Painting The Wall 期望DP Codeforces 398_B

B. Painting The Wall

time limit per test

1 second

memory limit per test

256 megabytes

input

standard input

output

standard output

User ainta decided to paint a wall. The wall consists of n2 tiles, that are arranged in an n × n table. Some tiles are painted, and the others are not. As he wants to paint it beautifully, he will follow the rules below.

1. Firstly user ainta looks at the wall. If there is at least one painted cell on each row and at least one painted cell on each column, he stops coloring. Otherwise, he goes to step 2.
2. User ainta choose any tile on the wall with uniform probability.
3. If the tile he has chosen is not painted, he paints the tile. Otherwise, he ignores it.
4. Then he takes a rest for one minute even if he doesn't paint the tile. And then ainta goes to step 1.

 

However ainta is worried if it would take too much time to finish this work. So he wants to calculate the expected time needed to paint the wall by the method above. Help him find the expected time. You can assume that choosing and painting any tile consumes no time at all.

Input

The first line contains two integers n and m (1 ≤ n ≤ 2·103; 0 ≤ m ≤ min(n2, 2·104)) — the size of the wall and the number of painted cells.

Next m lines goes, each contains two integers ri and ci (1 ≤ ri, ci ≤ n) — the position of the painted cell. It is guaranteed that the positions are all distinct. Consider the rows of the table are numbered from 1 to n. Consider the columns of the table are numbered from1 to n.

Output

In a single line print the expected time to paint the wall in minutes. Your answer will be considered correct if it has at most 10 - 4 absolute or relative error.

Sample test(s)

input

5 2
2 3
4 1

output

11.7669491886

input

2 2
1 1
1 2

output

2.0000000000

input

1 1
1 1

output

0.0000000000

基本上是第一次做期望的题。。。

令f[i][j]表示当前图中有i行j列没有涂过色，然后用标准方法乱搞就行了。

#include<iostream>
#include<cstdio>
#include<algorithm>
#include<cstring>
#include<iomanip>
using namespace std;
#define MAXN 2100

int ptc[MAXN],ptr[MAXN];
typedef long double real;
real dp[MAXN][MAXN];
int main()
{
       int i,j,x,y,z;
        real n;int m;
        cin>>n>>m;
        for (i=0;i<m;i++)
        {
                scanf("%d%d",&x,&y);
                ptc[x]=ptr[y]=1;
        }
        int tr,tc;
        tr=tc=0;
        for (i=1;i<=n;i++)
        {
                if (!ptc[i])tc++;
                if (!ptr[i])tr++;
        }
        dp[0][0]=0;
        real tot_p;
        for (i=0;i<=tr;i++)
        {
                dp[i+1][0]=dp[i][0]+1/(1-(n-i-1)/n);
        }
        for (i=0;i<=tc;i++)
        {
                dp[0][i+1]=dp[0][i]+1/(1-(n-i-1)/n);
        }
        for (i=1;i<=tr;i++)
        {
                for (j=1;j<=tc;j++)
                {
                     tot_p=(n-i)*j+(n-j)*i+i*j;
                        dp[i][j]=dp[i-1][j-1]*(i*j)/tot_p+dp[i][j-1]*(n-i)*j/tot_p+
                                dp[i-1][j]*(n-j)*i/tot_p+1/(1-(n-i)*(n-j)/(n*n));
                }
        }
        double ans=dp[tr][tc];
        printf("%.10f
",ans);
        return 0;
}