"className":15928,"style":3090},[3089],[25,15930,15931,15934],{"style":3996},[25,15932],{"className":15933,"style":548},[501],[25,15935,15937],{"className":15936},[81],[25,15938,55],{"className":15939},[81,82],[25,15941,519],{"className":15942},[518],[25,15944,15946],{"className":15945},[489],[25,15947,15949],{"className":15948,"style":4015},[493],[25,15950],{},[25,15952],{"className":15953},[94,721],[25,15955],{"className":15956,"style":902},[441],[25,15958,907,15960],{"className":15959},[906],[25,15961,911],{"style":910},[25,15963],{"className":15964,"style":902},[441],[25,15966,15968,15971,16024],{"className":15967},[81],[25,15969],{"className":15970},[87,721],[25,15972,15974],{"className":15973},[2916],[25,15975,15977,16016],{"className":15976},[484,485],[25,15978,15980,16013],{"className":15979},[489],[25,15981,15983,15994,16002],{"className":15982,"style":3973},[493],[25,15984,15985,15988],{"style":3976},[25,15986],{"className":15987,"style":548},[501],[25,15989,15991],{"className":15990},[81],[25,15992,5357],{"className":15993},[81],[25,15995,15996,15999],{"style":3082},[25,15997],{"className":15998,"style":548},[501],[25,16000],{"className":16001,"style":3090},[3089],[25,16003,16004,16007],{"style":3996},[25,16005],{"className":16006,"style":548},[501],[25,16008,16010],{"className":16009},[81],[25,16011,55],{"className":16012},[81,82],[25,16014,519],{"className":16015},[518],[25,16017,16019],{"className":16018},[489],[25,16020,16022],{"className":16021,"style":4015},[493],[25,16023],{},[25,16025],{"className":16026},[94,721],[25,16028],{"className":16029,"style":442},[441],[25,16031,944],{"className":16032},[446],[25,16034],{"className":16035,"style":442},[441],[25,16037,16039,16042,16045,16048,16051,16056,16059,16062,16065,16068,16071,16074],{"className":16038},[72],[25,16040],{"className":16041,"style":1000},[76],[25,16043,950],{"className":16044},[81,82],[25,16046,55],{"className":16047},[81,82],[25,16049],{"className":16050,"style":902},[441],[25,16052,907,16054],{"className":16053},[906],[25,16055,911],{"style":910},[25,16057],{"className":16058,"style":902},[441],[25,16060,55],{"className":16061},[81,82],[25,16063],{"className":16064,"style":442},[441],[25,16066],{"className":16067,"style":442},[441],[25,16069,15807],{"className":16070},[446],[25,16072],{"className":16073,"style":442},[441],[25,16075],{"className":16076,"style":442},[441],[25,16078,16080,16084,16146,16149,16154,16157,16160,16163,16166],{"className":16079},[72],[25,16081],{"className":16082,"style":16083},[76],"height:2.0074em;vertical-align:-0.686em;",[25,16085,16087,16090,16143],{"className":16086},[81],[25,16088],{"className":16089},[87,721],[25,16091,16093],{"className":16092},[2916],[25,16094,16096,16135],{"className":16095},[484,485],[25,16097,16099,16132],{"className":16098},[489],[25,16100,16102,16113,16121],{"className":16101,"style":4089},[493],[25,16103,16104,16107],{"style":3976},[25,16105],{"className":16106,"style":548},[501],[25,16108,16110],{"className":16109},[81],[25,16111,5357],{"className":16112},[81],[25,16114,16115,16118],{"style":3082},[25,16116],{"className":16117,"style":548},[501],[25,16119],{"className":16120,"style":3090},[3089],[25,16122,16123,16126],{"style":3996},[25,16124],{"className":16125,"style":548},[501],[25,16127,16129],{"className":16128},[81],[25,16130,5346],{"className":16131},[81],[25,16133,519],{"className":16134},[518],[25,16136,16138],{"className":16137},[489],[25,16139,16141],{"className":16140,"style":4015},[493],[25,16142],{},[25,16144],{"className":16145},[94,721],[25,16147,52],{"className":16148},[87],[25,16150,907,16152],{"className":16151},[906],[25,16153,911],{"style":910},[25,16155],{"className":16156,"style":902},[441],[25,16158,55],{"className":16159},[81,82],[25,16161],{"className":16162,"style":600},[441],[25,16164,2958],{"className":16165},[604],[25,16167],{"className":16168,"style":600},[441],[25,16170,16172,16175,16178,16181,16184,16187],{"className":16171},[72],[25,16173],{"className":16174,"style":77},[76],[25,16176,147],{"className":16177},[81],[25,16179,58],{"className":16180},[94],[25,16182],{"className":16183,"style":442},[441],[25,16185,944],{"className":16186},[446],[25,16188],{"className":16189,"style":442},[441],[25,16191,16193,16196,16199,16202,16207,16210,16213,16216,16219,16222,16225],{"className":16192},[72],[25,16194],{"className":16195,"style":1000},[76],[25,16197,950],{"className":16198},[81,82],[25,16200],{"className":16201,"style":902},[441],[25,16203,907,16205],{"className":16204},[906],[25,16206,911],{"style":910},[25,16208],{"className":16209,"style":902},[441],[25,16211,55],{"className":16212},[81,82],[25,16214],{"className":16215,"style":442},[441],[25,16217],{"className":16218,"style":442},[441],[25,16220,15807],{"className":16221},[446],[25,16223],{"className":16224,"style":442},[441],[25,16226],{"className":16227,"style":442},[441],[25,16229,16231,16235,16324,16327,16332,16335,16338,16341,16344],{"className":16230},[72],[25,16232],{"className":16233,"style":16234},[76],"height:2.4em;vertical-align:-0.95em;",[25,16236,16238,16244,16306,16309,16312,16315,16318],{"className":16237},[460],[25,16239,16241],{"className":16240,"style":465},[87,464],[25,16242,52],{"className":16243},[469,1033],[25,16245,16247,16250,16303],{"className":16246},[81],[25,16248],{"className":16249},[87,721],[25,16251,16253],{"className":16252},[2916],[25,16254,16256,16295],{"className":16255},[484,485],[25,16257,16259,16292],{"className":16258},[489],[25,16260,16262,16273,16281],{"className":16261,"style":4089},[493],[25,16263,16264,16267],{"style":3976},[25,16265],{"className":16266,"style":548},[501],[25,16268,16270],{"className":16269},[81],[25,16271,5357],{"className":16272},[81],[25,16274,16275,16278],{"style":3082},[25,16276],{"className":16277,"style":548},[501],[25,16279],{"className":16280,"style":3090},[3089],[25,16282,16283,16286],{"style":3996},[25,16284],{"className":16285,"style":548},[501],[25,16287,16289],{"className":16288},[81],[25,16290,5346],{"className":16291},[81],[25,16293,519],{"className":16294},[518],[25,16296,16298],{"className":16297},[489],[25,16299,16301],{"className":16300,"style":4015},[493],[25,16302],{},[25,16304],{"className":16305},[94,721],[25,16307],{"className":16308,"style":600},[441],[25,16310,2958],{"className":16311},[604],[25,16313],{"className":16314,"style":600},[441],[25,16316,950],{"className":16317},[81,82],[25,16319,16321],{"className":16320,"style":465},[94,464],[25,16322,58],{"className":16323},[469,1033],[25,16325],{"className":16326,"style":902},[441],[25,16328,907,16330],{"className":16329},[906],[25,16331,911],{"style":910},[25,16333],{"className":16334,"style":902},[441],[25,16336,55],{"className":16337},[81,82],[25,16339],{"className":16340,"style":442},[441],[25,16342,944],{"className":16343},[446],[25,16345],{"className":16346,"style":442},[441],[25,16348,16350,16353],{"className":16349},[72],[25,16351],{"className":16352,"style":16083},[76],[25,16354,16356,16359,16412],{"className":16355},[81],[25,16357],{"className":16358},[87,721],[25,16360,16362],{"className":16361},[2916],[25,16363,16365,16404],{"className":16364},[484,485],[25,16366,16368,16401],{"className":16367},[489],[25,16369,16371,16382,16390],{"className":16370,"style":4089},[493],[25,16372,16373,16376],{"style":3976},[25,16374],{"className":16375,"style":548},[501],[25,16377,16379],{"className":16378},[81],[25,16380,147],{"className":16381},[81],[25,16383,16384,16387],{"style":3082},[25,16385],{"className":16386,"style":548},[501],[25,16388],{"className":16389,"style":3090},[3089],[25,16391,16392,16395],{"style":3996},[25,16393],{"className":16394,"style":548},[501],[25,16396,16398],{"className":16397},[81],[25,16399,5346],{"className":16400},[81],[25,16402,519],{"className":16403},[518],[25,16405,16407],{"className":16406},[489],[25,16408,16410],{"className":16409,"style":4015},[493],[25,16411],{},[25,16413],{"className":16414},[94,721],[10,16416,16417,16418,16539,16540,270],{},"When ",[25,16419,16421,16443],{"className":16420,"translate":29},[28],[25,16422,16424],{"className":16423},[33],[35,16425,16426],{"xmlns":37},[39,16427,16428,16440],{},[42,16429,16430,16432,16434],{},[45,16431,950],{},[49,16433,2447],{},[2916,16435,16436,16438],{},[145,16437,5346],{},[145,16439,5357],{},[60,16441,16442],{"encoding":62},"c \\ge \\frac{3}{4}",[25,16444,16446,16465],{"className":16445,"ariaHidden":68},[67],[25,16447,16449,16453,16456,16459,16462],{"className":16448},[72],[25,16450],{"className":16451,"style":16452},[76],"height:0.7719em;vertical-align:-0.136em;",[25,16454,950],{"className":16455},[81,82],[25,16457],{"className":16458,"style":442},[441],[25,16460,2447],{"className":16461},[446],[25,16463],{"className":16464,"style":442},[441],[25,16466,16468,16471],{"className":16467},[72],[25,16469],{"className":16470,"style":3215},[76],[25,16472,16474,16477,16536],{"className":16473},[81],[25,16475],{"className":16476},[87,721],[25,16478,16480],{"className":16479},[2916],[25,16481,16483,16528],{"className":16482},[484,485],[25,16484,16486,16525],{"className":16485},[489],[25,16487,16489,16503,16511],{"className":16488,"style":3234},[493],[25,16490,16491,16494],{"style":3067},[25,16492],{"className":16493,"style":548},[501],[25,16495,16497],{"className":16496},[1031,1032,1033,1034],[25,16498,16500],{"className":16499},[81,1034],[25,16501,5357],{"className":16502},[81,1034],[25,16504,16505,16508],{"style":3082},[25,16506],{"className":16507,"style":548},[501],[25,16509],{"className":16510,"style":3090},[3089],[25,16512,16513,16516],{"style":3093},[25,16514],{"className":16515,"style":548},[501],[25,16517,16519],{"className":16518},[1031,1032,1033,1034],[25,16520,16522],{"className":16521},[81,1034],[25,16523,5346],{"className":16524},[81,1034],[25,16526,519],{"className":16527},[518],[25,16529,16531],{"className":16530},[489],[25,16532,16534],{"className":16533,"style":3115},[493],[25,16535],{},[25,16537],{"className":16538},[94,721],", this inequality always holds, and the additional condition is satisfied. Therefore, it can be applied to the third case of the master theorem, and the solution is ",[25,16541,16543,16579],{"className":16542,"translate":29},[28],[25,16544,16546],{"className":16545},[33],[35,16547,16548],{"xmlns":37},[39,16549,16550,16576],{},[42,16551,16552,16554,16556,16558,16560,16562,16564,16566,16568,16570,16572,16574],{},[45,16553,47],{},[49,16555,52],{"stretchy":51},[45,16557,55],{},[49,16559,58],{"stretchy":51},[49,16561,301],{},[45,16563,239],{"mathvariant":142},[49,16565,52],{"stretchy":51},[45,16567,55],{},[45,16569,842],{},[49,16571,845],{},[45,16573,55],{},[49,16575,58],{"stretchy":51},[60,16577,16578],{"encoding":62},"T(n) = \\Theta(n\\log n)",[25,16580,16582,16609],{"className":16581,"ariaHidden":68},[67],[25,16583,16585,16588,16591,16594,16597,16600,16603,16606],{"className":16584},[72],[25,16586],{"className":16587,"style":77},[76],[25,16589,47],{"className":16590,"style":83},[81,82],[25,16592,52],{"className":16593},[87],[25,16595,55],{"className":16596},[81,82],[25,16598,58],{"className":16599},[94],[25,16601],{"className":16602,"style":442},[441],[25,16604,301],{"className":16605},[446],[25,16607],{"className":16608,"style":442},[441],[25,16610,16612,16615,16618,16621,16624,16627,16632,16635,16638],{"className":16611},[72],[25,16613],{"className":16614,"style":77},[76],[25,16616,239],{"className":16617},[81],[25,16619,52],{"className":16620},[87],[25,16622,55],{"className":16623},[81,82],[25,16625],{"className":16626,"style":902},[441],[25,16628,907,16630],{"className":16629},[906],[25,16631,911],{"style":910},[25,16633],{"className":16634,"style":902},[441],[25,16636,55],{"className":16637},[81,82],[25,16639,58],{"className":16640},[94],[10,16642,16643],{},[13382,16644,16645],{},"Example 4:",[10,16647,16648],{},[25,16649,16651,16697],{"className":16650,"translate":29},[28],[25,16652,16654],{"className":16653},[33],[35,16655,16656],{"xmlns":37},[39,16657,16658,16694],{},[42,16659,16660,16662,16664,16666,16668,16670,16672,16674,16676,16678,16680,16682,16684,16686,16688,16690,16692],{},[45,16661,47],{},[49,16663,52],{"stretchy":51},[45,16665,55],{},[49,16667,58],{"stretchy":51},[49,16669,301],{},[145,16671,147],{},[45,16673,47],{},[49,16675,52],{"stretchy":51},[45,16677,55],{},[45,16679,143],{"mathvariant":142},[145,16681,147],{},[49,16683,58],{"stretchy":51},[49,16685,390],{},[45,16687,55],{},[45,16689,842],{},[49,16691,845],{},[45,16693,55],{},[60,16695,16696],{"encoding":62},"T(n) = 2T(n\u002F2) + n\\log n",[25,16698,16700,16727,16760],{"className":16699,"ariaHidden":68},[67],[25,16701,16703,16706,16709,16712,16715,16718,16721,16724],{"className":16702},[72],[25,16704],{"className":16705,"style":77},[76],[25,16707,47],{"className":16708,"style":83},[81,82],[25,16710,52],{"className":16711},[87],[25,16713,55],{"className":16714},[81,82],[25,16716,58],{"className":16717},[94],[25,16719],{"className":16720,"style":442},[441],[25,16722,301],{"className":16723},[446],[25,16725],{"className":16726,"style":442},[441],[25,16728,16730,16733,16736,16739,16742,16745,16748,16751,16754,16757],{"className":16729},[72],[25,16731],{"className":16732,"style":77},[76],[25,16734,147],{"className":16735},[81],[25,16737,47],{"className":16738,"style":83},[81,82],[25,16740,52],{"className":16741},[87],[25,16743,55],{"className":16744},[81,82],[25,16746,166],{"className":16747},[81],[25,16749,58],{"className":16750},[94],[25,16752],{"className":16753,"style":600},[441],[25,16755,390],{"className":16756},[604],[25,16758],{"className":16759,"style":600},[441],[25,16761,16763,16766,16769,16772,16777,16780],{"className":16762},[72],[25,16764],{"className":16765,"style":1000},[76],[25,16767,55],{"className":16768},[81,82],[25,16770],{"className":16771,"style":902},[441],[25,16773,907,16775],{"className":16774},[906],[25,16776,911],{"style":910},[25,16778],{"className":16779,"style":902},[441],[25,16781,55],{"className":16782},[81,82],[10,16784,14049,16785,9637,16836,9637,16887,2632,16972,9666,17239,15413,17324,17352,17353,9637,17381,9666,17409,15756],{},[25,16786,16788,16806],{"className":16787,"translate":29},[28],[25,16789,16791],{"className":16790},[33],[35,16792,16793],{"xmlns":37},[39,16794,16795,16803],{},[42,16796,16797,16799,16801],{},[45,16798,9290],{},[49,16800,301],{},[145,16802,147],{},[60,16804,16805],{"encoding":62},"a = 2",[25,16807,16809,16827],{"className":16808,"ariaHidden":68},[67],[25,16810,16812,16815,16818,16821,16824],{"className":16811},[72],[25,16813],{"className":16814,"style":121},[76],[25,16816,9290],{"className":16817},[81,82],[25,16819],{"className":16820,"style":442},[441],[25,16822,301],{"className":16823},[446],[25,16825],{"className":16826,"style":442},[441],[25,16828,16830,16833],{"className":16829},[72],[25,16831],{"className":16832,"style":1220},[76],[25,16834,147],{"className":16835},[81],[25,16837,16839,16857],{"className":16838,"translate":29},[28],[25,16840,16842],{"className":16841},[33],[35,16843,16844],{"xmlns":37},[39,16845,16846,16854],{},[42,16847,16848,16850,16852],{},[45,16849,9303],{},[49,16851,301],{},[145,16853,147],{},[60,16855,16856],{"encoding":62},"b = 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a} = n^{\\log_2 2} = 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it should consider the third case of the master theorem. Substituting 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\\le cn\\log n \\implies \\log n - 1 \\le c\\log n \\implies (1-c)\\log n \\le 