What is the optimal time complexity of solving Generate balanced parentheses?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Generate balanced parentheses?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Generate balanced parentheses?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Generate balanced parentheses?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Python BasicsEasy

Generate balanced parentheses

Learn how to solve the 'Generate balanced parentheses' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

Write a function generate_parentheses(n) that generates all combinations of n pairs of well-formed (balanced) parentheses. Return the result as a sorted list of strings. A string of parentheses is balanced if every opening '(' has a corresponding closing ')' and they are properly nested.

Constraints

•1 <= n <= 8

Examples

Example 1

Input

n = 3

Output

['((()))', '(()())', '(())()', '()(())', '()()()']

Explanation

All 5 valid ways to arrange 3 pairs of parentheses.

Example 2

Input

n = 1

Output

['()']

Explanation

Only one way to arrange 1 pair of parentheses.

Example 3

Input

n = 2

Output

['(())', '()()']

Explanation

Two valid arrangements: nested or sequential.

Need a Hint?

Use simple arithmetic operators (like modulo `%`, division `//`), conditional checks, or loops to inspect number properties.

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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