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 integers and edges represent subtracting a perfect square. I prefer starting with a Breadth-First Search approach because it guarantees finding the minimum depth immediately. We start at n and explore all possible subtractions of perfect squares. If we reach zero, the current depth is our answer. However, for larger inputs, a Dynamic Programming solution is often more predictable. We initialize an array where dp[i] stores the minimum squares for i. For each number from 1 to n, we iterate through all perfect squares less than or equal to i, updating dp[i] as the minimum of its current value and dp[i - square] + 1. For example, if n is 12, we check 1, 4, and 9. dp[12] becomes min(dp[11]+1, dp[8]+1, dp[3]+1). Given Airbnb's focus on efficiency and scalability, I would also mention checking Lagrange's Four-Square Theorem to quickly return results for specific cases without full computation, demonstrating both algorithmic knowledge and practical optimization skills.
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
Practice This Question with AI
Answer this question orally or via text and get instant AI-powered feedback on your response quality, structure, and delivery.