1",[25,17553,17555,17708,17749,17775,17794,17832,17853,17888],{"className":17554,"ariaHidden":68},[67],[25,17556,17558,17561,17564,17626,17629,17634,17637,17699,17702,17705],{"className":17557},[72],[25,17559],{"className":17560,"style":15888},[76],[25,17562,147],{"className":17563},[81],[25,17565,17567,17570,17623],{"className":17566},[81],[25,17568],{"className":17569},[87,721],[25,17571,17573],{"className":17572},[2916],[25,17574,17576,17615],{"className":17575},[484,485],[25,17577,17579,17612],{"className":17578},[489],[25,17580,17582,17593,17601],{"className":17581,"style":3973},[493],[25,17583,17584,17587],{"style":3976},[25,17585],{"className":17586,"style":548},[501],[25,17588,17590],{"className":17589},[81],[25,17591,147],{"className":17592},[81],[25,17594,17595,17598],{"style":3082},[25,17596],{"className":17597,"style":548},[501],[25,17599],{"className":17600,"style":3090},[3089],[25,17602,17603,17606],{"style":3996},[25,17604],{"className":17605,"style":548},[501],[25,17607,17609],{"className":17608},[81],[25,17610,55],{"className":17611},[81,82],[25,17613,519],{"className":17614},[518],[25,17616,17618],{"className":17617},[489],[25,17619,17621],{"className":17620,"style":4015},[493],[25,17622],{},[25,17624],{"className":17625},[94,721],[25,17627],{"className":17628,"style":902},[441],[25,17630,907,17632],{"className":17631},[906],[25,17633,911],{"style":910},[25,17635],{"className":17636,"style":902},[441],[25,17638,17640,17643,17696],{"className":17639},[81],[25,17641],{"className":17642},[87,721],[25,17644,17646],{"className":17645},[2916],[25,17647,17649,17688],{"className":17648},[484,485],[25,17650,17652,17685],{"className":17651},[489],[25,17653,17655,17666,17674],{"className":17654,"style":3973},[493],[25,17656,17657,17660],{"style":3976},[25,17658],{"className":17659,"style":548},[501],[25,17661,17663],{"className":17662},[81],[25,17664,147],{"className":17665},[81],[25,17667,17668,17671],{"style":3082},[25,17669],{"className":17670,"style":548},[501],[25,17672],{"className":17673,"style":3090},[3089],[25,17675,17676,17679],{"style":3996},[25,17677],{"className":17678,"style":548},[501],[25,17680,17682],{"className":17681},[81],[25,17683,55],{"className":17684},[81,82],[25,17686,519],{"className":17687},[518],[25,17689,17691],{"className":17690},[489],[25,17692,17694],{"className":17693,"style":4015},[493],[25,17695],{},[25,17697],{"className":17698},[94,721],[25,17700],{"className":17701,"style":442},[441],[25,17703,944],{"className":17704},[446],[25,17706],{"className":17707,"style":442},[441],[25,17709,17711,17714,17717,17720,17723,17728,17731,17734,17737,17740,17743,17746],{"className":17710},[72],[25,17712],{"className":17713,"style":1000},[76],[25,17715,950],{"className":17716},[81,82],[25,17718,55],{"className":17719},[81,82],[25,17721],{"className":17722,"style":902},[441],[25,17724,907,17726],{"className":17725},[906],[25,17727,911],{"style":910},[25,17729],{"className":17730,"style":902},[441],[25,17732,55],{"className":17733},[81,82],[25,17735],{"className":17736,"style":442},[441],[25,17738],{"className":17739,"style":442},[441],[25,17741,15807],{"className":17742},[446],[25,17744],{"className":17745,"style":442},[441],[25,17747],{"className":17748,"style":442},[441],[25,17750,17752,17755,17760,17763,17766,17769,17772],{"className":17751},[72],[25,17753],{"className":17754,"style":1000},[76],[25,17756,907,17758],{"className":17757},[906],[25,17759,911],{"style":910},[25,17761],{"className":17762,"style":902},[441],[25,17764,55],{"className":17765},[81,82],[25,17767],{"className":17768,"style":600},[441],[25,17770,2958],{"className":17771},[604],[25,17773],{"className":17774,"style":600},[441],[25,17776,17778,17782,17785,17788,17791],{"className":17777},[72],[25,17779],{"className":17780,"style":17781},[76],"height:0.7804em;vertical-align:-0.136em;",[25,17783,339],{"className":17784},[81],[25,17786],{"className":17787,"style":442},[441],[25,17789,944],{"className":17790},[446],[25,17792],{"className":17793,"style":442},[441],[25,17795,17797,17800,17803,17806,17811,17814,17817,17820,17823,17826,17829],{"className":17796},[72],[25,17798],{"className":17799,"style":1000},[76],[25,17801,950],{"className":17802},[81,82],[25,17804],{"className":17805,"style":902},[441],[25,17807,907,17809],{"className":17808},[906],[25,17810,911],{"style":910},[25,17812],{"className":17813,"style":902},[441],[25,17815,55],{"className":17816},[81,82],[25,17818],{"className":17819,"style":442},[441],[25,17821],{"className":17822,"style":442},[441],[25,17824,15807],{"className":17825},[446],[25,17827],{"className":17828,"style":442},[441],[25,17830],{"className":17831,"style":442},[441],[25,17833,17835,17838,17841,17844,17847,17850],{"className":17834},[72],[25,17836],{"className":17837,"style":77},[76],[25,17839,52],{"className":17840},[87],[25,17842,339],{"className":17843},[81],[25,17845],{"className":17846,"style":600},[441],[25,17848,2958],{"className":17849},[604],[25,17851],{"className":17852,"style":600},[441],[25,17854,17856,17859,17862,17865,17868,17873,17876,17879,17882,17885],{"className":17855},[72],[25,17857],{"className":17858,"style":77},[76],[25,17860,950],{"className":17861},[81,82],[25,17863,58],{"className":17864},[94],[25,17866],{"className":17867,"style":902},[441],[25,17869,907,17871],{"className":17870},[906],[25,17872,911],{"style":910},[25,17874],{"className":17875,"style":902},[441],[25,17877,55],{"className":17878},[81,82],[25,17880],{"className":17881,"style":442},[441],[25,17883,944],{"className":17884},[446],[25,17886],{"className":17887,"style":442},[441],[25,17889,17891,17894],{"className":17890},[72],[25,17892],{"className":17893,"style":1220},[76],[25,17895,339],{"className":17896},[81],[10,17898,16417,17899,17950,17951,18001,18002,18135,18136,270],{},[25,17900,17902,17920],{"className":17901,"translate":29},[28],[25,17903,17905],{"className":17904},[33],[35,17906,17907],{"xmlns":37},[39,17908,17909,17917],{},[42,17910,17911,17913,17915],{},[45,17912,950],{},[49,17914,2447],{},[145,17916,339],{},[60,17918,17919],{"encoding":62},"c \\ge 1",[25,17921,17923,17941],{"className":17922,"ariaHidden":68},[67],[25,17924,17926,17929,17932,17935,17938],{"className":17925},[72],[25,17927],{"className":17928,"style":16452},[76],[25,17930,950],{"className":17931},[81,82],[25,17933],{"className":17934,"style":442},[441],[25,17936,2447],{"className":17937},[446],[25,17939],{"className":17940,"style":442},[441],[25,17942,17944,17947],{"className":17943},[72],[25,17945],{"className":17946,"style":1220},[76],[25,17948,339],{"className":17949},[81],", this inequality always holds. However, the additional condition requires ",[25,17952,17954,17971],{"className":17953,"translate":29},[28],[25,17955,17957],{"className":17956},[33],[35,17958,17959],{"xmlns":37},[39,17960,17961,17969],{},[42,17962,17963,17965,17967],{},[45,17964,950],{},[49,17966,4605],{},[145,17968,339],{},[60,17970,11849],{"encoding":62},[25,17972,17974,17992],{"className":17973,"ariaHidden":68},[67],[25,17975,17977,17980,17983,17986,17989],{"className":17976},[72],[25,17978],{"className":17979,"style":1867},[76],[25,17981,950],{"className":17982},[81,82],[25,17984],{"className":17985,"style":442},[441],[25,17987,4605],{"className":17988},[446],[25,17990],{"className":17991,"style":442},[441],[25,17993,17995,17998],{"className":17994},[72],[25,17996],{"className":17997,"style":1220},[76],[25,17999,339],{"className":18000},[81],", so this case cannot be applied to the third case of the master theorem. In other words, not all recursive formulas of the form ",[25,18003,18005,18051],{"className":18004,"translate":29},[28],[25,18006,18008],{"className":18007},[33],[35,18009,18010],{"xmlns":37},[39,18011,18012,18048],{},[42,18013,18014,18016,18018,18020,18022,18024,18026,18028,18030,18032,18034,18036,18038,18040,18042,18044,18046],{},[45,18015,47],{},[49,18017,52],{"stretchy":51},[45,18019,55],{},[49,18021,58],{"stretchy":51},[49,18023,301],{},[45,18025,9290],{},[45,18027,47],{},[49,18029,52],{"stretchy":51},[45,18031,55],{},[45,18033,143],{"mathvariant":142},[45,18035,9303],{},[49,18037,58],{"stretchy":51},[49,18039,390],{},[45,18041,9310],{},[49,18043,52],{"stretchy":51},[45,18045,55],{},[49,18047,58],{"stretchy":51},[60,18049,18050],{"encoding":62},"T(n) = a T(n\u002Fb) + f(n)",[25,18052,18054,18081,18117],{"className":18053,"ariaHidden":68},[67],[25,18055,18057,18060,18063,18066,18069,18072,18075,18078],{"className":18056},[72],[25,18058],{"className":18059,"style":77},[76],[25,18061,47],{"className":18062,"style":83},[81,82],[25,18064,52],{"className":18065},[87],[25,18067,55],{"className":18068},[81,82],[25,18070,58],{"className":18071},[94],[25,18073],{"className":18074,"style":442},[441],[25,18076,301],{"className":18077},[446],[25,18079],{"className":18080,"style":442},[441],[25,18082,18084,18087,18090,18093,18096,18099,18102,18105,18108,18111,18114],{"className":18083},[72],[25,18085],{"className":18086,"style":77},[76],[25,18088,9290],{"className":18089},[81,82],[25,18091,47],{"className":18092,"style":83},[81,82],[25,18094,52],{"className":18095},[87],[25,18097,55],{"className":18098},[81,82],[25,18100,143],{"className":18101},[81],[25,18103,9303],{"className":18104},[81,82],[25,18106,58],{"className":18107},[94],[25,18109],{"className":18110,"style":600},[441],[25,18112,390],{"className":18113},[604],[25,18115],{"className":18116,"style":600},[441],[25,18118,18120,18123,18126,18129,18132],{"className":18119},[72],[25,18121],{"className":18122,"style":77},[76],[25,18124,9310],{"className":18125,"style":9468},[81,82],[25,18127,52],{"className":18128},[87],[25,18130,55],{"className":18131},[81,82],[25,18133,58],{"className":18134},[94]," can