What is the Longest Valid Parentheses Problem?

The Longest Valid Parentheses problem is a classic question you're likely to encounter in technical interviews, especially when evaluating a candidate's ability to solve string manipulation challenges. It asks you to find the length of the longest valid (well-formed) parentheses substring in a given string consisting of '(' and ')'.

For example, for the input string s = "(()())", the longest valid substring is the entire string itself, and the output would be 6. In contrast, if the input were s = "()(())", the answer would still be 6.

Understanding Valid Parentheses

A valid parentheses string is one where every opening parenthesis '(' has a corresponding closing parenthesis ')', and they are correctly nested. For instance:

• Valid: (), (()), (()())
• Invalid: ((), ())(

Approach to the Problem

There are several ways to approach this problem. The most intuitive methods include using a stack or employing dynamic programming. Below, we'll explore both methods.

Method 1: Using a Stack

The stack-based approach is often the simplest to implement. A stack can help track the indices of the parentheses. Here's how it works:

1. Initialize a stack that starts with -1 to handle the base case.
2. Iterate through the string, pushing the index of every '(' onto the stack.
3. When encountering a ')', pop the top of the stack. If the stack becomes empty after popping, push the current index onto the stack. Otherwise, calculate the length of the valid substring by subtracting the current index from the index at the new top of the stack.

Here’s the code for this approach:

PythonCopy code

Method 2: Using Dynamic Programming

The dynamic programming approach involves creating an array dp that stores the lengths of valid parentheses substrings ending at each index.

1. Initialize dp[i] to 0 for all i.
2. Traverse the string starting from the second character. If the character is ')' and the previous character is '(', update dp[i] as dp[i-2] + 2.
3. If the previous character is ')' and if the character before the last valid substring is an opening parenthesis, update accordingly.

Here’s the corresponding code:

PythonCopy code

Complexity Analysis

For both methods, the time complexity is O(n) since we traverse the string once. The space complexity is O(n) for the stack and O(n) for the dynamic programming array.

Understanding these methods equips you with the tools to answer the longest valid parentheses question effectively in your tech interviews. By practicing these techniques, you can improve your problem-solving skills and prepare for similar challenges.