SPOJ - AMR11A-（DP逆向思维）—— 个人觉得好题

Posted 2020-12-03 jealous-boy

原题链接：https://www.spoj.com/problems/AMR11A/en/

VJ暑期训练链接：https://vjudge.net/contest/237052#problem/A

AMR11A - Magic Grid

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Thanks a lot for helping Harry Potter in finding the Sorcerer‘s Stone of Immortality in October. Did we not tell you that it was just an online game ? uhhh! now here is the real onsite task for Harry. You are given a magrid S ( a magic grid ) having R rows and C columns. Each cell in this magrid has either a Hungarian horntail dragon that our intrepid hero has to defeat, or a flask of magic potion that his teacher Snape has left for him. A dragon at a cell (i,j) takes away |S[i][j]| strength points from him, and a potion at a cell (i,j) increases Harry‘s strength by S[i][j]. If his strength drops to 0 or less at any point during his journey, Harry dies, and no magical stone can revive him.

 

Harry starts from the top-left corner cell (1,1) and the Sorcerer‘s Stone is in the bottom-right corner cell (R,C). From a cell (i,j), Harry can only move either one cell down or right i.e., to cell (i+1,j) or cell (i,j+1) and he can not move outside the magrid. Harry has used magic before starting his journey to determine which cell contains what, but lacks the basic simple mathematical skill to determine what minimum strength he needs to start with to collect the Sorcerer‘s Stone. Please help him once again.

Thanks a lot for helping Harry Potter in finding the Sorcerer‘s Stone of Immortality in October. Did we not tell you that it was just an online game ? uhhh! now here is the real onsite task for Harry. You are given a magrid S ( a magic grid ) having R rows and C columns. Each cell in this magrid has either a Hungarian horntail dragon that our intrepid hero has to defeat, or a flask of magic potion that his teacher Snape has left for him. A dragon at a cell (i,j) takes away |S[i][j]| strength points from him, and a potion at a cell (i,j) increases Harry‘s strength by S[i][j]. If his strength drops to 0 or less at any point during his journey, Harry dies, and no magical stone can revive him.

Harry starts from the top-left corner cell (1,1) and the Sorcerer‘s Stone is in the bottom-right corner cell (R,C). From a cell (i,j), Harry can only move either one cell down or right i.e., to cell (i+1,j) or cell (i,j+1) and he can not move outside the magrid. Harry has used magic before starting his journey to determine which cell contains what, but lacks the basic simple mathematical skill to determine what minimum strength he needs to start with to collect the Sorcerer‘s Stone. Please help him once again.

 

Input (STDIN):

The first line contains the number of test cases T. T cases follow. Each test case consists of R C in the first line followed by the description of the grid in R lines, each containing C integers. Rows are numbered 1 to R from top to bottom and columns are numbered 1 to C from left to right. Cells with S[i][j] < 0 contain dragons, others contain magic potions.

Output (STDOUT):

Output T lines, one for each case containing the minimum strength Harry should start with from the cell (1,1) to have a positive strength through out his journey to the cell (R,C).

Constraints:

1 ≤ T ≤ 5

2 ≤ R, C ≤ 500

-10^3 ≤ S[i][j] ≤ 10^3

S[1][1] = S[R][C] = 0

 

Sample Input:

3
2 3
0 1 -3
1 -2 0
2 2
0 1
2 0
3 4
0 -2 -3 1
-1 4 0 -2
1 -2 -3 0

 

Sample Output:

2
1
2

Explanation:

Case 1 : If Harry starts with strength = 1 at cell (1,1), he cannot maintain a positive strength in any possible path. He needs at least strength = 2 initially.

Case 2 : Note that to start from (1,1) he needs at least strength = 1.

题意：求走到右下角最少需要一开始为多少体力，注意：在走的过程中体力为0或者为“负”时死亡。

思维：用逆向DP表示走到该点走到右下角最少需要多少体力！！！

 1 #include<stdio.h>
 2 int DP[505][505];
 3 int min(int a,int b)
 4 {
 5     if(a<b) return a;
 6     return b;
 7 }
 8 int main()
 9 {
10     int T;
11     scanf("%d",&T);
12     while (T--)
13     {
14         int R,C;
15         scanf("%d%d",&R,&C);
16         for (int i=0;i<R;i++)
17         for (int j=0;j<C;j++)
18         scanf("%d",&DP[i][j]);
19         for (int i=C-2;i>=0;i--)//最后一行先直接更新
20         {
21             int t=DP[R-1][i];
22             DP[R-1][i]=DP[R-1][i+1]-DP[R-1][i];
23             if(DP[R-1][i]<=0)//特殊1 
24             DP[R-1][i]=1;
25             if (t<0&&DP[R-1][i+1]==0)//右下角 
26             DP[R-1][i]++;//一开始特殊2 
27         } 
28         for (int i=R-2;i>=0;i--)//最后一列先直接更新
29         {
30             int t=DP[i][C-1];
31             DP[i][C-1]=DP[i+1][C-1]-DP[i][C-1];
32             if(DP[i][C-1]<=0)//特殊1 
33             DP[i][C-1]=1;
34             if (t<0&&DP[i+1][C-1]==0)//右下角 
35             DP[i][C-1]++;//一开始特殊2 
36         }
37         for (int i=R-2;i>=0;i--)//依次更新 
38         for (int j=C-2;j>=0;j--)
39         {
40             DP[i][j]=min(DP[i][j+1],DP[i+1][j])-DP[i][j];
41             if(DP[i][j]<=0)//特殊1 
42             DP[i][j]=1;
43         }
44         printf("%d\n",DP[0][0]);
45     }
46     return 0;
47  }