be solved using the master theorem; there is a gap between cases two and three. In such cases, we can only use the recursion tree method. Students can try it, and the result will be ",[25,18137,18139,18181],{"className":18138,"translate":29},[28],[25,18140,18142],{"className":18141},[33],[35,18143,18144],{"xmlns":37},[39,18145,18146,18178],{},[42,18147,18148,18150,18152,18154,18156,18158,18160,18162,18164,18174,18176],{},[45,18149,47],{},[49,18151,52],{"stretchy":51},[45,18153,55],{},[49,18155,58],{"stretchy":51},[49,18157,301],{},[45,18159,239],{"mathvariant":142},[49,18161,52],{"stretchy":51},[45,18163,55],{},[5367,18165,18166,18172],{},[42,18167,18168,18170],{},[45,18169,842],{},[49,18171,845],{},[145,18173,147],{},[45,18175,55],{},[49,18177,58],{"stretchy":51},[60,18179,18180],{"encoding":62},"T(n) = \\Theta(n \\log^2 n)",[25,18182,18184,18211],{"className":18183,"ariaHidden":68},[67],[25,18185,18187,18190,18193,18196,18199,18202,18205,18208],{"className":18186},[72],[25,18188],{"className":18189,"style":77},[76],[25,18191,47],{"className":18192,"style":83},[81,82],[25,18194,52],{"className":18195},[87],[25,18197,55],{"className":18198},[81,82],[25,18200,58],{"className":18201},[94],[25,18203],{"className":18204,"style":442},[441],[25,18206,301],{"className":18207},[446],[25,18209],{"className":18210,"style":442},[441],[25,18212,18214,18218,18221,18224,18227,18230,18263,18266,18269],{"className":18213},[72],[25,18215],{"className":18216,"style":18217},[76],"height:1.1484em;vertical-align:-0.25em;",[25,18219,239],{"className":18220},[81],[25,18222,52],{"className":18223},[87],[25,18225,55],{"className":18226},[81,82],[25,18228],{"className":18229,"style":902},[441],[25,18231,18233,18238],{"className":18232},[906],[25,18234,907,18236],{"className":18235},[906],[25,18237,911],{"style":910},[25,18239,18241],{"className":18240},[1010],[25,18242,18244],{"className":18243},[484],[25,18245,18247],{"className":18246},[489],[25,18248,18251],{"className":18249,"style":18250},[493],"height:0.8984em;",[25,18252,18254,18257],{"style":18253},"top:-3.1473em;margin-right:0.05em;",[25,18255],{"className":18256,"style":1027},[501],[25,18258,18260],{"className":18259},[1031,1032,1033,1034],[25,18261,147],{"className":18262},[81,1034],[25,18264],{"className":18265,"style":902},[441],[25,18267,55],{"className":18268},[81,82],[25,18270,58],{"className":18271},[94],[10,18273,18274],{},"After going through these four examples, have you mastered the master theorem? If you have any questions, please leave a comment, and I will reply as soon as possible.",[10,18276,18277],{},"The study of algorithms is endless, and the calculation of time complexity is fundamental and should not be underestimated.",[14,18279,18281],{"id":18280},"details","Details",[10,18283,18284,18285,18330,18331,18359,18360,11214,18410,18551,18552,18692,18693,18743],{},"Careful readers may notice that in the fourth case mentioned above, although ",[25,18286,18288,18308],{"className":18287,"translate":29},[28],[25,18289,18291],{"className":18290},[33],[35,18292,18293],{"xmlns":37},[39,18294,18295,18305],{},[42,18296,18297,18299,18301,18303],{},[45,18298,55],{},[45,18300,6133],{},[45,18302,911],{},[45,18304,55],{},[60,18306,18307],{"encoding":62},"nlgn",[25,18309,18311],{"className":18310,"ariaHidden":68},[67],[25,18312,18314,18317,18320,18324,18327],{"className":18313},[72],[25,18315],{"className":18316,"style":1000},[76],[25,18318,55],{"className":18319},[81,82],[25,18321,6133],{"className":18322,"style":18323},[81,82],"margin-right:0.01968em;",[25,18325,911],{"className":18326,"style":10758},[81,82],[25,18328,55],{"className":18329},[81,82]," grows asymptotically faster than ",[25,18332,18334,18347],{"className":18333,"translate":29},[28],[25,18335,18337],{"className":18336},[33],[35,18338,18339],{"xmlns":37},[39,18340,18341,18345],{},[42,18342,18343],{},[45,18344,55],{},[60,18346,55],{"encoding":62},[25,18348,18350],{"className":18349,"ariaHidden":68},[67],[25,18351,18353,18356],{"className":18352},[72],[25,18354],{"className":18355,"style":121},[76],[25,18357,55],{"className":18358},[81,82],", it is not asymptotically greater in the polynomial sense because there is no ",[25,18361,18363,18380],{"className":18362,"translate":29},[28],[25,18364,18366],{"className":18365},[33],[35,18367,18368],{"xmlns":37},[39,18369,18370,18378],{},[42,18371,18372,18374,18376],{},[45,18373,11176],{},[49,18375,411],{},[145,18377,322],{},[60,18379,11183],{"encoding":62},[25,18381,18383,18401],{"className":18382,"ariaHidden":68},[67],[25,18384,18386,18389,18392,18395,18398],{"className":18385},[72],[25,18387],{"className":18388,"style":1867},[76],[25,18390,11176],{"className":18391},[81,82],[25,18393],{"className":18394,"style":442},[441],[25,18396,411],{"className":18397},[446],[25,18399],{"className":18400,"style":442},[441],[25,18402,18404,18407],{"className":18403},[72],[25,18405],{"className":18406,"style":1220},[76],[25,18408,322],{"className":18409},[81],[25,18411,18413,18455],{"className":18412,"translate":29},[28],[25,18414,18416],{"className":18415},[33],[35,18417,18418],{"xmlns":37},[39,18419,18420,18452],{},[42,18421,18422,18424,18426,18428,18430,18432,18434],{},[45,18423,55],{},[45,18425,6133],{},[45,18427,911],{},[45,18429,55],{},[49,18431,301],{},[45,18433,5035],{"mathvariant":142},[42,18435,18436,18438,18450],{},[49,18437,52],{"fence":68},[5367,18439,18440,18442],{},[45,18441,55],{},[42,18443,18444,18446,18448],{},[145,18445,339],{},[49,18447,390],{},[45,18449,11176],{},[49,18451,58],{"fence":68},[60,18453,18454],{"encoding":62},"nlgn = \\Omega \\left( n^{1+\\epsilon} \\right)",[25,18456,18458,18485],{"className":18457,"ariaHidden":68},[67],[25,18459,18461,18464,18467,18470,18473,18476,18479,18482],{"className":18460},[72],[25,18462],{"className":18463,"style":1000},[76],[25,18465,55],{"className":18466},[81,82],[25,18468,6133],{"className":18469,"style":18323},[81,82],[25,18471,911],{"className":18472,"style":10758},[81,82],[25,18474,55],{"className":18475},[81,82],[25,18477],{"className":18478,"style":442},[441],[25,18480,301],{"className":18481},[446],[25,18483],{"className":18484,"style":442},[441],[25,18486,18488,18492,18495,18498],{"className":18487},[72],[25,18489],{"className":18490,"style":18491},[76],"height:1.2em;vertical-align:-0.35em;",[25,18493,5035],{"className":18494},[81],[25,18496],{"className":18497,"style":902},[441],[25,18499,18501,18507,18545],{"className":18500},[460],[25,18502,18504],{"className":18503,"style":465},[87,464],[25,18505,52],{"className":18506},[469,6165],[25,18508,18510,18513],{"className":18509},[81],[25,18511,55],{"className":18512},[81,82],[25,18514,18516],{"className":18515},[1010],[25,18517,18519],{"className":18518},[484],[25,18520,18522],{"className":18521},[489],[25,18523,18525],{"className":18524,"style":5473},[493],[25,18526,18527,18530],{"style":5476},[25,18528],{"className":18529,"style":1027},[501],[25,18531,18533],{"className":18532},[1031,1032,1033,1034],[25,18534,18536,18539,18542],{"className":18535},[81,1034],[25,18537,339],{"className":18538},[81,1034],[25,18540,390],{"className":18541},[604,1034],[25,18543,11176],{"className":18544},[81,82,1034],[25,18546,18548],{"className":18547,"style":465},[94,464],[25,18549,58],{"className":18550},[469,6165]," holds. On the contrary, ",[25,18553,18555,18597],{"className":18554,"translate":29},[28],[25,18556,18558],{"className":18557},[33],[35,18559,18560],{"xmlns":37},[39,18561,18562,18594],{},[42,18563,18564,18566,18568,18570,18572,18574,18576],{},[45,18565,55],{},[45,18567,6133],{},[45,18569,911],{},[45,18571,55],{},[49,18573,301],{},[45,18575,6137],{},[42,18577,18578,18580,18592],{},[49,18579,52],{"fence":68},[5367,18581,18582,18584],{},[45,18583,55],{},[42,18585,18586,18588,18590],{},[145,18587,339],{},[49,18589,390],{},[45,18591,11176],{},[49,18593,58],{"fence":68},[60,18595,18596],{"encoding":62},"nlgn = o \\left( n^{1+\\epsilon} \\right)",[25,18598,18600,18627],{"className":18599,"ariaHidden":68},[67],[25,18601,18603,18606,18609,18612,18615,18618,18621,18624],{"className":18602},[72],[25,18604],{"className":18605,"style":1000},[76],[25,18607,55],{"className":18608},[81,82],[25,18610,6133],{"className":18611,"style":18323},[81,82],[25,18613,911],{"className":18614,"style":10758},[81,82],[25,18616,55],{"className":18617},[81,82],[25,18619],{"className":18620,"style":442},[441],[25,18622,301],{"className":18623},[446],[25,18625],{"className":18626,"style":442},[441],[25,18628,18630,18633,18636,18639],{"className":18629},[72],[25,18631],{"className":18632,"style":18491},[76],[25,18634,6137],{"className":18635},[81,82],[25,18637],{"className":18638,"style":902},[441],[25,18640,18642,18648,18686],{"className":18641},[460],[25,18643,18645],{"className":18644,"style":465},[87,464],[25,18646,52],{"className":18647},[469,6165],[25,18649,18651,18654],{"className":18650},[81],[25,18652,55],{"className":18653},[81,82],[25,18655,18657],{"className":18656},[1010],[25,18658,18660],{"className":18659},[484],[25,18661,18663],{"className":18662},[489],[25,18664,18666],{"className":18665,"style":5473},[493],[25,18667,18668,18671],{"style":5476},[25,18669],{"className":18670,"style":1027},[501],[25,18672,18674],{"className":18673},[1031,1032,1033,1034],[25,18675,18677,18680,18683],{"className":18676},[81,1034],[25,18678,339],{"className":18679},[81,1034],[25,18681,390],{"className":18682},[604,1034],[25,18684,11176],{"className":18685},[81,82,1034],[25,18687,18689],{"className":18688,"style":465},[94,464],[25,18690,58],{"className":18691},[469,6165]," holds for any ",[25,18694,18696,18713],{"className":18695,"translate":29},[28],[25,18697,18699],{"className":18698},[33],[35,18700,18701],{"xmlns":37},[39,18702,18703,18711],{},[42,18704,18705,18707,18709],{},[45,18706,11176],{},[49,18708,411],{},[145,18710,322],{},[60,18712,11183],{"encoding":62},[25,18714,18716,18734],{"className":18715,"ariaHidden":68},[67],[25,18717,18719,18722,18725,18728,18731],{"className":18718},[72],[25,18720],{"className":18721,"style":1867},[76],[25,18723,11176],{"className":18724},[81,82],[25,18726],{"className":18727,"style":442},[441],[25,18729,411],{"className":18730},[446],[25,18732],{"className":18733,"style":442},[441],[25,18735,18737,18740],{"className":18736},[72],[25,18738],{"className":18739,"style":1220},[76],[25,18741,322],{"className":18742},[81],". So, in fact, this case does not need to be judged for regular conditions to draw conclusions in advance. However, in the third case in the previous text, I did not emphasize asymptotic greater in the polynomial sense, not because I forgot to write it, but because this condition is not needed at all.",[10,18745,18746,18747,18954,18955,18961],{},"For any recurrence relation that fits the master theorem format, if it satisfies the regular conditions in case three, then it must satisfy ",[25,18748,18750,18804],{"className":18749,"translate":29},[28],[25,18751,18753],{"className":18752},[33],[35,18754,18755],{"xmlns":37},[39,18756,18757,18801],{},[42,18758,18759,18761,18769,18771,18773],{},[45,18760,9310],{},[42,18762,18763,18765,18767],{},[49,18764,52],{"fence":68},[45,18766,55],{},[49,18768,58],{"fence":68},[49,18770,301],{},[45,18772,5035],{"mathvariant":142},[42,18774,18775,18777,18799],{},[49,18776,52],{"fence":68},[5367,18778,18779,18781],{},[45,18780,55],{},[42,18782,18783,18785,18787,18793,18795,18797],{},[45,18784,6133],{},[45,18786,6137],{},[946,18788,18789,18791],{},[45,18790,911],{},[45,18792,9303],{},[45,18794,9290],{},[49,18796,390],{},[45,18798,11176],{},[49,18800,58],{"fence":68},[60,18802,18803],{"encoding":62},"f\\left(n\\right) = \\Omega \\left( n^{log_b a + \\epsilon} \\right)",[25,18805,18807,18840],{"className":18806,"ariaHidden":68},[67],[25,18808,18810,18813,18816,18819,18831,18834,18837],{"className":18809},[72],[25,18811],{"className":18812,"style":77},[76],[25,18814,9310],{"className":18815,"style":9468},[81,82],[25,18817],{"className":18818,"style":902},[441],[25,18820,18822,18825,18828],{"className":18821},[460],[25,18823,52],{"className":18824,"style":465},[87,464],[25,18826,55],{"className":18827},[81,82],[25,18829,58],{"className":18830,"style":465},[94,464],[25,18832],{"className":18833,"style":442},[441],[25,18835,301],{"className":18836},[446],[25,18838],{"className":18839,"style":442},[441],[25,18841,18843,18846,18849,18852],{"className":18842},[72],[25,18844],{"className":18845,"style":18491},[76],[25,18847,5035],{"className":18848},[81],[25,18850],{"className":18851,"style":902},[441],[25,18853,18855,18861,18948],{"className":18854},[460],[25,18856,18858],{"className":18857,"style":465},[87,464],[25,18859,52],{"className":18860},[469,6165],[25,18862,18864,18867],{"className":18863},[81],[25,18865,55],{"className":18866},[81,82],[25,18868,18870],{"className":18869},[1010],[25,18871,18873],{"className":18872},[484],[25,18874,18876],{"className":18875},[489],[25,18877,18879],{"className":18878,"style":6094},[493],[25,18880,18881,18884],{"style":5476},[25,18882],{"className":18883,"style":1027},[501],[25,18885,18887],{"className":18886},[1031,1032,1033,1034],[25,18888,18890,18893,18896,18939,18942,18945],{"className":18889},[81,1034],[25,18891,6133],{"className":18892,"style":18323},[81,82,1034],[25,18894,6137],{"className":18895},[81,82,1034],[25,18897,18899,18902],{"className":18898},[81,1034],[25,18900,911],{"className":18901,"style":10758},[81,82,1034],[25,18903,18905],{"className":18904},[1010],[25,18906,18908,18930],{"className":18907},[484,485],[25,18909,18911,18927],{"className":18910},[489],[25,18912,18915],{"className":18913,"style":18914},[493],"height:0.3448em;",[25,18916,18918,18921],{"style":18917},"top:-2.3488em;margin-left:-0.0359em;margin-right:0.0714em;",[25,18919],{"className":18920,"style":6160},[501],[25,18922,18924],{"className":18923},[1031,6164,6165,1034],[25,18925,9303],{"className":18926},[81,82,1034],[25,18928,519],{"className":18929},[518],[25,18931,18933],{"className":18932},[489],[25,18934,18937],{"className":18935,"style":18936},[493],"height:0.1512em;",[25,18938],{},[25,18940,9290],{"className":18941},[81,82,1034],[25,18943,390],{"className":18944},[604,1034],[25,18946,11176],{"className":18947},[81,82,1034],[25,18949,18951],{"className":18950,"style":465},[94,464],[25,18952,58],{"className":18953},[469,6165],".\" The specific proof details can be found in ",[9290,18956,18960],{"href":18957,"rel":18958},"https:\u002F\u002Fcs.stackexchange.com\u002Fa\u002F121749\u002F128497",[18959],"nofollow","this answer"," on StackExchange, and are not detailed here.",[10,18963,18964,18965,19168],{},"Although regular conditions can imply ",[25,18966,18968,19021],{"className":18967,"translate":29},[28],[25,18969,18971],{"className":18970},[33],[35,18972,18973],{"xmlns":37},[39,18974,18975,19019],{},[42,18976,18977,18979,18987,18989,18991],{},[45,18978,9310],{},[42,18980,18981,18983,18985],{},[49,18982,52],{"fence":68},[45,18984,55],{},[49,18986,58],{"fence":68},[49,18988,301],{},[45,18990,5035],{"mathvariant":142},[42,18992,18993,18995,19017],{},[49,18994,52],{"fence":68},[5367,18996,18997,18999],{},[45,18998,55],{},[42,19000,19001,19003,19005,19011,19013,19015],{},[45,19002,6133],{},[45,19004,6137],{},[946,19006,19007,19009],{},[45,19008,911],{},[45,19010,9303],{},[45,19012,9290],{},[49,19014,390],{},[45,19016,11176],{},[49,19018,58],{"fence":68},[60,19020,18803],{"encoding":62},[25,19022,19024,19057],{"className":19023,"ariaHidden":68},[67],[25,19025,19027,19030,19033,19036,19048,19051,19054],{"className":19026},[72],[25,19028],{"className":19029,"style":77},[76],[25,19031,9310],{"className":19032,"style":9468},[81,82],[25,19034],{"className":19035,"style":902},[441],[25,19037,19039,19042,19045],{"className":19038},[460],[25,19040,52],{"className":19041,"style":465},[87,464],[25,19043,55],{"className":19044},[81,82],[25,19046,58],{"className":19047,"style":465},[94,464],[25,19049],{"className":19050,"style":442},[441],[25,19052,301],{"className":19053},[446],[25,19055],{"className":19056,"style":442},[441],[25,19058,19060,19063,19066,19069],{"className