Longest Valid Parentheses

This challenge asks you to find the length of the longest valid (well-formed) parentheses substring within a given string containing only '(' and ')' characters. Valid parentheses must be properly nested and balanced; for example, "()" is valid, but ")(" or "((" are not. This problem is a classic dynamic programming or stack-based problem, frequently encountered in coding interviews to assess problem-solving skills.

Problem Description

Given a string s consisting only of '(' and ')' characters, determine the length of the longest valid (well-formed) parentheses substring. A valid parentheses substring is one where each opening parenthesis has a corresponding closing parenthesis and they are properly nested.

What needs to be achieved:

• Identify the longest contiguous substring within s that represents a valid parentheses sequence.
• Return the length of this longest valid substring.

Key Requirements:

• The input string s will only contain '(' and ')' characters.
• The solution should handle cases with no valid parentheses, single parentheses, and complex nested structures.

Expected Behavior:

The function should return an integer representing the length of the longest valid parentheses substring. If no valid substring exists, it should return 0.

Edge Cases to Consider:

• Empty string: Should return 0.
• String with only opening or closing parentheses: Should return 0.
• String with unbalanced parentheses: Should correctly identify the longest valid substring, not just any substring.
• Nested parentheses: "(())" is valid, and should be handled correctly.
• Adjacent valid substrings: "()()" should return 4.

Examples

Example 1:

Input: "(()"
Output: 2
Explanation: The longest valid substring is "()", which has length 2.

Example 2:

Input: ")()())"
Output: 4
Explanation: The longest valid substring is "()()", which has length 4.

Example 3:

Input: ""
Output: 0
Explanation: The input string is empty, so there are no valid parentheses.

Example 4:

Input: "()(())"
Output: 6
Explanation: The entire string is a valid parentheses sequence.

Example 5:

Input: "(()())"
Output: 6
Explanation: The entire string is a valid parentheses sequence.

Constraints

• The length of the input string s will be between 0 and 100,000.
• The input string s will only contain '(' and ')' characters.
• The solution should ideally have a time complexity of O(n), where n is the length of the string. A solution with O(n^2) time complexity is acceptable, but less desirable.
• The solution should have a space complexity of O(n) or better.

Notes

Consider using a stack to keep track of the indices of unmatched opening parentheses. Alternatively, dynamic programming can be used to store the lengths of valid substrings ending at each index. Think about how to handle the case where a closing parenthesis is encountered without a corresponding opening parenthesis. Pay close attention to edge cases and ensure your solution handles them correctly. The key is to efficiently track the validity of parentheses as you iterate through the string.