POJ 3071 Football（概率DP）

Consider a single-elimination football tournament involving 2ⁿ teams, denoted 1, 2, …, 2ⁿ. In each round of the tournament, all teams still in the tournament are placed in a list in order of increasing index. Then, the first team in the list plays the second team, the third team plays the fourth team, etc. The winners of these matches advance to the next round, and the losers are eliminated. After n rounds, only one team remains undefeated; this team is declared the winner.

Given a matrix P = [p_ij] such that p_ij is the probability that team i will beat team j in a match determine which team is most likely to win the tournament.

Input

The input test file will contain multiple test cases. Each test case will begin with a single line containing n (1 ≤ n ≤ 7). The next 2ⁿ lines each contain 2ⁿ values; here, the jth value on the ith line represents p_ij. The matrix P will satisfy the constraints that p_ij = 1.0 − p_ji for all i ≠ j, and p_ii = 0.0 for all i. The end-of-file is denoted by a single line containing the number −1. Note that each of the matrix entries in this problem is given as a floating-point value. To avoid precision problems, make sure that you use either the double data type instead of float.

Output

The output file should contain a single line for each test case indicating the number of the team most likely to win. To prevent floating-point precision issues, it is guaranteed that the difference in win probability for the top two teams will be at least 0.01.

Sample Input

2
0.0 0.1 0.2 0.3
0.9 0.0 0.4 0.5
0.8 0.6 0.0 0.6
0.7 0.5 0.4 0.0
-1

Sample Output

2

Hint

In the test case above, teams 1 and 2 and teams 3 and 4 play against each other in the first round; the winners of each match then play to determine the winner of the tournament. The probability that team 2 wins the tournament in this case is

P(2 wins)

= P(2 beats 1)P(3 beats 4)P(2 beats 3) + P(2 beats 1)P(4 beats 3)P(2 beats 4) = p21p34p23 + p21p43p24 = 0.9 · 0.6 · 0.4 + 0.9 · 0.4 · 0.5 = 0.396.

The next most likely team to win is team 3, with a 0.372 probability of winning the tournament.

题意：2^n个队进行足球赛，每个队打败另外一个队都有一个概率。
问最后胜利的概率最大的是哪只球队。

 

经典的概率dp题目,注意这个寻找对手的思想

 

/*分析：假设dp[i][j]表示进行第i次比赛j赢得概率 
则主要在于计算第i次比赛可能与j相邻的队伍t并且本次是与t比赛  
然后dp[i][j]+=dp[i-1][j]*dp[i-1][t]*p[j][t] 
对于t如何算？ 
假设队伍id: 
0 1 2 3 4 5 6 7 
对于第一轮比赛相对上一轮比赛: 
0属于第一个赢家,1属于第二个赢家,2属于第三个赢家,3属于第四个赢家...  
第一次比赛后: 
0/1 2/3 4/5 6/7 
对于第二轮比赛相对上一轮比赛: 
0/1属于第一个赢家,2/3属于第二个赢家... 
可以发现规律都是: 
第奇数个赢家和前一个赢家比赛 
第偶数个赢家和后一个赢家比赛 
也不难发现规律，对于第j个队伍在第i次比赛 
属于第j/(2^(i-1))个赢家//从0开始  
*/  
#include <iostream>  
#include <cstdio>  
#include <cstdlib>  
#include <cstring>  
#include <string>  
#include <queue>  
#include <algorithm>  
#include <map>  
#include <cmath>  
#include <iomanip>  
#define INF 99999999  
typedef long long LL;  
using namespace std;  
  
const int MAX=(1<<7)+10;  
const int N=10+10;  
int n;  
double p[MAX][MAX],dp[N][MAX];//dp[i][j]表示第i回合j胜利的概率  
  
int main(){  
    while(~scanf("%d",&n),n != -1){  
        int bit=1<<n;  
        for(int i=0;i<bit;++i){  
            for(int j=0;j<bit;++j)scanf("%lf",&p[i][j]);  
        }  
        memset(dp,0,sizeof dp);  
        for(int i=0;i<bit;++i)dp[0][i]=1;  
        for(int i=1;i<=n;++i){  
            for(int j=0;j<bit;++j){  
                int t=j>>(i-1);//j经历i-1次属于第t个赢家   
                if(t&1){//j为奇数赢家，本次将与第j-1个赢家比赛   
                    for(int k=t*(1<<(i-1))-1;k>=(t-1)*(1<<(i-1));--k){  
                        dp[i][j]+=dp[i-1][j]*dp[i-1][k]*p[j][k];  
                    }  
                }else{//j为偶数赢家,本次将与第j+1个赢家比赛   
                    for(int k=(t+1)*(1<<(i-1));k<(t+2)*(1<<(i-1));++k){  
                        dp[i][j]+=dp[i-1][j]*dp[i-1][k]*p[j][k];  
                    }  
                }  
            }  
        }  
        int id=0;  
        for(int i=0;i<bit;++i){  
            if(dp[n][i]>dp[n][id])id=i;  
        }  
        printf("%d
",id+1);  
    }  
    return 0;  
}