hdu 6199 沈阳网络赛---gems gems gems（DP）

题目链接

 

Problem Description

Now there are n gems, each of which has its own value. Alice and Bob play a game with these n gems.
They place the gems in a row and decide to take turns to take gems from left to right. 
Alice goes first and takes 1 or 2 gems from the left. After that, on each turn a player can take k or k+1 gems if the other player takes k gems in the previous turn. The game ends when there are no gems left or the current player can't take k or k+1 gems.
Your task is to determine the difference between the total value of gems Alice took and Bob took. Assume both players play optimally. Alice wants to maximize the difference while Bob wants to minimize it.

 

 

Input

The first line contains an integer T (1≤T≤10), the number of the test cases. 
For each test case:
the first line contains a numbers n (1≤n≤20000);
the second line contains n numbers: V1,V2…Vn. (−100000≤Vi≤100000)

 

 

Output

For each test case, print a single number in a line: the difference between the total value of gems Alice took and the total value of gems Bob took.

 

 

Sample Input

1

3

1 3 2

 

 

Sample Output

4

 

 

题意：现在有n块宝石排成一行，价值为v[1~n]。 现在Alice和Bob 从左至右轮流取宝石，Alice先取，可以取1或2个，然后Bob取……如果前一个人取了k个，那么当前这个人只能取k或者k+1个宝石。（结束条件，如果前一个人取了k个宝石，而剩下的少于k个则停止不用取；如果剩余k个，那么这个人必须取k个），现在Alice和Bob都想得到的宝石价值和比对方尽量多，求Alice得到的价值和 - Bob得到的价值和的最大值？

 

思路：动态规划，定义dp[i][k][j], k取值0和1（0表示Alice取，1表示Bob取），i表示第i块宝石（1<=i<=20000），j表示从第i块宝石开始向右取j块宝石（1<=j<=200, 因为递增的取宝石，一个人一次最多取200块宝石），所以dp[i][k][j]表示取完i~n块宝石时（由后向前推），第k个人从i开始向右连续取k个宝石两人的价值差；

         另外，本题空间限制比较紧，所以需要使用滚动数组，第一维设为205即可。

 

代码如下：

#include <iostream>
#include <algorithm>
#include <cstdio>
#include <cstring>
using namespace std;
typedef long long LL;
const int N=20005;
const int M=203;
int sum[N];
int dp[M][2][205];

int main()
{
    int T; cin>>T;
    while(T--)
    {
       sum[0]=0;
       memset(dp,0,sizeof(dp));
       int n; scanf("%d",&n);
       for(int i=1;i<=n;i++) scanf("%d",&sum[i]),sum[i]+=sum[i-1];
       for(int i=n;i>=1;i--)
       {
           for(int j=1;j<M;j++)
           {
               if(i+j-1>n) break;

               int tmp=sum[i+j-1]-sum[i-1], t;
               if(i+j+j<=n) t=min(dp[(i+j)%M][1][j],dp[(i+j)%M][1][j+1]);
               else if(i+j+j-1<=n) t=dp[(i+j)%M][1][j];
               else t=0;
               dp[i%M][0][j]=tmp+t;

               if(i+j+j<=n) t=max(dp[(i+j)%M][0][j],dp[(i+j)%M][0][j+1]);
               else if(i+j+j-1<=n) t=dp[(i+j)%M][0][j];
               else t=0;
               dp[i%M][1][j]=t-tmp;
           }
       }
       int ans;
       if(n>=2) ans=max(dp[1][0][1],dp[1][0][2]);
       else ans=dp[1][0][1];
       printf("%d\n",ans);
    }
    return 0;
}