Cheapest Palindrome【区间dp】

Cheapest Palindrome

Time Limit: 2000MS		Memory Limit: 65536K
Total Submissions: 11768		Accepted: 5571

Description

Keeping track of all the cows can be a tricky task so Farmer John has installed a system to automate it. He has installed on each cow an electronic ID tag that the system will read as the cows pass by a scanner. Each ID tag's contents are currently a single string with length M (1 ≤ M ≤ 2,000) characters drawn from an alphabet of N (1 ≤ N ≤ 26) different symbols (namely, the lower-case roman alphabet).

Cows, being the mischievous creatures they are, sometimes try to spoof the system by walking backwards. While a cow whose ID is "abcba" would read the same no matter which direction the she walks, a cow with the ID "abcb" can potentially register as two different IDs ("abcb" and "bcba").

FJ would like to change the cows's ID tags so they read the same no matter which direction the cow walks by. For example, "abcb" can be changed by adding "a" at the end to form "abcba" so that the ID is palindromic (reads the same forwards and backwards). Some other ways to change the ID to be palindromic are include adding the three letters "bcb" to the begining to yield the ID "bcbabcb" or removing the letter "a" to yield the ID "bcb". One can add or remove characters at any location in the string yielding a string longer or shorter than the original string.

Unfortunately as the ID tags are electronic, each character insertion or deletion has a cost (0 ≤ cost ≤ 10,000) which varies depending on exactly which character value to be added or deleted. Given the content of a cow's ID tag and the cost of inserting or deleting each of the alphabet's characters, find the minimum cost to change the ID tag so it satisfies FJ's requirements. An empty ID tag is considered to satisfy the requirements of reading the same forward and backward. Only letters with associated costs can be added to a string.

Input

Line 1: Two space-separated integers:  N and  M 
Line 2: This line contains exactly  M characters which constitute the initial ID string 
Lines 3.. N+2: Each line contains three space-separated entities: a character of the input alphabet and two integers which are respectively the cost of adding and deleting that character.

Output

Line 1: A single line with a single integer that is the minimum cost to change the given name tag.

Sample Input

3 4
abcb
a 1000 1100
b 350 700
c 200 800

Sample Output

900

Hint

If we insert an "a" on the end to get "abcba", the cost would be 1000. If we delete the "a" on the beginning to get "bcb", the cost would be 1100. If we insert "bcb" at the begining of the string, the cost would be 350 + 200 + 350 = 900, which is the minimum.

Source

USACO 2007 Open Gold

思路：

区间dp。可以从小到大的枚举区间，dp[i][j]记录将区间 [ i , j ] 变为回文的最小花费。

由于是从小到大的枚举区间，所以只需要考虑边界是否不同即可，如果不同则可以从更小范围的dp加上最少花费转移过来。

对于相同的字符，我们可以在一边减去一个或者在另一边加上一个，都可将这段区间变为回文。所以我们就不需要考虑是删除还是添加了，只需要枚举两端点的字符即可。存的时候只需要存min(add,del)即可。

1、如果str[i]==str[j]：dp[i][j]=dp[i+1][j-1];

2、str[i]!=str[[j]：

     dp[i][j]=min(dp[i][j] , dp[i+1][j]+add[ str[i-'a'] ] );//操作左边的字符

     dp[i][j]=min(dp[i][j] , dp[i][j-1] +add[ str[j-'a'] ] );//操作右边的字符

代码：

#include<stdio.h>
#include<string.h>
#include<algorithm>
#include<math.h>
using namespace std;
#define inf 1<<29
int n,m;
char str[2005];
int add[30];
int dp[2005][2005];
int main()
{
    while(~scanf("%d%d",&n,&m))
    {
        scanf("%s",str);
        for(int i=0;i<n;i++)
        {
            char c[2];
            int x,y;
            scanf("%s%d%d",c,&x,&y);
            add[c[0]-'a']=min(x,y);
        }
        for(int k=1;k<m;k++)
        {
            for(int i=0,j=k;j<m;i++,j++)
            {
                dp[i][j]=inf;
                if(str[i]!=str[j])
                {
                    dp[i][j]=min(dp[i][j],dp[i+1][j]+add[str[i]-'a']);
                    dp[i][j]=min(dp[i][j],dp[i][j-1]+add[str[j]-'a']);
                }
                else
                    dp[i][j]=dp[i+1][j-1];
            }
        }
        printf("%d\n",dp[0][m-1]);
    }
    return 0;
}