Cheapest Palindrome（DP）

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.

一个字符串，每个字母增添删除都有花费，问最少花费多少，通过增添删除使其成为回文串。

记忆化搜索撒。l和r表示区间
如果s[i] == s[j]那么不用花费，直接到l + 1,r + 1
否则，对于l来说，要么在右面加一个s[i]，要么删除s[i]，然后转移到l + 1,r
要么同理改变r，然后转移到l,r - 1
在每一个情况中，比较增加和删除同一个字符的最小值，再加上递归的结果，即可，最后两种情况再比较一下。
dp[l][r] = min(dfs(l + 1,r) + min(add,del),dfs(l,r - 1) + min(add,del));

#include<cstdio>
#include<cstring>
#include<iostream>
#include<queue>
#include<vector>
#include<algorithm>
#include<string>
#include<cmath>
#include<set>
#include<map>
#include<vector>
#include<stack>
#include<utility>
#include<sstream>
using namespace std;
typedef long long ll;
const int inf = 0x3f3f3f3f;
const int maxn = 1005;
int n,m,dp[2005][2005];
char s[2005];
map<char,int>ad,de;
int dfs(int l,int r)
{
    if(dp[l][r] != -1)return dp[l][r];
    if(l >= r)return dp[l][r] = 0;
    if(s[l] == s[r])
        return dp[l][r] = dfs(l + 1,r - 1);
    else
        return dp[l][r] = min(dfs(l,r - 1) +  min(ad[s[r]],de[s[r]]),dfs(l + 1,r) + min(ad[s[l]],de[s[l]]));
}
int main()
{
    #ifdef LOCAL
    freopen("C:\\Users\\ΡΡ\\Desktop\\in.txt","r",stdin);
    //freopen("C:\\Users\\ΡΡ\\Desktop\\out.txt","w",stdout);
    #endif // LOCAL
    scanf("%d%d",&n,&m);
    scanf("%s",s);
    for(int i = 1;i <= n;i++)
    {
        int add,del;
        char c[10];
        scanf("%s%d%d",&c,&add,&del);
        ad[c[0]] = add;
        de[c[0]] = del;
    }
    memset(dp,-1,sizeof(dp));
    printf("%d\n",dfs(0,m - 1));
    return 0;
}