Cracking the Code: Understanding the Valid Parentheses Problem
Welcome back to the ongoing series on pivotal computer algorithms, where we dissect and solve commonly encountered coding problems. Today, I dive into the “Valid Parentheses” problem, an essential challenge often found in coding interviews and essential for understanding stack operations in computer science. Previously, we explored the “Two Sum”و “Reverse Integer”, “Palindrome Number”, “Roman to Integer”, and “Longest Common Prefix” problems, each shedding light on different algorithmic concepts. This series aims to equip you with robust problem-solving skills and a deep understanding of how algorithms function under various constraints.
About the Valid Parentheses Problem
The “Valid Parentheses” problem requires determining if a string made up solely of bracket characters — namely ‘()’, ‘[]’, and ‘{}’ — is valid. A valid string must adhere to specific rules:
- Each opening bracket must be closed by the same type of brackets.
- Opening brackets must be closed in the correct order.
- Every closing bracket must have a corresponding opening bracket of the same type.
This problem is not just about matching brackets but also about ensuring that the nested structure is correct, which is where it gains its complexity.
Examples:
- Example 1:
- Input:
s = "()" - Output:
true - Explanation: The brackets are correctly paired and nested.
- Example 2:
- Input:
s = "()[]{}" - Output:
true - Explanation: Multiple types of brackets are used, but all are correctly paired and ordered.
- Example 3:
- Input:
s = "(]" - Output:
false - Explanation: The brackets are incorrectly nested, as the ‘)’ should not come directly after a ‘[‘.
This problem is a classic example of using data structures like stacks to keep track of bracket types and their order, providing a foundational exercise in understanding stack operations within the realm of computer science algorithms.
Solutions for “Valid Parentheses”
1. Simplest Solution: Using a Stack
The stack is an ideal data structure for this type of problem because it allows for checking the balancing of symbols in a last-in, first-out manner.
def isValid(s):
# Mapping of closing to opening brackets
bracket_map = {')': '(', '}': '{', ']': '['}
stack = []
for char in s:
if char in bracket_map:
top_element = stack.pop() if stack else '#'
if bracket_map[char] != top_element:
return False
else:
stack.append(char)
return not stack # If stack is empty, all brackets are balancedOptimized Solution: Early Exit and Stack Optimization
This solution optimizes the use of the stack by implementing an early exit strategy and reducing unnecessary operations.
def isValid_optimized(s):
# Early exit if the string length is odd
if len(s) % 2 != 0:
return False
bracket_map = {')': '(', '}': '{', ']': '['}
stack = []
for char in s:
if char in bracket_map:
top_element = stack.pop() if stack else '#'
if bracket_map[char] != top_element:
return False
else:
stack.append(char)
return not stackTime and Memory Complexity
Simplest Solution:
- Time Complexity: O(n) — where nnn is the length of the string. Each character in the string is processed once.
- Space Complexity: O(n)— in the worst case, all characters are opening brackets, and they will all go into the stack.
Optimized Solution:
- Time Complexity: O(n)— similar to the simplest solution but with the potential for early termination if the string length is odd.
- Space Complexity: O(n)— same as the simplest solution, although it often uses less space due to early exits.
Explanation of the Stack Usage Method
The stack is a fundamental data structure in computer science used for various types of problems, especially those involving nested structures or sequence reversal. In the context of the “Valid Parentheses” problem, the stack is particularly useful for tracking unmatched opening brackets. As we iterate through the string, every time we encounter a closing bracket, we check the element at the top of the stack. If it matches the corresponding opening bracket, we pop the stack. If not, or if the stack is empty (indicating no matching opening bracket), the string is immediately invalid. This method efficiently manages the sequence and ensures that each closing bracket has a preceding matching opening bracket in the correct order.
Conclusion
The “Valid Parentheses” problem is a classic example of using a stack to manage related pairs in sequence. Both the simplest and the optimized solutions provide efficient means to validate the structure of strings containing brackets. This problem not only reinforces the utility of stacks in solving real-world problems but also highlights important algorithmic strategies like early termination and conditional processing based on input characteristics. These concepts are essential for writing clean, efficient, and maintainable code in various programming and scripting tasks.
Visit my GitHub repository for code snippets and more
This exploration not only prepares us for interviews but also sharpens our problem-solving skills in developing efficient and effective software solutions.
