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b1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, 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