A Mini Locomotive【简单动态规划】

桃扇骨 2022-06-01 11:05 168阅读 0赞

## A Mini Locomotive ##

> A train has a locomotive that pulls the train with its many passenger coaches. If the locomotive breaks down, there is no way to pull the train. Therefore, the office of railroads decided to distribute three mini locomotives to each station. A mini locomotive can pull only a few passenger coaches. If a locomotive breaks down, three mini locomotives cannot pull all passenger coaches. So, the office of railroads made a decision as follows:
> 
> 1.  Set the number of maximum passenger coaches a mini locomotive can pull, and a mini locomotive will not pull over the number. The number is same for all three locomotives.
> 2.  With three mini locomotives, let them transport the maximum number of passengers to destination. The office already knew the number of passengers in each passenger coach, and no passengers are allowed to move between coaches.
> 3.  Each mini locomotive pulls consecutive passenger coaches. Right after the locomotive, passenger coaches have numbers starting from 1.
> 
> For example, assume there are 7 passenger coaches, and one mini locomotive can pull a maximum of 2 passenger coaches. The number of passengers in the passenger coaches, in order from 1 to 7, is 35, 40, 50, 10, 30, 45, and 60.
> 
> If three mini locomotives pull passenger coaches 1-2, 3-4, and 6-7, they can transport 240 passengers. In this example, three mini locomotives cannot transport more than 240 passengers.
> 
> Given the number of passenger coaches, the number of passengers in each passenger coach, and the maximum number of passenger coaches which can be pulled by a mini locomotive, write a program to find the maximum number of passengers which can be transported by the three mini locomotives.

#### Input ####

> The first line of the input contains a single integer t (1 <= t <= 11), the number of test cases, followed by the input data for each test case. The input for each test case will be as follows:  
> The first line of the input file contains the number of passenger coaches, which will not exceed 50,000. The second line contains a list of space separated integers giving the number of passengers in each coach, such that the i th number of in this line is the number of passengers in coach i. No coach holds more than 100 passengers. The third line contains the maximum number of passenger coaches which can be pulled by a single mini locomotive. This number will not exceed 1/3 of the number of passenger coaches.

#### Output ####

> There should be one line per test case, containing the maximum number of passengers which can be transported by the three mini locomotives.

#### Sample Input ####

> 1  
> 7  
> 35 40 50 10 30 45 60  
> 2

#### Sample Output ####

> 240

#### 题意概括： ####

有n辆客车，每辆客车都搭载有固定的人数，现在有3辆牵引车，每辆牵引车都能牵引m辆客车，但只能牵引连续m辆客车，客车的顺序就是数据给出的顺序。

### 解题分析： ###

这道题是一个简单的动态规划，需要推出动态转移方程 dp\[i\]\[j\] = max(dp\[i-1\]\[j\], dp\[i-m\]\[j-1\]+b\[i\]);

#### AC代码： ####

#include<stdio.h>
    #include<string.h>
    #include<algorithm>
    using namespace std;
    
    #define N 50010
    
    int main()
    {
        int i, j, t, n, m, sum;
        int a[N], b[N], dp[N][4];
        scanf("%d", &t);
        while(t--){
            scanf("%d", &n);
            memset(dp, 0, sizeof(dp));
            sum = 0;
            for(i = 1; i <= n; i++) scanf("%d", &a[i]);
            scanf("%d", &m);
            for(i = 1; i <= n; i++){
                sum += a[i];
                if(i >= m){
                    b[i] = sum;//用来记录连续m辆客车的人数
                    sum -= a[i-m+1];
                }
            }
            for(i = m; i <= n; i++){
                for(j = 1; j <= 3; j++)
                    dp[i][j] = max(dp[i-1][j], dp[i-m][j-1]+b[i]);
            }
            printf("%d\n", dp[n][3]);
        }
        return 0;
    }