":19059},[72],[25,19061],{"className":19062,"style":18491},[76],[25,19064,5035],{"className":19065},[81],[25,19067],{"className":19068,"style":902},[441],[25,19070,19072,19078,19162],{"className":19071},[460],[25,19073,19075],{"className":19074,"style":465},[87,464],[25,19076,52],{"className":19077},[469,6165],[25,19079,19081,19084],{"className":19080},[81],[25,19082,55],{"className":19083},[81,82],[25,19085,19087],{"className":19086},[1010],[25,19088,19090],{"className":19089},[484],[25,19091,19093],{"className":19092},[489],[25,19094,19096],{"className":19095,"style":6094},[493],[25,19097,19098,19101],{"style":5476},[25,19099],{"className":19100,"style":1027},[501],[25,19102,19104],{"className":19103},[1031,1032,1033,1034],[25,19105,19107,19110,19113,19153,19156,19159],{"className":19106},[81,1034],[25,19108,6133],{"className":19109,"style":18323},[81,82,1034],[25,19111,6137],{"className":19112},[81,82,1034],[25,19114,19116,19119],{"className":19115},[81,1034],[25,19117,911],{"className":19118,"style":10758},[81,82,1034],[25,19120,19122],{"className":19121},[1010],[25,19123,19125,19145],{"className":19124},[484,485],[25,19126,19128,19142],{"className":19127},[489],[25,19129,19131],{"className":19130,"style":18914},[493],[25,19132,19133,19136],{"style":18917},[25,19134],{"className":19135,"style":6160},[501],[25,19137,19139],{"className":19138},[1031,6164,6165,1034],[25,19140,9303],{"className":19141},[81,82,1034],[25,19143,519],{"className":19144},[518],[25,19146,19148],{"className":19147},[489],[25,19149,19151],{"className":19150,"style":18936},[493],[25,19152],{},[25,19154,9290],{"className":19155},[81,82,1034],[25,19157,390],{"className":19158},[604,1034],[25,19160,11176],{"className":19161},[81,82,1034],[25,19163,19165],{"className":19164,"style":465},[94,464],[25,19166,58],{"className":19167},[469,6165],", the reverse is not true. We can construct the following function:",[10,19170,19171],{},[25,19172,19174,19233],{"className":19173,"translate":29},[28],[25,19175,19177],{"className":19176},[33],[35,19178,19179],{"xmlns":37},[39,19180,19181,19230],{},[42,19182,19183,19185,19193,19195,19197,19209,19211,19213],{},[45,19184,47],{},[42,19186,19187,19189,19191],{},[49,19188,52],{"fence":68},[45,19190,55],{},[49,19192,58],{"fence":68},[49,19194,301],{},[45,19196,47],{},[42,19198,19199,19201,19203,19205,19207],{},[49,19200,52],{"fence":68},[45,19202,55],{},[45,19204,143],{"mathvariant":142},[145,19206,147],{},[49,19208,58],{"fence":68},[49,19210,390],{},[45,19212,55],{},[42,19214,19215,19217,19219,19221,19224,19226,19228],{},[49,19216,52],{"fence":68},[145,19218,147],{},[49,19220,2958],{},[45,19222,19223],{},"cos",[49,19225,845],{},[45,19227,55],{},[49,19229,58],{"fence":68},[60,19231,19232],{"encoding":62},"T\\left(n\\right) = T\\left(n\u002F2\\right) + n\\left( 2 - \\cos n \\right)",[25,19234,19236,19269,19305],{"className":19235,"ariaHidden":68},[67],[25,19237,19239,19242,19245,19248,19260,19263,19266],{"className":19238},[72],[25,19240],{"className":19241,"style":77},[76],[25,19243,47],{"className":19244,"style":83},[81,82],[25,19246],{"className":19247,"style":902},[441],[25,19249,19251,19254,19257],{"className":19250},[460],[25,19252,52],{"className":19253,"style":465},[87,464],[25,19255,55],{"className":19256},[81,82],[25,19258,58],{"className":19259,"style":465},[94,464],[25,19261],{"className":19262,"style":442},[441],[25,19264,301],{"className":19265},[446],[25,19267],{"className":19268,"style":442},[441],[25,19270,19272,19275,19278,19281,19296,19299,19302],{"className":19271},[72],[25,19273],{"className":19274,"style":77},[76],[25,19276,47],{"className":19277,"style":83},[81,82],[25,19279],{"className":19280,"style":902},[441],[25,19282,19284,19287,19290,19293],{"className":19283},[460],[25,19285,52],{"className":19286,"style":465},[87,464],[25,19288,55],{"className":19289},[81,82],[25,19291,166],{"className":19292},[81],[25,19294,58],{"className":19295,"style":465},[94,464],[25,19297],{"className":19298,"style":600},[441],[25,19300,390],{"className":19301},[604],[25,19303],{"className":19304,"style":600},[441],[25,19306,19308,19311,19314,19317],{"className":19307},[72],[25,19309],{"className":19310,"style":77},[76],[25,19312,55],{"className":19313},[81,82],[25,19315],{"className":19316,"style":902},[441],[25,19318,19320,19323,19326,19329,19332,19335,19338,19341,19344],{"className":19319},[460],[25,19321,52],{"className":19322,"style":465},[87,464],[25,19324,147],{"className":19325},[81],[25,19327],{"className":19328,"style":600},[441],[25,19330,2958],{"className":19331},[604],[25,19333],{"className":19334,"style":600},[441],[25,19336,19223],{"className":19337},[906],[25,19339],{"className":19340,"style":902},[441],[25,19342,55],{"className":19343},[81,82],[25,19345,58],{"className":19346,"style":465},[94,464],[10,19348,5495,19349,9637,19400,9666,19451,18692,19805,19856],{},[25,19350,19352,19370],{"className":19351,"translate":29},[28],[25,19353,19355],{"className":19354},[33],[35,19356,19357],{"xmlns":37},[39,19358,19359,19367],{},[42,19360,19361,19363,19365],{},[45,19362,9290],{},[49,19364,301],{},[145,19366,339],{},[60,19368,19369],{"encoding":62},"a=1",[25,19371,19373,19391],{"className":19372,"ariaHidden":68},[67],[25,19374,19376,19379,19382,19385,19388],{"className":19375},[72],[25,19377],{"className":19378,"style":121},[76],[25,19380,9290],{"className":19381},[81,82],[25,19383],{"className":19384,"style":442},[441],[25,19386,301],{"className":19387},[446],[25,19389],{"className":19390,"style":442},[441],[25,19392,19394,19397],{"className":19393},[72],[25,19395],{"className":19396,"style":1220},[76],[25,19398,339],{"className":19399},[81],[25,19401,19403,19421],{"className":19402,"translate":29},[28],[25,19404,19406],{"className":19405},[33],[35,19407,19408],{"xmlns":37},[39,19409,19410,19418],{},[42,19411,19412,19414,19416],{},[45,19413,9303],{},[49,19415,301],{},[145,19417,147],{},[60,19419,19420],{"encoding":62},"b=2",[25,19422,19424,19442],{"className":19423,"ariaHidden":68},[67],[25,19425,19427,19430,19433,19436,19439],{"className":19426},[72],[25,19428],{"className":19429,"style":9556},[76],[25,19431,9303],{"className":19432},[81,82],[25,19434],{"className":19435,"style":442},[441],[25,19437,301],{"className":19438},[446],[25,19440],{"className":19441,"style":442},[441],[25,19443,19445,19448],{"className":19444},[72],[25,19446],{"className":19447,"style":1220},[76],[25,19449,147],{"className":19450},[81],[25,19452,19454,19544],{"className":19453,"translate":29},[28],[25,19455,19457],{"className":19456},[33],[35,19458,19459],{"xmlns":37},[39,19460,19461,19541],{},[42,19462,19463,19465,19473,19475,19477,19493,19495,19497,19525,19527,19529],{},[45,19464,9310],{},[42,19466,19467,19469,19471],{},[49,19468,52],{"fence":68},[45,19470,55],{},[49,19472,58],{"fence":68},[49,19474,301],{},[45,19476,55],{},[42,19478,19479,19481,19483,19485,19487,19489,19491],{},[49,19480,52],{"fence":68},[145,19482,147],{},[49,19484,2958],{},[45,19486,19223],{},[49,19488,845],{},[45,19490,55],{},[49,19492,58],{"fence":68},[49,19494,301],{},[45,19496,5035],{"mathvariant":142},[42,19498,19499,19501,19523],{},[49,19500,52],{"fence":68},[5367,19502,19503,19505],{},[45,19504,55],{},[42,19506,19507,19509,19511,19517,19519,19521],{},[45,19508,6133],{},[45,19510,6137],{},[946,19512,19513,19515],{},[45,19514,911],{},[145,19516,147],{},[145,19518,339],{},[49,19520,390],{},[45,19522,11176],{},[49,19524,58],{"fence":68},[49,19526,301],{},[45,19528,5035],{"mathvariant":142},[42,19530,19531,19533,19539],{},[49,19532,52],{"fence":68},[5367,19534,19535,19537],{},[45,19536,55],{},[45,19538,11176],{},[49,19540,58],{"fence":68},[60,19542,19543],{"encoding":62},"f\\left(n\\right) = n \\left( 2 - \\cos n \\right) = \\Omega \\left( n^{log_2 1+ \\epsilon} \\right) = \\Omega \\left( n^\\epsilon \\right)",[25,19545,19547,19580,19631,19754],{"className":19546,"ariaHidden":68},[67],[25,19548,19550,19553,19556,19559,19571,19574,19577],{"className":19549},[72],[25,19551],{"className":19552,"style":77},[76],[25,19554,9310],{"className":19555,"style":9468},[81,82],[25,19557],{"className":19558,"style":902},[441],[25,19560,19562,19565,19568],{"className":19561},[460],[25,19563,52],{"className":19564,"style":465},[87,464],[25,19566,55],{"className":19567},[81,82],[25,19569,58],{"className":19570,"style":465},[94,464],[25,19572],{"className":19573,"style":442},[441],[25,19575,301],{"className":19576},[446],[25,19578],{"className":19579,"style":442},[441],[25,19581,19583,19586,19589,19592,19622,19625,19628],{"className":19582},[72],[25,19584],{"className":19585,"style":77},[76],[25,19587,55],{"className":19588},[81,82],[25,19590],{"className":19591,"style":902},[441],[25,19593,19595,19598,19601,19604,19607,19610,19613,19616,19619],{"className":19594},[460],[25,19596,52],{"className":19597,"style":465},[87,464],[25,19599,147],{"className":19600},[81],[25,19602],{"className":19603,"style":600},[441],[25,19605,2958],{"className":19606},[604],[25,19608],{"className":19609,"style":600},[441],[25,19611,19223],{"className":19612},[906],[25,19614],{"className":19615,"style":902},[441],[25,19617,55],{"className":19618},[81,82],[25,19620,58],{"className":19621,"style":465},[94,464],[25,19623],{"className":19624,"style":442},[441],[25,19626,301],{"className":19627},[446],[25,19629],{"className":19630,"style":442},[441],[25,19632,19634,19637,19640,19643,19745,19748,19751],{"className":19633},[72],[25,19635],{"className":19636,"style":18491},[76],[25,19638,5035],{"className":19639},[81],[25,19641],{"className":19642,"style":902},[441],[25,19644,19646,19652,19739],{"className":19645},[460],[25,19647,19649],{"className":19648,"style":465},[87,464],[25,19650,52],{"className":19651},[469,6165],[25,19653,19655,19658],{"className":19654},[81],[25,19656,55],{"className":19657},[81,82],[25,19659,19661],{"className":19660},[1010],[25,19662,19664],{"className":19663},[484],[25,19665,19667],{"className":19666},[489],[25,19668,19670],{"className":19669,"style":6094},[493],[25,19671,19672,19675],{"style":5476},[25,19673],{"className":19674,"style":1027},[501],[25,19676,19678],{"className":19677},[1031,1032,1033,1034],[25,19679,19681,19684,19687,19730,19733,19736],{"className":19680},[81,1034],[25,19682,6133],{"className":19683,"style":18323},[81,82,1034],[25,19685,6137],{"className":19686},[81,82,1034],[25,19688,19690,19693],{"className":19689},[81,1034],[25,19691,911],{"className":19692,"style":10758},[81,82,1034],[25,19694,19696],{"className":19695},[1010],[25,19697,19699,19721],{"className":19698},[484,485],[25,19700,19702,19718],{"className":19701},[489],[25,19703,19706],{"className":19704,"style":19705},[493],"height:0.3173em;",[25,19707,19709,19712],{"style":19708},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[25,19710],{"className":19711,"style":6160},[501],[25,19713,19715],{"className":19714},[1031,6164,6165,1034],[25,19716,147],{"className":19717},[81,1034],[25,19719,519],{"className":19720},[518],[25,19722,19724],{"className":19723},[489],[25,19725,19728],{"className":19726,"style":19727},[493],"height:0.143em;",[25,19729],{},[25,19731,339],{"className":19732},[81,1034],[25,19734,390],{"className":19735},[604,1034],[25,19737,11176],{"className":19738},[81,82,1034],[25,19740,19742],{"className":19741,"style":465},[94,464],[25,19743,58],{"className":19744},[469,6165],[25,19746],{"className":19747,"style":442},[441],[25,19749,301],{"className":19750},[446],[25,19752],{"className":19753,"style":442},[441],[25,19755,19757,19760,19763,19766],{"className":19756},[72],[25,19758],{"className":19759,"style":77},[76],[25,19761,5035],{"className":19762},[81],[25,19764],{"className":19765,"style":902},[441],[25,19767,19769,19772,19802],{"className":19768},[460],[25,19770,52],{"className":19771,"style":465},[87,464],[25,19773,19775,19778],{"className":19774},[81],[25,19776,55],{"className":19777},[81,82],[25,19779,19781],{"className":19780},[1010],[25,19782,19784],{"className":19783},[484],[25,19785,19787],{"className":19786},[489],[25,19788,19791],{"className":19789,"style":19790},[493],"height:0.6644em;",[25,19792,19793,19796],{"style":5476},[25,19794],{"className":19795,"style":1027},[501],[25,19797,19799],{"className":19798},[1031,1032,1033,1034],[25,19800,11176],{"className":19801},[81,82,1034],[25,19803,58],{"className":19804,"style":465},[94,464],[25,19806,19808,19826],{"className":19807,"translate":29},[28],[25,19809,19811],{"className":19810},[33],[35,19812,19813],{"xmlns":37},[39,19814,19815,19823],{},[42,19816,19817,19819,19821],{},[45,19818,11176],{},[49,19820,944],{},[145,19822,339],{},[60,19824,19825],{"encoding":62},"\\epsilon \\le1",[25,19827,19829,19847],{"className":19828,"ariaHidden":68},[67],[25,19830,19832,19835,19838,19841,19844],{"className":19831},[72],[25,19833],{"className":19834,"style":16452},[76],[25,19836,11176],{"className":19837},[81,82],[25,19839],{"className":19840,"style":442},[441],[25,19842,944],{"className":19843},[446],[25,19845],{"className":19846,"style":442},[441],[25,19848,19850,19853],{"className":19849},[72],[25,19851],{"className":19852,"style":1220},[76],[25,19854,339],{"className":19855},[81],". However, when applying the regular condition:",[10,19858,19859],{},[25,19860,19862,19980],{"className":19861,"translate":29},[28],[25,19863,19865],{"className":19864},[33],[35,19866,19867],{"xmlns":37},[39,19868,19869,19977],{},[308,19870,19871,19915],{"rowspacing":310,"columnalign":3580,"columnspacing":3581},[314,19872,19873,19893],{},[317,19874,19875],{},[320,19876,19877],{"scriptlevel":322,"displaystyle":68},[42,19878,19879,19881],{},[45,19880,9310],{},[42,19882,19883,19885,19887,19889,19891],{},[49,19884,52],{"fence":68},[45,19886,55],{},[45,19888,143],{"mathvariant":142},[145,19890,147],{},[49,19892,58],{"fence":68},[317,19894,19895],{},[320,19896,19897],{"scriptlevel":322,"displaystyle":68},[42,19898,19899,19901,19903,19905,19907],{},[42,19900],{},[49,19902,944],{},[45,19904,950],{},[45,19906,9310],{},[42,19908,19909,19911,19913],{},[49,19910,52],{"fence":68},[45,19912,55],{},[49,19914,58],{"fence":68},[314,19916,19917,19949],{},[317,19918,19919],{},[320,19920,19921],{"scriptlevel":322,"displaystyle":68},[42,19922,19923,19929],{},[2916,19924,19925,19927],{},[45,19926,55],{},[145,19928,147],{},[42,19930,19931,19933,19935,19937,19939,19941,19947],{},[49,19932,52],{"fence":68},[145,19934,147],{},[49,19936,2958],{},[45,19938,19223],{},[49,19940,845],{},[2916,19942,19943,19945],{},[45,19944,55],{},[145,19946,147],{},[49,19948,58],{"fence":68},[317,19950,19951],{},[320,19952,19953],{"scriptlevel":322,"displaystyle":68},[42,19954,19955,19957,19959,19961],{},[42,19956],{},[49,19958,944],{},[45,19960,950],{},[42,19962,19963,19965,19967,19969,19971,19973,19975],{},[49,19964,52],{"fence":68},[145,19966,147],{},[49,19968,2958],{},[45,19970,19223],{},[49,19972,845],{},[45,19974,55],{},[49,19976,58],{"fence":68},[60,19978,19979],{"encoding":62},"\\begin{aligned} f\\left(n \u002F 2\\right) &\\le c f\\left(n\\right) \\\\ \\frac{n}{2} \\left(2 - \\cos \\frac{n}{2} \\right) & \\le c \\left( 2 - \\cos n \\right) \\end{aligned}",[25,19981,19983],{"className":19982,"ariaHidden":68},[67],[25,19984,19986,19990],{"className":19985},[72],[25,19987],{"className":19988,"style":19989},[76],"height:3.636em;vertical-align:-1.568em;",[25,19991,19993],{"className":19992},[81],[25,19994,19996,20221],{"className":19995},[308],[25,19997,19999],{"className":19998},[480],[25,20000,20002,20212],{"className":20001},[484,485],[25,20003,20005,20209],{"className":20004},[489],[25,20006,20009,20040],{"className":20007,"style":20008},[493],"height:2.068em;",[25,20010,20012,20016],{"style":20011},"top:-4.378em;",[25,20013],{"className":20014,"style":20015},[501],"height:3.15em;",[25,20017,20019,20022,20025],{"className":20018},[81],[25,20020,9310],{"className":20021,"style":9468},[81,82],[25,20023],{"className":20024,"style":902},[441],[25,20026,20028,20031,20034,20037],{"className":20027},[460],[25,20029,52],{"className":20030,"style":465},[87,464],[25,20032,55],{"className":20033},[81,82],[25,20035,166],{"className":20036},[81],[25,20038,58],{"className":20039,"style":465},[94,464],[25,20041,20043,20046],{"style":20042},"top:-2.568em;",[25,20044],{"className":20045,"style":20015},[501],[25,20047,20049,20111,20114],{"className":20048},[81],[25,20050,20052,20055,20108],{"className":20051},[81],[25,20053],{"className":20054},[87,721],[25,20056,20058],{"className":20057},[2916],[25,20059,20061,20100],{"className":20060},[484,485],[25,20062,20064