题目链接：http://poj.org/problem?id=2955

Brackets

Description

We give the following inductive definition of a “regular brackets” sequence:

• the empty sequence is a regular brackets sequence,
• if s is a regular brackets sequence, then (s) and [s] are regular brackets sequences, and
• if a and b are regular brackets sequences, then ab is a regular brackets sequence.
• no other sequence is a regular brackets sequence

For instance, all of the following character sequences are regular brackets sequences:

(), [], (()), ()[], ()[()]

while the following character sequences are not:

(, ], )(, ([)], ([(]

Given a brackets sequence of characters a₁a₂ … a_n, your goal is to find the length of the longest regular brackets sequence that is a subsequence of s. That is, you wish to find the largest m such that for indices i₁, i₂, …, i_m where 1 ≤ i₁ < i₂ < … < i_m ≤ n, a_i1a_i2 … a_im is a regular brackets sequence.

Given the initial sequence ([([]])], the longest regular brackets subsequence is [([])].

Input

The input test file will contain multiple test cases. Each input test case consists of a single line containing only the characters (, ), [, and ]; each input test will have length between 1 and 100, inclusive. The end-of-file is marked by a line containing the word “end” and should not be processed.

Output

For each input case, the program should print the length of the longest possible regular brackets subsequence on a single line.

Sample Input

((()))
()()()
([]])
)[)(
([][][)
end

Sample Output

6
6
4
0
6

Source

Stanford Local 2004  
合法序列就是括号可以两两匹配的。
思路就是区间DP的思想了。
我的代码是采用记忆化搜索写出来的。  
状态转移方程dp[i][j]=max(dp[i+1][j],2+dp[i+1][k-1]+dp[k+1][j])       i和j是一对括号 && i<k<=j
其实就是看第i个括号的情况。
舍弃第i个括号，就是dp[i+1][j],或者是往后找和i匹配的，然后就分成了两部分了。

//============================================================================
// Name        : POJ.cpp
// Author      : 
// Version     :
// Copyright   : Your copyright notice
// Description : Hello World in C++, Ansi-style
//============================================================================

#include <iostream>
#include <string.h>
#include <stdio.h>
#include <algorithm>
using namespace std;
const int MAXN=110;
char str[MAXN];
int dp[MAXN][MAXN];
int solve(int i,int j)
{
    if(dp[i][j]!=-1)return dp[i][j];
    if(j<=i)return dp[i][j]=0;
    else if(j==i+1)
    {
        if( (str[i]=='('&&str[j]==')')||(str[i]=='['&&str[j]==']') )return dp[i][j]=2;
        else return dp[i][j]=0;
    }
    dp[i][j]=solve(i+1,j);
    for(int k=i+1;k<=j;k++)
        if( (str[i]=='('&&str[k]==')')||(str[i]=='['&&str[k]==']') )
            dp[i][j]=max(dp[i][j],2+solve(i+1,k-1)+solve(k+1,j));
    return dp[i][j];
}

int main()
{
//    freopen("in.txt","r",stdin);
//    freopen("out.txt","w",stdout);
    while(scanf("%s",str)==1)
    {
        if(strcmp(str,"end")==0)break;
        memset(dp,-1,sizeof(dp));
        int n=strlen(str);
        printf("%d\n",solve(0,n-1));
    }
    return 0;
}