P1005 [NOIP 2007 提高组] 矩阵取数游戏

题目描述

帅帅经常跟同学玩一个矩阵取数游戏：对于一个给定的 $n\times m$ 的矩阵，矩阵中的每个元素 $a_{i,j}$ 均为非负整数。游戏规则如下：

1. 每次取数时须从每行各取走一个元素，共 $n$ 个。经过 $m$ 次后取完矩阵内所有元素；
2. 每次取走的各个元素只能是该元素所在行的行首或行尾；
3. 每次取数都有一个得分值，为每行取数的得分之和，每行取数的得分 = 被取走的元素值 $\times 2^i$，其中 $i$ 表示第 $i$ 次取数（从 $1$ 开始编号）；
4. 游戏结束总得分为 $m$ 次取数得分之和。

帅帅想请你帮忙写一个程序，对于任意矩阵，可以求出取数后的最大得分。

输入格式

输入文件包括 $n+1$ 行：

第一行为两个用空格隔开的整数 $n$ 和 $m$。

第 $2\sim n+1$ 行为 $n\times m$ 矩阵，其中每行有 $m$ 个用单个空格隔开的非负整数。

输出格式

输出文件仅包含 $1$ 行，为一个整数，即输入矩阵取数后的最大得分。

输入输出样例 #1

输入 #1

2 3
1 2 3
3 4 2

输出 #1

82

说明/提示

【数据范围】

对于 $60\%$ 的数据，满足 $1\le n,m\le 30$，答案不超过 $10^{16}$。
对于 $100\%$ 的数据，满足 $1\le n,m\le 80$，$0\le a_{i,j}\le 1000$。

【题目来源】

NOIP 2007 提高第三题。

答案

C++

#include <iostream>
#include <vector>
#include <algorithm>
using namespace std;typedef long long ll;int main() {int n, m;cin >> n >> m;vector<vector<ll>> matrix(n, vector<ll>(m));for (int i = 0; i < n; i++) {for (int j = 0; j < m; j++) {cin >> matrix[i][j];}}vector<ll> pow2(m + 1);pow2[0] = 1;for (int i = 1; i <= m; i++) {pow2[i] = pow2[i - 1] * 2;}ll total = 0;for (int i = 0; i < n; i++) {vector<vector<ll>> dp(m, vector<ll>(m, 0));for (int len = 0; len < m; len++) {for (int l = 0; l + len < m; l++) {int r = l + len;int k = m - len;if (l == r) {dp[l][r] = matrix[i][l] * pow2[k];} else {dp[l][r] = max(matrix[i][l] * pow2[k] + dp[l + 1][r],matrix[i][r] * pow2[k] + dp[l][r - 1]);}}}total += dp[0][m - 1];}cout << total << endl;return 0;
}

C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>typedef long long ll;ll max(ll a, ll b) {return a > b ? a : b;
}int main() {int n, m;scanf("%d %d", &n, &m);ll** matrix = (ll**)malloc(n * sizeof(ll*));for (int i = 0; i < n; i++) {matrix[i] = (ll*)malloc(m * sizeof(ll));for (int j = 0; j < m; j++) {scanf("%lld", &matrix[i][j]);}}ll* pow2 = (ll*)malloc((m + 1) * sizeof(ll));pow2[0] = 1;for (int i = 1; i <= m; i++) {pow2[i] = pow2[i - 1] * 2;}ll total = 0;ll** dp = (ll**)malloc(m * sizeof(ll*));for (int i = 0; i < m; i++) {dp[i] = (ll*)malloc(m * sizeof(ll));}for (int i = 0; i < n; i++) {memset(dp, 0, m * m * sizeof(ll));for (int len = 0; len < m; len++) {for (int l = 0; l + len < m; l++) {int r = l + len;int k = m - len;if (l == r) {dp[l][r] = matrix[i][l] * pow2[k];} else {dp[l][r] = max(matrix[i][l] * pow2[k] + dp[l + 1][r],matrix[i][r] * pow2[k] + dp[l][r - 1]);}}}total += dp[0][m - 1];}printf("%lld\n", total);for (int i = 0; i < n; i++) {free(matrix[i]);}free(matrix);free(pow2);for (int i = 0; i < m; i++) {free(dp[i]);}free(dp);return 0;
}

Python

n, m = map(int, input().split())
matrix = []
for _ in range(n):row = list(map(int, input().split()))matrix.append(row)pow2 = [1] * (m + 1)
for i in range(1, m + 1):pow2[i] = pow2[i-1] * 2total = 0
for i in range(n):dp = [[0] * m for _ in range(m)]for length in range(m):for l in range(m - length):r = l + lengthk = m - lengthif l == r:dp[l][r] = matrix[i][l] * pow2[k]else:dp[l][r] = max(matrix[i][l] * pow2[k] + dp[l+1][r],matrix[i][r] * pow2[k] + dp[l][r-1])total += dp[0][m-1]print(total)

Java

import java.util.Scanner;public class MatrixGame {public static void main(String[] args) {Scanner scanner = new Scanner(System.in);int n = scanner.nextInt();int m = scanner.nextInt();long[][] matrix = new long[n][m];for (int i = 0; i < n; i++) {for (int j = 0; j < m; j++) {matrix[i][j] = scanner.nextLong();}}long[] pow2 = new long[m + 1];pow2[0] = 1;for (int i = 1; i <= m; i++) {pow2[i] = pow2[i - 1] * 2;}long total = 0;long[][] dp = new long[m][m];for (int i = 0; i < n; i++) {for (int len = 0; len < m; len++) {for (int l = 0; l + len < m; l++) {int r = l + len;int k = m - len;if (l == r) {dp[l][r] = matrix[i][l] * pow2[k];} else {dp[l][r] = Math.max(matrix[i][l] * pow2[k] + dp[l + 1][r],matrix[i][r] * pow2[k] + dp[l][r - 1]);}}}total += dp[0][m - 1];for (int a = 0; a < m; a++) {for (int b = 0; b < m; b++) {dp[a][b] = 0;}}}System.out.println(total);scanner.close();}
}

Go

package mainimport ("fmt"
)func max(a, b int64) int64 {if a > b {return a}return b
}func main() {var n, m intfmt.Scan(&n, &m)matrix := make([][]int64, n)for i := range matrix {matrix[i] = make([]int64, m)for j := range matrix[i] {fmt.Scan(&matrix[i][j])}}pow2 := make([]int64, m+1)pow2[0] = 1for i := 1; i <= m; i++ {pow2[i] = pow2[i-1] * 2}var total int64 = 0dp := make([][]int64, m)for i := range dp {dp[i] = make([]int64, m)}for i := 0; i < n; i++ {for len := 0; len < m; len++ {for l := 0; l+len < m; l++ {r := l + lenk := m - lenif l == r {dp[l][r] = matrix[i][l] * pow2[k]} else {dp[l][r] = max(matrix[i][l]*pow2[k]+dp[l+1][r],matrix[i][r]*pow2[k]+dp[l][r-1],)}}}total += dp[0][m-1]for a := 0; a < m; a++ {for b := 0; b < m; b++ {dp[a][b] = 0}}}fmt.Println(total)
}