HDU5492 Find a path[DP 方差]

Posted 2020-08-12 Candy?

Find a path

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)
Total Submission(s): 1536    Accepted Submission(s): 673

Problem Description

Frog fell into a maze. This maze is a rectangle containing N rows and M columns. Each grid in this maze contains a number, which is called the magic value. Frog now stays at grid (1, 1), and he wants to go to grid (N, M). For each step, he can go to either the grid right to his current location or the grid below his location. Formally, he can move from grid (x, y) to (x + 1, y) or (x, y +1), if the grid he wants to go exists.
Frog is a perfectionist, so he‘d like to find the most beautiful path. He defines the beauty of a path in the following way. Let’s denote the magic values along a path from (1, 1) to (n, m) as A1,A2,…AN+M−1, and Aavg is the average value of all Ai. The beauty of the path is (N+M–1) multiplies the variance of the values:(N+M−1)∑N+M−1i=1(Ai−Aavg)2
In Frog‘s opinion, the smaller, the better. A path with smaller beauty value is more beautiful. He asks you to help him find the most beautiful path.  

 

 

Input

The first line of input contains a number T indicating the number of test cases (T≤50).
Each test case starts with a line containing two integers N and M (1≤N,M≤30). Each of the next N lines contains M non-negative integers, indicating the magic values. The magic values are no greater than 30.

 

 

Output

For each test case, output a single line consisting of “Case #X: Y”. X is the test case number starting from 1. Y is the minimum beauty value.

 

 

Sample Input

1 2 2 1 2 3 4

 

 

Sample Output

Case #1: 14

 

 

Source

2015 ACM/ICPC Asia Regional Hefei Online

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题意：(1,1)到(n,m)向右向下的最小方差路径

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把方差的式子化简推理，得到

(n+m-1)*Σai^2-(Σai)^2   i belong path

注意两者平方位置演算时写清了

f[i][j][k]表示到(i,j)和为k的最小平方和，递推就行了

//
//  main.cpp
//  hdu5492
//
//  Created by Candy on 10/2/16.
//  Copyright © 2016 Candy. All rights reserved.
//

#include <iostream>
#include <cstdio>
#include <algorithm>
#include <cstring>
#include <set>
using namespace std;
const int N=35,INF=1e9;
inline int read(){
    char c=getchar();int x=0,f=1;
    while(c<‘0‘||c>‘9‘){if(c==‘-‘)f=-1;c=getchar();}
    while(c>=‘0‘&&c<=‘9‘){x=x*10+c-‘0‘;c=getchar();}
    return x;
}
int T,n,m,l,a[N][N],f[N][N][N*2*30];
int dp(){
    memset(f,127,sizeof(f));
    f[1][1][a[1][1]]=a[1][1]*a[1][1];
    for(int i=1;i<=n;i++)
        for(int j=1;j<=m;j++)
            for(int k=0;k<=l*30;k++)
                if(f[i][j][k]<=INF){
                    int x=i+1,y=j,w=a[x][y];
                    f[x][y][k+w]=min(f[x][y][k+w],f[i][j][k]+w*w);
                    x=i;y=j+1;w=a[x][y];
                    f[x][y][k+w]=min(f[x][y][k+w],f[i][j][k]+w*w);
                }
    int ans=INF;
    for(int k=0;k<=l*30;k++)
        if(f[n][m][k]<INF)
            ans=min(ans,l*f[n][m][k]-k*k);
    return ans;
}
int main(int argc, const char * argv[]) {
    T=read();int cas=0;
    while(T--){
        n=read();m=read();l=n+m-1;
        for(int i=1;i<=n;i++)
            for(int j=1;j<=m;j++) a[i][j]=read();
        printf("Case #%d: %d\n",++cas,dp());
    }

    return 0;
}