Median

我们知道排序的开销是nlogn, 

证明如下：n 个节点的permutation是n!

binary tree的高度至少是 logn!

logn! >nlogn  [这儿实际上是log_{2^{n   ]}}  

---

现在我们要找到n个元素中的median,通过排序肯定是可以的，缺点是O(nlogn).

而我们的理想算法是接近线性的。应该是可以的，因为排序明显是做的有点过了。

我只需要找到median，我可不管你最后是不是排序的。

randomized divide-and-conquer algorithm for selection

for any number u, image splitting the list U into three categories:

• elements smaller than u
• those equal to u
• elements greater than u

the search can instantly be narrowed down to one of these sublists.

                           | -        selection(SL,k)                           if (k < |SL|)

selection(S,k) =  |          v

                           | -        selection(SR,k-|SL|-|v|)

 

sum up:

Quicksort is a sorting algorithm that splits the array in exactly the same way as the

median algorithm. Its worst-case performance is O(n²), like that of median finding.

But it can be proved that its average case is O(nlogn).

Furthermore, empirically it outperforms other sorting algorithms.

This has made quicksort a favorite in many apps.