Perfect Squares
Given a positive integer $n$, find the least number of perfect square numbers (e.g., 1, 4, 9, 16, ...) which sum up to $n$. Use BFS or DP.
Why Interviewers Ask This
Airbnb asks this to evaluate a candidate's ability to model real-world optimization problems, such as minimizing transaction costs or resource allocation. They assess proficiency in dynamic programming and breadth-first search, specifically testing if you can identify overlapping subproblems and make optimal local choices that lead to a global minimum efficiently.
How to Answer This Question
1. Clarify the problem constraints: Confirm if n is large enough to require O(n) space or if mathematical optimizations like Lagrange's Four-Square Theorem apply. 2. Propose two distinct strategies: First, outline the BFS approach as it naturally finds the shortest path (minimum squares) in an unweighted graph of remainders. Second, describe the Dynamic Programming approach using a bottom-up array where dp[i] represents the min squares for i. 3. Analyze trade-offs: Explain that BFS may be faster for small depths but uses more memory, while DP is space-efficient but requires iterating through all previous states. 4. Implement the chosen solution: Write clean code, initializing the base case dp[0]=0 and iterating through perfect squares to update the array. 5. Optimize: Mention mathematical shortcuts for edge cases where the answer is guaranteed to be 1, 2, or 3 based on number theory properties.
Key Points to Cover
- Explicitly identifying this as a shortest-path problem in an implicit graph
- Demonstrating clear understanding of overlapping subproblems for DP
- Comparing time and space complexity between BFS and DP approaches
- Handling edge cases like perfect square inputs returning 1 immediately
- Mentioning mathematical optimizations like Legendre's Three-Square Theorem
Sample Answer
To solve the Perfect Squares problem, I would first analyze the requirements to ensure we are minimizing the count of numbers summing to n. This is essentially finding the shortest path in a graph where nodes are integer…
Common Mistakes to Avoid
- Using a naive recursive solution without memoization, leading to exponential time complexity
- Failing to initialize the DP array correctly, causing incorrect base case handling
- Not considering the worst-case scenario where n is very large and requiring O(sqrt(n)) iteration
- Ignoring mathematical properties that could reduce the search space significantly before coding
Sound confident on this question in 5 minutes
Answer once and get a 30-second AI critique of your structure, content, and delivery. First attempt is free — no signup needed.