HDU1394 Minimum Inversion Number（线段树OR归并排序）

Minimum Inversion Number

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 15113    Accepted Submission(s): 9230

Problem Description

The inversion number of a given number sequence a1, a2, ..., an is the number of pairs (ai, aj) that satisfy i < j and ai > aj.

For a given sequence of numbers a1, a2, ..., an, if we move the first m >= 0 numbers to the end of the seqence, we will obtain another sequence. There are totally n such sequences as the following:

a1, a2, ..., an-1, an (where m = 0 - the initial seqence)
a2, a3, ..., an, a1 (where m = 1)
a3, a4, ..., an, a1, a2 (where m = 2)
...
an, a1, a2, ..., an-1 (where m = n-1)

You are asked to write a program to find the minimum inversion number out of the above sequences.

 

Input

The input consists of a number of test cases. Each case consists of two lines: the first line contains a positive integer n (n <= 5000); the next line contains a permutation of the n integers from 0 to n-1.

 

Output

For each case, output the minimum inversion number on a single line.

 

Sample Input

10 1 3 6 9 0 8 5 7 4 2

 

Sample Output

16

题意：求不同数列中最小的逆序对数。

思路：

   方法一：转：http://blog.csdn.net/wiking__acm/article/details/7920429

      统计a[i]前面的，且比它大的数

      这样做的话，就可以利用输入的时效性，每输入一个数，就把这个数的num[i]值为1，

   然后统计比这个数大的数的num和,

   因为这里的和一定是在这个数列中比a[i]大，且在它前面出现的数之和，

   然后把把这个和加到总逆序数sum里。

   这样做的话直接的暴力作法依然是n2，但是，

   我们可以在，统计比这个数大的数的num和这一步进行优化，利用线段树求区间域值的复杂度是logn，

   所以总体复杂度就降到了nlogn。

   再来看这道题，求得初始数列的逆序数后，再求其他排列的逆序数有一个规律，就是

   sum = sum + (n - 1 - a[i]) - a[i];比如原来的逆序数是sum,把a[0]移到最后后，减少逆序数a[0]，同时增加逆序数n-a[0]-1个。

#include <cstdio>
#include <iostream>
#include <cstdlib>
#include <algorithm>
#include <ctime>
#include <cmath>
#include <string>
#include <cstring>
#include <stack>
#include <queue>
#include <list>
#include <vector>
#include <map>
#include <set>
using namespace std;

const int INF=0x3f3f3f3f;
const double eps=1e-10;
const double PI=acos(-1.0);
#define maxn 5500
int tre[4*maxn];

void update(int num, int le, int ri, int x)
{
    if(le == ri)
    {
        tre[num] = 1;
        return;
    }
    int mid = (le+ri)/2;
    if(x<=mid)
        update(num*2,le,mid,x);
    else
        update(num*2+1,mid+1,ri,x);
    tre[num] = tre[num*2] + tre[num*2+1];
}
int query(int num, int le, int ri, int x, int y)
{
    if(x<=le&&y>=ri)
    {
        return tre[num];
    }
    int mid = (le+ri)/2;
    int ans = 0;
    if(x<=mid)
        ans += query(num*2,le,mid,x,y);
    if(y>mid)
        ans += query(num*2+1,mid+1,ri,x,y);
    return ans;
}
int a[maxn];
int main()
{
    int n;
    while(~scanf("%d", &n))
    {
        memset(tre, 0, sizeof tre);
        int sum = 0;

        for(int i = 0; i < n; i++)
        {
            scanf("%d", &a[i]);
            update(1,0,n-1,a[i]);

            sum += query(1,0,n-1,a[i]+1,n-1);
        }
      
        int ans = sum;
        for(int i = 0; i < n; i++)
        {
            sum = sum - a[i] + (n-1-a[i]);
            ans = min(ans, sum);
        }
        printf("%d
", ans);
    }
    return 0;
}

方法二：归并排序求逆序对。

 

#include <cstdio>
#include <iostream>
#include <cstdlib>
#include <algorithm>
#include <ctime>
#include <cmath>
#include <string>
#include <cstring>
#include <stack>
#include <queue>
#include <list>
#include <vector>
#include <map>
#include <set>
using namespace std;

const int INF=0x3f3f3f3f;
const double eps=1e-10;
const double PI=acos(-1.0);
#define maxn 5500

int cnt;
int t[maxn],a[maxn],b[maxn];
void merge_sort(int x,int y)//归并排序模板
{
    if(y-x>1)
    {
        int m=x+(y-x)/2;
        int p=x,q=m,i=x;
        merge_sort(x,m);
        merge_sort(m,y);
        while(p<m||q<y)
        {
            if(q>=y||(p<m&&a[p]<=a[q]))
                t[i++]=a[p++];
            else
                {
                    t[i++]=a[q++];
                    cnt+=m-p;
                }
        }
        for(i=x;i<y;i++)
            a[i]=t[i];
    }
}
int main()
{
    int n;
    while(~scanf("%d", &n))
    {
        for(int i = 0; i < n; i++)
        {
            scanf("%d", &a[i]);
            b[i] = a[i];
        }
        cnt = 0;
        merge_sort(0,n);
        int ans = cnt;
        for(int i = 0; i < n; i++)
        {
            cnt = cnt - b[i] + (n-1-b[i]);
            ans = min(ans, cnt);
        }
        printf("%d
", ans);
    }
    return 0;
}