1433F - Zero Remainder Sum

1433F - Zero Remainder Sum

---

\(~~~~\)动态规划，\(dp[i][j][p][q]\)表示在第\(i\)行，第\(j\)列，已经在这一行选取了\(p\)个元素，且目前的元素和对\(k\)取模是\(q\)时的最大和，首先有\(dp[1][0][0][0]=0\)。可以采用递推的方式由当前状态向下一个状态递推，结果即\(dp[n+1][0][0][0]\)。
\(~~~~\)当更新到某个状态时，对下一列的状态进行递推：如果不选取\(a[i][j+1]\)，有转移方程
\(~~~~\)\(dp[i][j+1][p][q] = max(dp[i][j+1][p][q], dp[i][j][p][q])\)
\(~~~~\)如果选取，有转移方程(p大于0并且\(dp[i][j][p-1][q]\)被更新过的前提下)
\(~~~~\)\(dp[i][j+1][p][(q+a[i][j+1])\%k] = max(dp[i][j+1][p][(q+a[i][j+1])\%k], dp[i][j][p-1][q]+a[i][j+1])\)
更新完下一列的状态后，对\(dp[i+1][0][0][0]\)更新，方法类似
\(~~~~\)\(dp[i+1][0][0][q] = max(dp[i+1][0][0][q], dp[i][j+1][p][q]) (0<q<k)\)

---

代码

#include<iostream>
#include<stdio.h>
#include<cstdio>
#include<string.h>
#include<algorithm>
#include<vector>
#include<queue>
#include<map>
#include<cmath>
#define INF 0x3f3f3f3f
typedef long long ll;
using namespace std;
inline void read(int &p)
{
    p=0;int flag=1;char c=getchar();
    while(!isdigit(c)) {if(c=='-') flag=-1;c=getchar();}
    while(isdigit(c)) {p=p*10+c-'0';c=getchar();}p*=flag;
}
int dp[80][80][80][80];
int a[80][80];
int main()
{
    #ifdef LOCAL
    freopen("in.txt","r",stdin);
    freopen("out.txt","w",stdout);
    #endif
    int m, n, k;
    cin >> n >> m >> k;
    for(int i=1; i<=n; i++){
        for(int j=1; j<=m; j++){
            read(a[i][j]);
        }
    }
    memset(dp, -1, sizeof(dp));
    dp[1][0][0][0] = 0;
    for(int i=1; i<=n; i++){
        for(int j=0; j<m; j++){
            for(int p=0; p<=min(j+1, m/2); p++){
                for(int q=0; q<k; q++){
                    dp[i][j+1][p][q] = max(dp[i][j+1][p][q], dp[i][j][p][q]);
                    if(p && dp[i][j][p-1][q]!=-1){
                        dp[i][j+1][p][(q+a[i][j+1])%k] = max(dp[i][j+1][p][(q+a[i][j+1])%k], dp[i][j][p-1][q]+a[i][j+1]);
                    }
                }
                for(int q=0; q<k; q++){
                    dp[i+1][0][0][q] = max(dp[i+1][0][0][q], dp[i][j+1][p][q]);
                }
            }
        }
    }
    cout << dp[n+1][0][0][0] << endl;
    return 0;
}