ICPC Asia Nanning 2017 I. Rake It In (DFS+贪心 或 对抗搜索+Alpha-Beta剪枝)

题目链接：Rake It In

比赛链接：ICPC Asia Nanning 2017

Description

The designers have come up with a new simple game called “Rake It In”. Two players, Alice and Bob, initially select an integer k and initialize a score indicator. An \(4 \times 4\) board is created with 16 values placed on the board. Starting with player Alice, each player in a round selects a \(2 \times 2\) region of the board, adding the sum of values in the region to the score indicator, and then rotating these four values \(90\) degrees counterclockwise.

After \(2k\) rounds in total, each player has made decision in k times. The ultimate goal of Alice is to maximize the final score. However for Bob, his goal is to minimize the final score.

In order to test how good this game is, you are hired to write a program which can play the game. Specifically, given the starting configuration, they would like a program to determine the final score when both players are entirely rational.

Input

The input contains several test cases and the first line provides an integer \(t (1 \le t \le 200)\) which is the number of test cases.

Each case contains five lines. The first line provides the integer \(k (1 \le k \le 3)\). Each of the following four lines contains four integers indicating the values on the board initially. All values are integers between \(1\) to \(10\).

Output

For each case, output an integer in a line which is the predicted final score.

Sample Input

4
1
1 1 2 2
1 1 2 2
3 3 4 4
3 3 4 4
2
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
3
1 1 4 4
4 4 1 1
1 1 4 4
1 4 1 4
3
1 2 3 4
5 1 2 3
4 5 1 2
3 4 5 1

Sample Output

20
40
63
71

Solution

题意

有一块 \(4\times 4\) 的板，Alice 和 Bob 每次选择 \(2\times 2\) 的区域并逆时针旋转 \(90\) 度，这个区域的和累加到总分上。现在 Alice 先手，有 \(k\) 轮游戏，Alice 想要分数最大化，Bob 想要分数最小化，求最终的分数。

题解

DFS 贪心

比较好的解法是对抗搜索 与 \(Alpha-Beta\) 剪枝。

题解给出是上分支定界和启发式搜索。

但是用贪心 + 爆搜竟然过了。

关于对抗搜索和 \(Alpha-Beta\) 剪枝以后再更新。

Code

DFS + 贪心

#include <bits/stdc++.h>
using namespace std;
const int inf = 1000;

int k;
int mt[10][10];

// 交换两数
void swap(int &a, int &b) {
    a = a ^ b;
    b = a ^ b;
    a = a ^ b;
}

// 逆时针旋转
void rote(int x, int y) {
    swap(mt[x][y], mt[x + 1][y]);
    swap(mt[x][y + 1], mt[x + 1][y + 1]);
    swap(mt[x][y], mt[x + 1][y + 1]);
}

// 顺时针旋转
void rerote(int x, int y) {
    swap(mt[x][y], mt[x + 1][y + 1]);
    swap(mt[x][y + 1], mt[x + 1][y + 1]);
    swap(mt[x][y], mt[x + 1][y]);
}

// 求和
int sum(int x, int y) {
    return mt[x][y] + mt[x + 1][y] + mt[x][y + 1] + mt[x + 1][y + 1];
}

int dfs(int step) {
    if(step == 2 * k) { // 最后一步
        int ans = inf;
        for(int i = 0; i < 3; ++i) {
            for(int j = 0; j < 3; ++j) {
                ans = min(ans, sum(i, j));
            }
        }
        return ans;
    } else {
        // 奇数步选择最大 偶数步选择最小
        int ans = (step & 1)? 0: inf;
        for(int i = 0; i < 3; ++i) {
            for(int j = 0; j < 3; ++j) {
                rote(i, j); // 逆时针旋转
                if(step & 1) {
                    ans = max(ans, sum(i, j) + dfs(step + 1));
                } else {
                    ans = min(ans, sum(i, j) + dfs(step + 1));
                }
                rerote(i, j); // 回溯时转回来
            }
        }
        return ans;
    }
}

int main() {
    int T;
    cin >> T;
    while(T--) {
        scanf("%d", &k);
        for(int i = 0; i < 4; ++i) {
            for(int j = 0; j < 4; ++j) {
                scanf("%d", &mt[i][j]);
            }
        }
        int ans = dfs(1);
        printf("%d\n", ans);
    }
    return 0;
}

对抗搜索 + \(Alpha-Beta\) 剪枝

#include <bits/stdc++.h>
using namespace std;
const int inf = 1000;

int k;
int mt[10][10];

void swap(int &a, int &b) {
    a = a ^ b;
    b = a ^ b;
    a = a ^ b;
}

void rote(int x, int y) {
    swap(mt[x][y], mt[x + 1][y]);
    swap(mt[x][y + 1], mt[x + 1][y + 1]);
    swap(mt[x][y], mt[x + 1][y + 1]);
}

void rerote(int x, int y) {
    swap(mt[x][y], mt[x + 1][y + 1]);
    swap(mt[x][y + 1], mt[x + 1][y + 1]);
    swap(mt[x][y], mt[x + 1][y]);
}

int cnt(int x, int y) {
    return mt[x][y] + mt[x + 1][y] + mt[x][y + 1] + mt[x + 1][y + 1];
}

int dfs(int sum, int step, int alpha, int beta) {
    if(step == 2 * k + 1) {
        return sum;
    } else {
        for(int i = 0; i < 3; ++i) {
            for(int j = 0; j < 3; ++j) {
                rote(i, j);
                if(step & 1) {
                    alpha = max(alpha, dfs(sum + cnt(i, j), step + 1, alpha, beta));
                } else {
                    beta = min(beta, dfs(sum + cnt(i, j), step + 1, alpha, beta));
                }
                rerote(i, j);
                if(beta <= alpha) break;
            }
            if(beta <= alpha) break;
        }
        return (step & 1)? alpha: beta;
    }
}

int main() {
    int T;
    cin >> T;
    while(T--) {
        scanf("%d", &k);
        for(int i = 0; i < 4; ++i) {
            for(int j = 0; j < 4; ++j) {
                scanf("%d", &mt[i][j]);
            }
        }
        int ans = dfs(0, 1, -inf, inf);
        printf("%d\n", ans);
    }
    return 0;
}