What is the optimal time complexity of solving Ugly Numbers?

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 Ugly Numbers?

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 Ugly Numbers?

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 Ugly Numbers?

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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Competitive ProgrammingEasy

Ugly Numbers

Learn how to solve the 'Ugly Numbers' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

Write a function nth_ugly_number(n) that returns the nth ugly number. Ugly numbers are positive numbers whose only prime factors are 2, 3, or 5. By convention, 1 is treated as an ugly number.

Constraints

•1 <= n <= 500

Examples

Example 1

Input

nth_ugly_number(10)

Output

Explanation

The first 10 ugly numbers are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12.

Example 2

Input

nth_ugly_number(15)

Output

Explanation

The first 15 ugly numbers are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24.

Need a Hint?

Define subproblem states, establish the recurrence relation, and use memoization (top-down) or tabulation (bottom-up).

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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