Valid Parentheses

Background Knowledge

The "Valid Parentheses" problem involves working with strings and stack data structures. A string is a sequence of characters, and in this case, we're dealing with a specific set of characters: '(', ')', '{', '}', '[', and ']'. To solve this problem, you should understand the basics of string manipulation and iteration. Additionally, familiarity with stack operations (push, pop, peek) is crucial, as a stack can be used to keep track of the opening brackets encountered so far.

The concept of a stack is essential here. A stack is a Last-In-First-Out (LIFO) data structure, meaning the last element added to the stack will be the first one to be removed. This property makes a stack particularly useful for parsing and validating nested structures, such as parentheses in an expression. By pushing opening brackets onto the stack and popping them off when a matching closing bracket is encountered, we can efficiently determine if the brackets are properly nested and matched.

Understanding the matching rules for brackets is also vital. The rules are straightforward: every opening bracket must have a corresponding closing bracket of the same type, and these brackets must be properly nested. For example, ({[]}) is valid, but ({[}]) is not. Recognizing these patterns and applying them to the input string will help you develop an effective solution.

Algorithm/Approach

The general approach to solving the "Valid Parentheses" problem involves using a stack-based algorithm. The idea is to iterate through the input string, pushing opening brackets onto the stack and popping them off when a matching closing bracket is encountered. If a closing bracket is encountered when the stack is empty or the top of the stack does not match the closing bracket, the string is invalid. This approach ensures that the brackets are properly nested and matched.

Step-by-Step Strategy

To implement the solution:

1. Initialize an empty stack to store the opening brackets.
2. Iterate through each character in the input string.
3. If the character is an opening bracket ('(', '{', or '['), push it onto the stack.
4. If the character is a closing bracket (')', '}', or ']'), check if the stack is empty or if the top of the stack does not match the closing bracket. If either condition is true, return "invalid". Otherwise, pop the opening bracket from the stack.
5. After iterating through the entire string, check if the stack is empty. If it is, the string is valid. If the stack is not empty, the string is invalid because there are unmatched opening brackets.

Common Pitfalls

When implementing the solution, watch out for the following:

• Forgetting to handle the case where the stack is empty when encountering a closing bracket.
• Not properly matching the opening and closing brackets (e.g., pushing '(' onto the stack and trying to pop it with ']').
• Failing to check if the stack is empty after iterating through the entire string, which can lead to incorrect results for strings with unmatched opening brackets.

Time & Space Complexity

The expected time complexity for this solution is $O(n)$, where $n$ is the length of the input string, because we make a single pass through the string. The space complexity is also $O(n)$, as in the worst-case scenario (e.g., a string of all opening brackets), the stack will grow to the size of the input string.