Day 18 C. Coin Rows

最新推荐文章于 2024-11-08 16:29:45 发布

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最新推荐文章于 2024-11-08 16:29:45 发布

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分类专栏： Codeforces 文章标签：博弈论矩阵最优策略路径规划动态规划

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这是一个关于 Alice 和 Bob 在二维矩阵中进行博弈的问题，目标是通过最优策略决定路径，使得 Bob 收集到的硬币数量最少。Alice 先行，Bob 后行，每步只能向右或向下移动。Alice 想要最小化 Bob 的得分，而 Bob 则希望最大化得分。输入包括矩阵的列数和每行的硬币数量，输出是双方最优策略下 Bob 的得分。示例给出了三种情况的解答。算法需要找到每行的前缀和与后缀和的最大值的最小值，以此作为 Bob 的得分。

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Problem：
Alice and Bob are playing a game on a matrix, consisting of 2 rows and m columns. The cell in the i-th row in the j-th column contains ai,j coins in it.

Initially, both Alice and Bob are standing in a cell (1,1). They are going to perform a sequence of moves to reach a cell (2,m).

The possible moves are:

Move right — from some cell (x,y) to (x,y+1);
Move down — from some cell (x,y) to (x+1,y).
First, Alice makes all her moves until she reaches (2,m). She collects the coins in all cells she visit (including the starting cell).

When Alice finishes, Bob starts his journey. He also performs the moves to reach (2,m) and collects the coins in all cells that he visited, but Alice didn’t.

The score of the game is the total number of coins Bob collects.

Alice wants to minimize the score. Bob wants to maximize the score. What will the score of the game be if both players play optimally?

Input
The first line contains a single integer t (1≤t≤10^4) — the number of testcases.

Then the descriptions of t testcases follow.

The first line of the testcase contains a single integer m (1≤m≤10^5) — the number of columns of the matrix.

The i-th of the next 2 lines contain m integers a(i,1),a(i,2),…,a(i,m) (1≤a(i,j)≤10^4) — the number of coins in the cell in the i-th row in the j-th column of the matrix.

The sum of m over all testcases doesn’t exceed 10^5.

Output
For each testcase print a single integer — the score of the game if both players play optimally.

Example
input
3

3
1 3 7
3 5 1

3
1 3 9
3 5 1

1
4
7
output
7
8
0

Note
The paths for the testcases are shown on the following pictures. Alice’s path is depicted in red and Bob’s path is depicted in blue.

在这里插入图片描述
这题的大致意思为：Alice想要让Bob得到的分数最少而Bob想得到在Alice干扰下的最高分数而且规定两人前进的道路只能向下或者向右并且游戏棋盘最大为2*m 所以只需要考虑下面一行的前缀和与上面一行的后缀和两者相比中的最大值的最小值

#include <iostream>
#include <vector>
#define ll long long
using namespace std;
int main()
{
    int t;
    cin >> t;
    while (t--) 
    {
        int m;
        cin >> m;
        vector<int> a(m);
        for (int i = 0; i < m; i++) 
        {
            cin >> a[i];
        }
        vector<int> b(m);
        for (int i = 0; i < m; i++)
        {
            cin >> b[i];
        }
        vector<int> pref(m + 1);
        for (int i = 0; i < m; i++) 
        {
            pref[i + 1] = pref[i] + b[i];
        }
        vector<int> suf(m + 1);
        for (int i = m - 1; i >= 0; i--) 
        {
            suf[i] = suf[i + 1] + a[i];
        }
        int ans = (int)1e9;
        for (int i = 0; i < m; i++)
        {
            ans = min(ans, max(pref[i], suf[i + 1]));
        }
        cout << ans << endl;;
    }
    return 0;
}