poj 3579 Median （二分搜索之查找第k大的值）

最新推荐文章于 2021-02-11 12:22:46 发布

转载最新推荐文章于 2021-02-11 12:22:46 发布 · 83 阅读

本文介绍了一种高效算法，通过二分搜索和排序解决数对差值中位数的问题。针对大量数据输入，该算法避免了暴力枚举的超时风险，适用于竞赛及实际场景。

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Description

Given N numbers, X1, X2, ... , XN, let us calculate the difference of every pair of numbers: ∣Xi - Xj∣ (1 ≤ i ＜ j ≤ N). We can get C(N,2) differences through this work, and now your task is to find the median of the differences as quickly as you can!

Note in this problem, the median is defined as the (m/2)-th  smallest number if m,the amount of the differences, is even. For example, you have to find the third smallest one in the case of m = 6.

Input

The input consists of several test cases.
In each test case, N will be given in the first line. Then N numbers are given, representing X1, X2, ... , XN, ( Xi ≤ 1,000,000,000  3 ≤ N ≤ 1,00,000 )

Output

For each test case, output the median in a separate line.

Sample Input

Sample Output

1
8

Source

POJ Founder Monthly Contest – 2008.04.13, Lei Tao

这道题直接暴力枚举肯定是超时。

可以采用二分枚举一个数mid。对a数组排序后，与a_i的差大于mid（也就是某个数大于X_i + mid）的那些数的个数如果小于N / 2的话，说明mid太大了。以此为条件进行第一重二分搜索，第二重二分搜索是对a的搜索，直接用lower_bound实现。

 1 #include<iostream>
 2 #include<cstdio>
 3 #include<cstring>
 4 #include<algorithm>
 5 #include<math.h>
 6 #include<stdlib.h>
 7 using namespace std;
 8 #define N 100006
 9 int n,m;
10 int a[N];
11 bool solve(int mid){
12     int cnt=0;
13     for(int i=0;i<n;i++){
14         int tmp=n-(lower_bound(a,a+n,a[i]+mid)-a);//a[i]加上mid看看有多少个
15         cnt+=tmp;
16     }
17     if(cnt>m) return true;//如果太多，则要调整mid向上，>还是>=有时候要调整
18     return false;
19 }
20 int main()
21 {
22     while(scanf("%d",&n)==1){
23         m=n*(n-1)/4;
24         for(int i=0;i<n;i++){
25             scanf("%d",&a[i]);
26         }
27         sort(a,a+n);
28         int low=0;
29         int high=a[n-1]-a[0];
30         while(low<high){
31             int mid=(low+high)>>1;
32             if(solve(mid)){
33                 low=mid+1;
34             }
35             else{
36                 high=mid;
37             }
38         }
39         printf("%d\n",low-1);//有时要调整，减1或不减
40     }
41     return 0;
42 }