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最新推荐文章于 2020-02-07 11:05:28 发布

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本文探讨了如何寻找给定括号序列中最长的有效括号子序列的问题，并提供了一种使用动态规划解决此问题的方法。文章包含了一个详细的C++实现案例。

Brackets

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 4686		Accepted: 2484

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

这道题用dp可以解决，dp方程可以从代码中获得。

以ac的代码：

#include<stdio.h>
#include<string.h>
#include<iostream>
using namespace std;
#define N 110

char ch[N];
int dp[N][N];

int main(){
    int length;
    while(scanf("%s",ch)!=EOF&&ch[0]!='e'){
        length=strlen(ch);
        memset(dp,0,sizeof(dp));
        for(int i=2;i<=length;i++){
            for(int j=0;j<=length-i;j++){
                dp[j][j+i-1]=dp[j][j+i-2];

                if(ch[j+i-1]=='('||ch[j+i-1]=='['){
                    continue;
                }
                for(int k=j;k<j+i-1;k++){
                    if((ch[k]=='('&&ch[j+i-1]==')')||(ch[k]=='['&&ch[j+i-1]==']')){
                        if(k!=i+j-2&&k!=j){
                            dp[j][j+i-1]=max(dp[j][j+i-1],dp[j][k-1]+dp[k+1][i+j-2]+2);
                        }
                        else if(k!=j){
                            dp[j][j+i-1]=max(dp[j][j+i-1],dp[j][k-1]+2);
                        }
                        else if(k!=i+j-2){
                            dp[j][j+i-1]=max(dp[j][j+i-1],dp[j+1][j+i-2]+2);
                        }
                        else{
                            dp[j][j+i-1]=max(dp[j][j+i-1],2);
                        }
                    }
                }
            }
        }

        printf("%d\n",dp[0][length-1]);
    }

    return 0;
}

参考链接：

http://www.cnblogs.com/ziyi--caolu/archive/2013/08/04/3236035.html