题目

{% fold 点击显/隐题目 %}

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 = [pij] such that pij is the probability that team i will beat team j in a match determine which team is most likely to win the tournament.
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 pij. The matrix P will satisfy the constraints that pij = 1.0 ? pji for all i ≠ j, and pii = 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.
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.
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
2
{% endfold %}

题解

画出来比赛的流程图,可以发现能够用位运算方便表示出如何进行比赛
如队伍i的第j场比赛的对手是对i的j位取反,后面的数为0到为1的范围内的数

然后使用dp[i][j]表示第i场比赛,队伍j获胜的概率
计算出所有的值然后选出最大的即可

代码

{% fold 点击显/隐代码 %}```cpp Football https://github.com/OhYee/sourcecode/tree/master/ACM 代码备份
//
#define debug
#include
///
#include
#include
#include
using namespace std;

const int maxn = 8;
const int maxm = 2 << 8;
double Pro[maxm][maxm];
double dp[maxn][maxm];

int getTeam(int number, int bit, int flag) {
int ans = number ^ (1 << bit); //把bit位取反

ans = (ans >> (bit)) << (bit);
if (flag)
    ans += (1 << bit) - 1;

return ans;

}

int main() {
#ifdef debug
freopen("in.txt", "r", stdin);
int START = clock();
#endif
cin.tie(0);
cin.sync_with_stdio(false);

int n;
while (cin >> n) {
    if (n == -1)
        continue;
    for (int i = 0; i < (1 << n); i++)
        for (int j = 0; j < (1 << n); j++)
            cin >> Pro[i][j];

    for (int i = 0; i < n; i++) {
        // cout << "Game round" << i + 1 << ": " << endl;
        for (int j = 0; j < (1 << n); j++) {
            if (i == 0) {
                // cout << "\t" << j << " vs. " << getTeam(j, i, 1) << endl;
                dp[i][j] = Pro[j][getTeam(j, i, 0)];
            } else {
                dp[i][j] = 0;
                for (int k = getTeam(j, i, 0); k <= getTeam(j, i, 1); k++) {
                    // cout << "\t" << j << " vs. " << k << endl;
                    dp[i][j] += dp[i - 1][k] * Pro[j][k];
                }
                dp[i][j] *= dp[i - 1][j];
            }
            // cout << "\t\tdp[i][j]=" << dp[i][j] << endl;
        }
    }

    int pos = 0;
    for (int i = 1; i < (1 << n); i++) {
        // cout << dp[n - 1][i] << endl;
        if (dp[n - 1][i] > dp[n - 1][pos])
            pos = i;
    }
    cout << pos + 1 << endl;
}

#ifdef debug
printf("Time:%.3fs.\n", double(clock() - START) / CLOCKS_PER_SEC);
#endif
return 0;
}

{% endfold %}