,20097],{"className":20063},[489],[25,20065,20067,20078,20086],{"className":20066,"style":3973},[493],[25,20068,20069,20072],{"style":3976},[25,20070],{"className":20071,"style":548},[501],[25,20073,20075],{"className":20074},[81],[25,20076,147],{"className":20077},[81],[25,20079,20080,20083],{"style":3082},[25,20081],{"className":20082,"style":548},[501],[25,20084],{"className":20085,"style":3090},[3089],[25,20087,20088,20091],{"style":3996},[25,20089],{"className":20090,"style":548},[501],[25,20092,20094],{"className":20093},[81],[25,20095,55],{"className":20096},[81,82],[25,20098,519],{"className":20099},[518],[25,20101,20103],{"className":20102},[489],[25,20104,20106],{"className":20105,"style":4015},[493],[25,20107],{},[25,20109],{"className":20110},[94,721],[25,20112],{"className":20113,"style":902},[441],[25,20115,20117,20123,20126,20129,20132,20135,20138,20141,20203],{"className":20116},[460],[25,20118,20120],{"className":20119,"style":465},[87,464],[25,20121,52],{"className":20122},[469,3954],[25,20124,147],{"className":20125},[81],[25,20127],{"className":20128,"style":600},[441],[25,20130,2958],{"className":20131},[604],[25,20133],{"className":20134,"style":600},[441],[25,20136,19223],{"className":20137},[906],[25,20139],{"className":20140,"style":902},[441],[25,20142,20144,20147,20200],{"className":20143},[81],[25,20145],{"className":20146},[87,721],[25,20148,20150],{"className":20149},[2916],[25,20151,20153,20192],{"className":20152},[484,485],[25,20154,20156,20189],{"className":20155},[489],[25,20157,20159,20170,20178],{"className":20158,"style":3973},[493],[25,20160,20161,20164],{"style":3976},[25,20162],{"className":20163,"style":548},[501],[25,20165,20167],{"className":20166},[81],[25,20168,147],{"className":20169},[81],[25,20171,20172,20175],{"style":3082},[25,20173],{"className":20174,"style":548},[501],[25,20176],{"className":20177,"style":3090},[3089],[25,20179,20180,20183],{"style":3996},[25,20181],{"className":20182,"style":548},[501],[25,20184,20186],{"className":20185},[81],[25,20187,55],{"className":20188},[81,82],[25,20190,519],{"className":20191},[518],[25,20193,20195],{"className":20194},[489],[25,20196,20198],{"className":20197,"style":4015},[493],[25,20199],{},[25,20201],{"className":20202},[94,721],[25,20204,20206],{"className":20205,"style":465},[94,464],[25,20207,58],{"className":20208},[469,3954],[25,20210,519],{"className":20211},[518],[25,20213,20215],{"className":20214},[489],[25,20216,20219],{"className":20217,"style":20218},[493],"height:1.568em;",[25,20220],{},[25,20222,20224],{"className":20223},[532],[25,20225,20227,20333],{"className":20226},[484,485],[25,20228,20230,20330],{"className":20229},[489],[25,20231,20233,20274],{"className":20232,"style":20008},[493],[25,20234,20235,20238],{"style":20011},[25,20236],{"className":20237,"style":20015},[501],[25,20239,20241,20244,20247,20250,20253,20256,20259,20262],{"className":20240},[81],[25,20242],{"className":20243},[81],[25,20245],{"className":20246,"style":442},[441],[25,20248,944],{"className":20249},[446],[25,20251],{"className":20252,"style":442},[441],[25,20254,950],{"className":20255},[81,82],[25,20257,9310],{"className":20258,"style":9468},[81,82],[25,20260],{"className":20261,"style":902},[441],[25,20263,20265,20268,20271],{"className":20264},[460],[25,20266,52],{"className":20267,"style":465},[87,464],[25,20269,55],{"className":20270},[81,82],[25,20272,58],{"className":20273,"style":465},[94,464],[25,20275,20276,20279],{"style":20042},[25,20277],{"className":20278,"style":20015},[501],[25,20280,20282,202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any ",[25,20345,20347,20365],{"className":20346,"translate":29},[28],[25,20348,20350],{"className":20349},[33],[35,20351,20352],{"xmlns":37},[39,20353,20354,20362],{},[42,20355,20356,20358,20360],{},[45,20357,950],{},[49,20359,4605],{},[145,20361,339],{},[60,20363,20364],{"encoding":62},"c \\lt 1",[25,20366,20368,20386],{"className":20367,"ariaHidden":68},[67],[25,20369,20371,20374,20377,20380,20383],{"className":20370},[72],[25,20372],{"className":20373,"style":1867},[76],[25,20375,950],{"className":20376},[81,82],[25,20378],{"className":20379,"style":442},[441],[25,20381,4605],{"className":20382},[446],[25,20384],{"className":20385,"style":442},[441],[25,20387,20389,20392],{"className":20388},[72],[25,20390],{"className":20391,"style":1220},[76],[25,20393,339],{"className":20394},[81],", the right side must be ",[25,20397,20399,20433],{"className":20398,"translate":29},[28],[25,20400,20402],{"className":20401},[33],[35,20403,20404],{"xmlns":37},[39,20405,20406,20430],{},[42,20407,20408,20410,20426,20428],{},[45,20409,950],{},[42,20411,20412,20414,20416,20418,20420,20422,20424],{},[49,20413,52],{"fence":68},[145,20415,147],{},[49,20417,2958],{},[45,20419,19223],{},[49,20421,845],{},[45,20423,55],{},[49,20425,58],{"fence":68},[49,20427,4605],{},[145,20429,5346],{},[60,20431,20432],{"encoding":62},"c \\left( 2 - \\cos n \\right) \\lt 3",[25,20434,20436,20487],{"className":20435,"ariaHidden":68},[67],[25,20437,20439,20442,20445,20448,20478,20481,20484],{"className":20438},[72],[25,20440],{"className":20441,"style":77},[76],[25,20443,950],{"className":20444},[81,82],[25,20446],{"className":20447,"style":902},[441],[25,20449,20451,20454,20457,20460,20463,20466,20469,20472,20475],{"className":20450},[460],[25,20452,52],{"className":20453,"style":465},[87,464],[25,20455,147],{"className":20456},[81],[25,20458],{"className":20459,"style":600},[441],[25,20461,2958],{"className":20462},[604],[25,20464],{"className":20465,"style":600},[441],[25,20467,19223],{"className":20468},[906],[25,20470],{"className":20471,"style":902},[441],[25,20473,55],{"className":20474},[81,82],[25,20476,58],{"className":20477,"style":465},[94,464],[25,20479],{"className":20480,"style":442},[441],[25,20482,4605],{"className":20483},[446],[25,20485],{"className":20486,"style":442},[441],[25,20488,20490,20493],{"className":20489},[72],[25,20491],{"className":20492,"style":1220},[76],[25,20494,5346],{"className":20495},[81],". However, the left side, when ",[25,20498,20500,20513],{"className":20499,"translate":29},[28],[25,20501,20503],{"className":20502},[33],[35,20504,20505],{"xmlns":37},[39,20506,20507,20511],{},[42,20508,20509],{},[45,20510,55],{},[60,20512,55],{"encoding":62},[25,20514,20516],{"className":20515,"ariaHidden":68},[67],[25,20517,20519,20522],{"className":20518},[72],[25,20520],{"className":20521,"style":121},[76],[25,20523,55],{"className":20524},[81,82]," is large enough, must be greater than ",[25,20527,20529,20542],{"className":20528,"translate":29},[28],[25,20530,20532],{"className":20531},[33],[35,20533,20534],{"xmlns":37},[39,20535,20536,20540],{},[42,20537,20538],{},[145,20539,5346],{},[60,20541,5346],{"encoding":62},[25,20543,20545],{"className":20544,"ariaHidden":68},[67],[25,20546,20548,20551],{"className":20547},[72],[25,20549],{"className":20550,"style":1220},[76],[25,20552,5346],{"className":20553},[81],". Therefore, this equation cannot hold, and the regular condition is not satisfied.",[14,20556,20558],{"id":20557},"reference","Reference",[10,20560,20561,20562,270],{},"These cases are from the Wikipedia entry on ",[9290,20563,20566],{"href":20564,"rel":20565},"https:\u002F\u002Fen.wikipedia.org\u002Fw\u002Findex.php?title=Master_theorem_(analysis_of_algorithms)&section=7#Inadmissible_equations",[18959],"Master theorem",{"title":9715,"searchDepth":20568,"depth":20568,"links":20569},2,[20570,20571,20572,20573,20574,20575],{"id":16,"depth":20568,"text":17},{"id":803,"depth":20568,"text":804},{"id":5317,"depth":20568,"text":5318},{"id":9245,"depth":20568,"text":9246},{"id":18280,"depth":20568,"text":18281},{"id":20557,"depth":20568,"text":20558},false,true,"md",null,{},"2023-12-18","\u002Fblog\u002F2023\u002F2023-12-18-the-master-theroy-for-time-complexity",{"title":5,"description":12},"blog\u002F2023\u002F2023-12-18-The-Master-Theroy-For-Time-Complexity","In this article, we will model the time complexity of divide and conquer using mathematical methods, analyze its asymptotic properties, and provide three methods of calculation.",[20587,20588,20589],"algorithm","mathematics","computation","E_mUE2BC_bi1SeO-VjPgymxDon8IifXO_I8JuDSrzFs",1777090295691]