Master公式秒算递归时间复杂度：原理与实战

从以往情况来看, 递归算法的时间复杂度分析总是算法学习里一块难啃 的 需要花费 大量精力与智慧去攻克的硬骨头 , 好多开发者在遇到递归函数时, 只能凭借猜测的模糊方式或者手动展开的有点繁琐的办法去大概进行性能估算 , 这样做效率很低而且特别容易出现错误 , Master公式恰恰是为了处理解决这一让人头疼的痛点而产生形成的 , 它给出提供了一套有条理的系统方法 , 能够以快速且准确的方式计算出大部分多数递归算法的时间复杂度。

什么是Master公式以及它如何解决你的时间复杂度难题

主定理, 也就是Master公式, 是专门用来分析一种递归式的, 这种递归式形如T(n) = aT(n/b) + f(n)。其中, a代表子问题数量, , b表示子问题规模缩小的倍数, 关于这样规模怎样缩小怎样缩小的倍数, f(n)是分解以及合并子问题所耗费的时间。你能够把它看成一个“三段式”判定规则: 对n^log_b(a)和f(n)开展比较。要是 f(n) 的增长更为缓慢, 那么时间复杂度便由 n^log_b(a) 来确定；要是两者的增长情况相当, 那就乘以 log n；要是 f(n) 的增长更快, 那就直接选取 f(n)。此规则听起来颇为抽象, 然而一旦你领会了它的判断逻辑, 好多递归函数都能够在十秒内处理妥当。

拿个例子来讲, 二分查找的递归实现能够写成 T(n) = T(n/2) + O(1)。在此处 a 等于 1, b 是 2, n 的以 b 为底 a 的对数次方也就是 n^0 等于 1, 而 f(n)=O(1), 这两者属于同阶, 因而时间复杂度是 O(log n)。又好比归并排序 T(n) 等于 2T(n/2) + O(n), a 为 2, b 为 2, n 的以 2 为底 2 的对数次方等于 n, 跟 f(n)=O(n) 同阶, 所以时间复杂度是 O(n log n)。你看，不需要层层展开递归树，一个公式就能得出结果。

用Master公式分析常见递归算法的具体步骤和陷阱

在实际进行操作的这个时候, 重点在于要把递归算法相关的代码转变为标准的Master公式形式。你得先去明确好递归函数的结构, 它自身调用了多少次哪? 每次调用在处理数据的时候规模缩小了多少? 除开递归调用之外, 还有多少另外的操作? 这三个变量分别是跟a、b以及f(n)相对应的。就好像在快速排序情况里, 要是每次所选取的基准值都能够把数组大致上进行平分这件事能做到, 那么T(n) = 2T(n/2) + O(n), 这和归并排序是一样的情况, 复杂度同样也便是 O(n log n)。然而, 在实际情形当中的快速排序, 有可能会碰到非常极端的状况, 举例来说, 每一次仅仅划分出来一个元素, 在这个时候, 递归式就转变成了T(n)=T(n - 1)+O(n), 这种情况并不满足Master公式当中的b>1这个条件, 进而就需要运用其他的方法来加以分析。

存在着一个常见的陷阱, 当你碰到 T(n) = T(n/2) + O(n) 这般的递归式时, a 等于 1, b 等于 2, n 的以 b 为底 a 的对数次方等于 1, 而 f(n) 是 O(n), 显然 f(n) 增长得更为迅速。依据第三条规则, 时间复杂度便是 O(n)。这是很容易理解的, 因为仅有一个子问题, 每次仅仅处理一半的数据, 然而合并步骤却是线性的, 整体呈现的就是线性时间。要是你手动去展开这个递归, 就会发觉总时间是一个等比数列求和, 其结果的确是 O(n)。

当你掌握了Master公式之后, 你便能够迅速地判断出一个递归算法是不是高效的。举个例子来说, 如若看到T(n) = 4T(n/2) + O(n^2) , 那么计算n^log_2(4)=n^2 , 其与f(n) 是同阶的, 答案是O(n^2 log n)。然而要是看到T(n) = 4T(n/2) + O(n) , 那么n^2 比n 增长得快, 复杂度便是O(n^2)。一旦这种直觉得以建立, 那么再去评估算法的好坏就会变得清晰许多了。意见是, 为了你在平常编码期间, 于繁复的这个转化进程里, 增添更多的练习机会, 在着手撰写递归函数之前, 预先运用Master公式来对复杂度予以估算, 从而防止在性能瓶颈处, 出现遭遇较大挫折的情况。

