POJ - 1976 A Mini Locomotive（背包DP）

A Mini LocomotiveDescriptionA 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 locomotivecan 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 transportedby the three mini locomotives. InputThe 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 ith number of in thisline 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 passengercoaches. OutputThere 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：有3个车头，可以连续拉k节车厢，问如何截取火车，使拉的人数最大解题思路：稍作思考，可以转化为背包问题，把连续k节车厢的人数和作为价值，每个车头占一个容量，背包大小为3. 因此有dp[i][j]数组，代表前i节车厢用j个车头去拉能得到的最大价值，相当于背包问题里的，前i个物品用背包容量大小为j的背包去放能获得的最大价值。因此有转移方程，先把a[i]转为前缀和形式

dp[i][j]=max(dp[i-1][j],dp[i-k][j-1]+a[i]-a[i-k]);

#include <stdio.h>#include <iostream>#include <algorithm>#include <string.h>using namespace std;int a[50005];int dp[50005][4];int main(){int n,t,k;scanf("%d",&t);while(t--){a[0]=0;memset(dp,0,sizeof(dp));scanf("%d",&n);for(int i=1;i<=n;i++) {scanf("%d",&a[i]);a[i]+=a[i-1];}scanf("%d",&k);for(int i=k;i<=n;i++){for(int j=1;j<=3;j++){dp[i][j]=max(dp[i-1][j],dp[i-k][j-1]+a[i]-a[i-k]);}}printf("%d\n",dp[3]);}return 0;}