Xiao Ming facevalue

TimeLimit:4000MS MemoryLimit:256MB

64-bit integer IO format: %lld

Problem Description

One day, Xiao Ming accidentally picked up a magic lamp. The lamp god said to Xiao Ming that he could satisfy his wish. Xiao Ming made a wish to the lamp god to make himself more handsome, so the lamp god promised him.

But the lamp god did not simply realize this wish for him. He gave Xiao Ming a sequence A containing n integers, and let him pick m numbers in A. The sum of the m numbers is handsome value which Xiao Ming will finally get.

Of course, Xiao Ming definitely thinks that the higher the handsome value, the better, but things are not so simple, the exact rules are defined as follows:

1. Xiao Ming selected a sequence b, b1, b2, b3 … bm (bi denotes the subscript in A, 1<=b1<b2<b3<… <bm<=n)

2. Any b[i] must satisfy b[i-1]+k>=b[i].(1<i<=m)

3. Xiao Ming's handsome value S=A[b1]+A[b2]+A[b3]+...+A[bm]

For the stupid Xiao Ming, he can only pick m number randomly, but he wants to get the most handsome value, so he asked you to help. Can you help him?

Input

The first line contain an integer t indicates the number of questions asked (1 <= t <= 20).

Next, the first line of each query contains two spaces separated by an integer n, m, k.

The next n integers are the n integers of sequence A.

(1<=k,m<=n<=3000. 0<=A[i]<=1e5)

(guarantee all inquiries ∑n<=3000)

Output

For each query, output a line of integers indicating the value of the maximum handsome value.

SampleInput

2
5 2 2
20 1 18 2 19
13 4 3
10 0 0 0 8 0 0 0 9 0 0 0 7

SampleOutput

38
19

Submit

题目地址：http://www.fjutacm.com/Contest.jsp?cid=659#P3

题意：该题的意思是给出一个t，代表t组，然后有n个数组，你可以选m个，每个选择的相邻的数字之间的下标差不能超过k

即i = 1， k = 3

那么[2, 4]的数都能选择1.

那么我们可以得到一个状态转移方程、

设状态为dp[i][j] 第一维i为当前选择了几个数字，第二维j为当前选择的是哪一个数字的下标

故可得状态转移方程为dp[i][j] = max(dp[i - 1][j], j 属于 [j - k, j - 1]) + dp[1][j];

那么我们可以得到一个O(n ^ 3)的dp

但是参考题目数据是无法通过该题的，可以采用ST表优化，优化查询[dp[i - 1][j - k], dp[i - 1][j - 1]]的最大值

复杂度可以降为O(n ^ 2 log n)

那么在比赛的时候可以通过该题

其实正解是可以优化到O(n ^ 2)的，使用单调队列优化该DP

先附上ST表优化的代码：

#include <iostream>
#include <cstdio>
#include <cstring>
#include <cmath>
#include <algorithm>
using namespace std;
typedef long long ll;
const int MAXN = 3005;
int t;
ll dp[MAXN][MAXN];
ll f[MAXN][20];
void ST_prework(int n, int step){
    for(int i = 1; i <= n; i ++) f[i][0] = dp[step - 1][i];
    int t = log(n) / log(2) + 1;
    for(int j = 1; j < t; j ++){
        for(int i = 1; i <= n - (1 << j) + 1; i ++){
            f[i][j] = max(f[i][j - 1], f[i + (1 << (j - 1))][j - 1]);
        }
    }
}
ll ST_query(int l, int r){
    int k = log(r - l + 1) / log(2);
    return max(f[l][k], f[r - (1 << k) + 1][k]);
}
int main(){
    scanf("%d", &t);
    while(t --){
        int n, m, k;
        scanf("%d%d%d", &n, &m, &k);
        for(int i = 1; i <= n; i ++){
            scanf("%lld", &dp[1][i]);
        }
        for(int i = 2; i <= m; i ++){
            ST_prework(n, i);
            for(int j = i; j <= n; j ++){
                int l = j - k;
                if(l <= 0){
                    l = 1;
                }
                dp[i][j] = ST_query(l, j - 1) + dp[1][j];
            }
        }
        ll re = 0;
        for(int i = 1; i <= n; i ++){
            re = max(re, dp[m][i]);
        }
        printf("%lld\n", re);
    }
    return 0;
}

以下为单调队列优化：

#include <iostream>
#include <cstdio>
#include <cstring>
#include <cmath>
#include <algorithm>
#include <queue>
using namespace std;
typedef long long ll;
const int MAXN = 3005;
int t;
ll dp[MAXN][MAXN];
deque<ll>que;
int main(){
    scanf("%d", &t);
    while(t --){
        int n, m, k;
        scanf("%d%d%d", &n, &m, &k);
        for(int i = 1; i <= n; i ++){
            scanf("%lld", &dp[1][i]);
        }
        for(int i = 2; i <= m; i ++){
            while(!que.empty()){
                que.pop_back();
            }
            int l = i - k;
            if(l <= 0){
                l = 1;
            }
            for(int j = l; j < i; j ++){
                if(que.empty()){
                    que.push_back(j);
                }
                else{
                    if(dp[i - 1][j] >= dp[i - 1][que.front()]){
                        while(!que.empty()){
                            que.pop_back();
                        }
                        que.push_back(j);
                    }
                    else{
                        while(dp[i - 1][j] >= dp[i - 1][que.back()]){
                            que.pop_back();
                        }
                        que.push_back(j);
                    }
                }
            }
            for(int j = i; j <= n; j ++){
                if(que.front() < j - k){
                    que.pop_front();
                }
                dp[i][j] = dp[i - 1][que.front()] + dp[1][j];
                if(dp[i - 1][j] >= dp[i - 1][que.front()]){
                    while(!que.empty()){
                        que.pop_back();
                    }
                    que.push_back(j);
                }
                else{
                    while(dp[i - 1][j] >= dp[i - 1][que.back()]){
                        que.pop_back();
                    }
                    que.push_back(j);
                }
            }
        }
        ll re = 0;
        for(int i = 1; i <= n; i ++){
            re = max(re, dp[m][i]);
        }
        printf("%lld\n", re);
    }
    return 0;
}

2019 fjnu acm校赛 D:Xiao Ming facevalue

Xiao Ming facevalue

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