https://www.hongxiu.com/booklist/1505260
https://www.hongxiu.com/booklist/1505259
https://www.hongxiu.com/booklist/1505258
https://www.hongxiu.com/booklist/1505256
https://www.hongxiu.com/booklist/1505257
https://www.hongxiu.com/booklist/1505255
https://www.hongxiu.com/booklist/1505254
https://www.hongxiu.com/booklist/1505253
https://www.hongxiu.com/booklist/1505252
https://www.hongxiu.com/booklist/1505251
https://www.hongxiu.com/booklist/1505250
https://www.hongxiu.com/booklist/1505248
https://www.hongxiu.com/booklist/1505249
https://www.hongxiu.com/booklist/1505246
https://www.hongxiu.com/booklist/1505247
https://www.hongxiu.com/booklist/1505245
https://www.hongxiu.com/booklist/1505244
https://www.hongxiu.com/booklist/1505243
https://www.hongxiu.com/booklist/1505242
https://www.hongxiu.com/booklist/1505241
https://www.hongxiu.com/booklist/1505240
https://www.hongxiu.com/booklist/1505239
https://www.hongxiu.com/booklist/1505238
https://www.hongxiu.com/booklist/1505237
https://www.hongxiu.com/booklist/1505236
https://www.hongxiu.com/booklist/1505235
https://www.hongxiu.com/booklist/1505234
https://www.hongxiu.com/booklist/1505233
https://www.hongxiu.com/booklist/1505232
https://www.hongxiu.com/booklist/1505231
https://www.hongxiu.com/booklist/1505230
https://www.hongxiu.com/booklist/1505229
https://www.hongxiu.com/booklist/1505228
https://www.hongxiu.com/booklist/1505227
https://www.hongxiu.com/booklist/1505226
https://www.hongxiu.com/booklist/1505225
https://www.hongxiu.com/booklist/1505224
https://www.hongxiu.com/booklist/1505223
https://www.hongxiu.com/booklist/1505222
https://www.hongxiu.com/booklist/1505221
https://www.hongxiu.com/booklist/1505220
https://www.hongxiu.com/booklist/1505219
https://www.hongxiu.com/booklist/1505218
https://www.hongxiu.com/booklist/1505217
https://www.hongxiu.com/booklist/1505216
https://www.hongxiu.com/booklist/1505215
https://www.hongxiu.com/booklist/1505214
https://www.hongxiu.com/booklist/1505213
https://www.hongxiu.com/booklist/1505212
https://www.hongxiu.com/booklist/1505211
https://www.hongxiu.com/booklist/1505210
https://www.hongxiu.com/booklist/1505209
https://www.hongxiu.com/booklist/1505208
https://www.hongxiu.com/booklist/1505207
https://www.hongxiu.com/booklist/1505206
https://www.hongxiu.com/booklist/1505205
https://www.hongxiu.com/booklist/1505204
https://www.hongxiu.com/booklist/1505203
https://www.hongxiu.com/booklist/1505202
https://www.hongxiu.com/booklist/1505201
https://www.hongxiu.com/booklist/1505200
https://www.hongxiu.com/booklist/1505199
https://www.hongxiu.com/booklist/1505198
https://www.hongxiu.com/booklist/1505197
https://www.hongxiu.com/booklist/1505196
https://www.hongxiu.com/booklist/1505195
https://www.hongxiu.com/booklist/1505194
https://www.hongxiu.com/booklist/1505193
https://www.hongxiu.com/booklist/1505192
https://www.hongxiu.com/booklist/1505191
https://www.hongxiu.com/booklist/1505190
https://www.hongxiu.com/booklist/1505189
https://www.hongxiu.com/booklist/1505188
https://www.hongxiu.com/booklist/1505187
https://www.hongxiu.com/booklist/1505186
https://www.hongxiu.com/booklist/1505185
https://www.hongxiu.com/booklist/1505184
https://www.hongxiu.com/booklist/1505183
https://www.hongxiu.com/booklist/1505182
https://www.hongxiu.com/booklist/1505181
https://www.hongxiu.com/booklist/1505180
https://www.hongxiu.com/booklist/1505179
https://www.hongxiu.com/booklist/1505178
https://www.hongxiu.com/booklist/1505177
https://www.hongxiu.com/booklist/1505176
https://www.hongxiu.com/booklist/1505175
https://www.hongxiu.com/booklist/1505174
https://www.hongxiu.com/booklist/1505173
https://www.hongxiu.com/booklist/1505172
https://www.hongxiu.com/booklist/1505171
https://www.hongxiu.com/booklist/1505170
https://www.hongxiu.com/booklist/1505169
https://www.hongxiu.com/booklist/1505168
https://www.hongxiu.com/booklist/1505167
https://www.hongxiu.com/booklist/1505166
https://www.hongxiu.com/booklist/1505165
https://www.hongxiu.com/booklist/1505164
https://www.hongxiu.com/booklist/1505163
https://www.hongxiu.com/booklist/1505162
https://www.hongxiu.com/booklist/1505161
https://www.hongxiu.com/booklist/1505160
https://www.hongxiu.com/booklist/1505159
https://www.hongxiu.com/booklist/1505158
https://www.hongxiu.com/booklist/1505157
https://www.hongxiu.com/booklist/1505156
https://www.hongxiu.com/booklist/1505155
https://www.hongxiu.com/booklist/1505154
https://www.hongxiu.com/booklist/1505153
https://www.hongxiu.com/booklist/1505152
https://www.hongxiu.com/booklist/1505151
https://www.hongxiu.com/booklist/1505150
https://www.hongxiu.com/booklist/1505149
https://www.hongxiu.com/booklist/1505148
https://www.hongxiu.com/booklist/1505147
https://www.hongxiu.com/booklist/1505146
https://www.hongxiu.com/booklist/1505145
https://www.hongxiu.com/booklist/1505144
https://www.hongxiu.com/booklist/1505143
https://www.hongxiu.com/booklist/1505142
https://www.hongxiu.com/booklist/1505141
https://www.hongxiu.com/booklist/1505140
https://www.hongxiu.com/booklist/1505139
https://www.hongxiu.com/booklist/1505138
https://www.hongxiu.com/booklist/1505137
https://www.hongxiu.com/booklist/1505136
https://www.hongxiu.com/booklist/1505135
https://www.hongxiu.com/booklist/1505134
https://www.hongxiu.com/booklist/1505133
https://www.hongxiu.com/booklist/1505132
https://www.hongxiu.com/booklist/1505131
https://www.hongxiu.com/booklist/1505130
https://www.hongxiu.com/booklist/1505129
https://www.hongxiu.com/booklist/1505128
https://www.hongxiu.com/booklist/1505127
https://www.hongxiu.com/booklist/1505126
https://www.hongxiu.com/booklist/1505125
https://www.hongxiu.com/booklist/1505124
https://www.hongxiu.com/booklist/1505123
https://www.hongxiu.com/booklist/1505122
https://www.hongxiu.com/booklist/1505121
https://www.hongxiu.com/booklist/1505120
https://www.hongxiu.com/booklist/1505119

