Codeforces Round #677 (Div. 3) F. Zero Remainder Sum

it2025-05-19  12

题意

n(n<=70)行m(m <= 70)列的元素,每行最多取m / 2(向下取整)个元素,使最后sum%k==0的sum最大,sum可为0 k <= 70, aij <= 70

解题思路:

设dp[i][j][l][mod] 第i行第j个为止,选l个模数为mod的最大值 分析dp方程: (1).第i行第0列取0个模数为mod的dp值,由上一行到任意一列j取g个模数为mod的dp最大值: d p [ i ] [ 0 ] [ 0 ] [ m o d ] = m a x ( d p [ i ] [ 0 ] [ 0 ] [ m o d ] , d p [ i − 1 ] [ j ] [ g ] [ m o d ] ) ; dp[i][0][0][mod] = max(dp[i][0][0][mod], dp[i - 1][j][g][mod]); dp[i][0][0][mod]=max(dp[i][0][0][mod],dp[i1][j][g][mod]); (2).当前如果不选,那么就是由当前行前一列递推过来: d p [ i ] [ j ] [ l ] [ m o d ] = m a x ( d p [ i ] [ j ] [ l ] [ m o d ] , d p [ i ] [ j − 1 ] [ l ] [ m o d ] ) ; dp[i][j][l][mod] = max(dp[i][j][l][mod], dp[i][j - 1][l][mod]); dp[i][j][l][mod]=max(dp[i][j][l][mod],dp[i][j1][l][mod]); (3).当前如果选:(l!=0 && dp[i][j - 1][l - 1][mod] != -1(即要从已知状态递推过来)) d p [ i ] [ j ] [ l ] [ ( m o d + a [ i ] [ j ] ) % k ] = m a x ( d p [ i ] [ j ] [ l ] [ ( m o d + a [ i ] [ j ] ) % k ] , d p [ i ] [ j − 1 ] [ l − 1 ] [ m o d ] + a [ i ] [ j ] ) ; dp[i][j][l][(mod + a[i][j]) \% k] = max(dp[i][j][l][(mod + a[i][j]) \% k], dp[i][j - 1][l - 1][mod] + a[i][j]); dp[i][j][l][(mod+a[i][j])%k]=max(dp[i][j][l][(mod+a[i][j])%k],dp[i][j1][l1][mod]+a[i][j]); 最后答案,枚举找最后一行模数为0的dp最大值即可 因为数据范围为70,所以时间空间都没问题

WA的原因:

没有搞清谁是主次,for循环遍历顺序不对,先遍历的模数,明明是应该先遍历列,在这基础上再遍历模数。如果先遍历模数再遍历列,则某些列的dp值会无法计算,因为要用到之间列的模数(未遍历到)

以及没有考虑到(1)的情况

AC代码:

#include <bits/stdc++.h> using namespace std; typedef long long ll; const int maxn = 75; int n, m, k; int a[maxn][maxn]; int dp[maxn][maxn][maxn][maxn];//dp[i][j][l][mod] 第i行第j个为止,选l个模数为mod的最大值 int main() { cin >> n >> m >> k; for (int i = 1; i <= n; i++) for (int j = 1; j <= m; j++) cin >> a[i][j]; memset(dp, -1, sizeof(dp)); for (int j = 1; j <= m; j++) for (int l = 0; l <= m / 2; l++) dp[0][j][l][0] = 0; for (int i = 1; i <= n; i++) { for (int j = 1; j <= m; j++) for (int g = 0; g <= m / 2; g++) for (int mod = 0; mod < k; mod++) dp[i][0][0][mod] = max(dp[i][0][0][mod], dp[i - 1][j][g][mod]); for (int j = 1; j <= m; j++) for (int l = 0; l <= m / 2; l++) for (int mod = 0; mod < k; mod++) { dp[i][j][l][mod] = max(dp[i][j][l][mod], dp[i][j - 1][l][mod]); if (l != 0 && dp[i][j - 1][l - 1][mod] != -1) dp[i][j][l][(mod + a[i][j]) % k] = max(dp[i][j][l][(mod + a[i][j]) % k], dp[i][j - 1][l - 1][mod] + a[i][j]); } } int ans = 0; for (int j = 1; j <= m; j++) for (int l = 0; l <= m / 2; l++) ans = max(ans, dp[n][j][l][0]); cout << ans << endl; }
最新回复(0)