【POJ3071】Football - 状态压缩+期望 DP

Description

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.

题目大意

给你$2^n$个球队，必须相邻的球队才可以比赛，在告诉你i球队战胜j球队的概率，问最后哪支球队赢得概率最大。

思路

设$f_{i,j}$表示第$i$次比赛队伍$j$获胜的概率，枚举与$j$比赛的$k$，可得$f_{i,j} = \sum_{k=1}^{2^n}f_{i-1,j}*f_{i-1,k}*p_{j,k}*[$ $\text{j与k相邻} ]$

判断相邻先<<(i-)再用异或(^)运算判断一下即可

/************************************************

*Author        :  lrj124

*Created Time  :  2018.09.30.19:13

*Mail          :  1584634848@qq.com

*Problem       :  poj3071

************************************************/

#include <iostream>

#include <cstring>

#include <cstdio>

using namespace std;

const int maxn = (1<<7) + 10;

double p[maxn][maxn],f[maxn][maxn];

int n;

int main() {

    while (cin >> n) {

        if (n == -1) break;

        memset(f,0,sizeof(f));

        for (int i = 1;i <= (1<<n);i++) {

            f[0][i] = 1;

            for (int j = 1;j <= (1<<n);j++) scanf("%lf",&p[i][j]);

        }

        for (int i = 1;i <= (1<<n);i++)

            for (int j = 1;j <= (1<<n);j++)

                for (int k = 1;k <= (1<<n);k++)

                    if ((((j-1)>>(i-1))^1) == ((k-1)>>(i-1)))

                        f[i][j] += f[i-1][j]*f[i-1][k]*p[j][k];

        double Max = 0;

        int ans;

        for (int i = 1;i <= (1<<n);i++)

            if (f[n][i] > Max) {

                Max = f[n][i];

                ans = i;

            }

        printf("%d\n",ans);

    }

    return 0;

}