https://www.hongxiu.com/booklist/1505118
https://www.hongxiu.com/booklist/1505117
https://www.hongxiu.com/booklist/1505116
https://www.hongxiu.com/booklist/1505115
https://www.hongxiu.com/booklist/1505114
https://www.hongxiu.com/booklist/1505113
https://www.hongxiu.com/booklist/1505112
https://www.hongxiu.com/booklist/1505111
https://www.hongxiu.com/booklist/1505110
https://www.hongxiu.com/booklist/1505109
https://www.hongxiu.com/booklist/1505108
https://www.hongxiu.com/booklist/1505107
https://www.hongxiu.com/booklist/1505106
https://www.hongxiu.com/booklist/1505105
https://www.hongxiu.com/booklist/1505104
https://www.hongxiu.com/booklist/1505103
https://www.hongxiu.com/booklist/1505102
https://www.hongxiu.com/booklist/1505101
https://www.hongxiu.com/booklist/1505100
https://www.hongxiu.com/booklist/1505099
https://www.hongxiu.com/booklist/1505098
https://www.hongxiu.com/booklist/1505097
https://www.hongxiu.com/booklist/1505096
https://www.hongxiu.com/booklist/1505095
https://www.hongxiu.com/booklist/1505094
https://www.hongxiu.com/booklist/1505093
https://www.hongxiu.com/booklist/1505092
https://www.hongxiu.com/booklist/1505091
https://www.hongxiu.com/booklist/1505090
https://www.hongxiu.com/booklist/1505089
https://www.hongxiu.com/booklist/1505088
https://www.hongxiu.com/booklist/1505087
https://www.hongxiu.com/booklist/1505086
https://www.hongxiu.com/booklist/1505085
https://www.hongxiu.com/booklist/1505084
https://www.hongxiu.com/booklist/1505083
https://www.hongxiu.com/booklist/1505082
https://www.hongxiu.com/booklist/1505081
https://www.hongxiu.com/booklist/1505080
https://www.hongxiu.com/booklist/1505079
https://www.hongxiu.com/booklist/1505078
https://www.hongxiu.com/booklist/1505077
https://www.hongxiu.com/booklist/1505076
https://www.hongxiu.com/booklist/1505075
https://www.hongxiu.com/booklist/1505074
https://www.hongxiu.com/booklist/1505073
https://www.hongxiu.com/booklist/1505072
https://www.hongxiu.com/booklist/1505071
https://www.hongxiu.com/booklist/1505070
https://www.hongxiu.com/booklist/1505069
https://www.hongxiu.com/booklist/1505068
https://www.hongxiu.com/booklist/1505067
https://www.hongxiu.com/booklist/1505066
https://www.hongxiu.com/booklist/1505065
https://www.hongxiu.com/booklist/1505064
https://www.hongxiu.com/booklist/1505063
https://www.hongxiu.com/booklist/1505062
https://www.hongxiu.com/booklist/1505061
https://www.hongxiu.com/booklist/1505060
https://www.hongxiu.com/booklist/1505059
https://www.hongxiu.com/booklist/1505058
https://www.hongxiu.com/booklist/1505057
https://www.hongxiu.com/booklist/1505056
https://www.hongxiu.com/booklist/1505055
https://www.hongxiu.com/booklist/1505054
https://www.hongxiu.com/booklist/1505053
https://www.hongxiu.com/booklist/1505052