如何证明均匀分区是快速排序算法的最佳案例? [英] How to prove the evenly partition is the best case for quick sort algorithm?
问题描述
为什么我们说快速排序的最佳情况是每次执行分区时,我们会将列表分为几乎相等的两个部分"?以及如何证明这正是所谓的最佳案例"?
Why we say the bestcase for quicksort is that "each time we perform a partition we divide the list into two nearly equal pieces"? And how to prove this is exactly the so called "best case"?
推荐答案
我创建了一个程序,而不是尝试进行分析.我比较了1/2、1/2(50%50%)分割与1/4、3/4(25%75%)分割的情况,随着n变大,似乎要多花22%的运算.代码设置为1/4,3/4分割:对于1/2,1/2分割,将行从左=(n + 3)/4更改为左=(n + 1)/2.向上舍入的目的是确保left> = 1,以避免无限递归.
I created a program rather than trying to do an analysis. I compared the case of 1/2, 1/2 (50% 50%) split versus 1/4, 3/4 (25% 75%) split, which appears to approach taking 22% more operations as n becomes large. The code is set for the 1/4,3/4 split: for 1/2,1/2 split, change the line from left = (n+3)/4 to left = (n+1)/2. The point of rounding left up is to make sure left >= 1, to avoid infinite recursion.
#include <stdio.h>
typedef unsigned long long uint64_t;
static uint64_t sum;
void qsa(uint64_t n)
{
uint64_t left, right;
if(n < 2)
return;
sum += n;
left = (n+3)/4; /* or left = (n+1)/2 */
right = n - left;
qsa(left);
qsa(right);
}
int main()
{
qsa(1024*1024);
printf("%llu\n", sum);
return(0);
}
结果
n = 1024*1024
20971520 1/2 1/2 n log2(n)
25331387 1/4 3/4 ~1.208 n log2(n)
n = 16*1024*1024
402653184 1/2 1/2 n log2(n)
488049677 1/4 3/4 ~1.212 n log2(n)
n = 1024*1024*1024
32212254720 1/2 1/2 n log2(n)
39180282211 1/4 3/4 ~1.216 n log2(n)
n = 16*1024*1024*1024
584115552256 1/2 1/2 n log2(n)
711608157825 1/4 3/4 ~1.218 n log2(n)
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