Question

Given question:

A string of parentheses is said to be
balanced if the left- and right-parentheses in the string can be paired off properly. For example, the strings "(())" and "()()" are both balanced, while the string "(()(" is not
balanced.
Given a string S of length n consisting of parentheses, suppose you want to find the longest subsequence of S that is balanced. Using dynamic programming, design an algorithm that finds the longest balanced subsequence of S in O(n^3) time.

My approach:
Suppose given string: S[1 2 … n]
A valid sub-sequence can end at S[i] iff S[i] == ‘)’ i.e. S[i] is a closing brace and there exists at least one unused opening brace previous to S[i]. which could be implemented in O(N).

#include<iostream>
using namespace std;
int main(){
    string s;
    cin >> s;
    int n = s.length(), o_count = 0, len = 0;
    for(int i=0; i<n; ++i){
        if(s[i] == '('){
            ++o_count;
            continue;
        }
        else if(s[i] == ')' && o_count > 0){
            ++len;
            --o_count;
        }
    }
    cout << len << endl;
    return 0;
}

I tried a couple of test cases and they seem to be working fine. Am I missing something here? If not, then how can I also design an O(n^3) Dynamic Programming solution for this problem?

This is the definition of subsequence that I’m using.

Thanks!

Answer 1

For O(n^3) DP this should work I think:

dp[i, j] = longest balanced subsequence in [i .. j]
dp[i, i] = 0
dp[i, i + 1] = 2 if [i, i + 1] == "()", 0 otherwise

dp[i, j] = max{dp[i, k] + dp[k + 1, j] : j > i + 1} in general

This can be implemented similar to how optimal matrix chain multiplication is.

Your algorithm also seems correct to me, see for example this problem:

http://xorswap.com/questions/107-implement-a-function-to-balance-parentheses-in-a-string-using-the-minimum-nu

Where the solutions are basically the same as yours.

You are only ignoring the extra brackets, so I don’t see why it wouldn’t work.