Number of Boomerangs
Direct Answer
Given $n$ points in the plane, return the number of 'boomerangs'. A boomerang is a triplet of points $(i, j, k)$ such that the distance between $i$ and $j$ is equal to the distance between $i$ and $k$. Use a HashMap for counting distances.
Why Interviewers Ask This
Apple interviewers ask this to evaluate your ability to optimize brute-force solutions using hash maps for constant-time lookups. They specifically test if you can recognize geometric patterns, handle edge cases like duplicate points or floating-point precision, and efficiently calculate combinatorial permutations without redundant calculations.
How to Answer This Question
1. Clarify the problem by defining a boomerang as an ordered triplet where the center point is equidistant from two others, noting that order matters for (j, k). 2. Propose a brute-force O(n^3) approach first to establish a baseline, then immediately pivot to an optimized O(n^2) solution using a HashMap. 3. Explain the core logic: iterate through each point acting as the 'center', calculate distances to all other points, and store frequency counts in a map. 4. Detail the calculation formula: for any distance with count 'c', the number of valid permutations is c * (c - 1), adding this to your total result. 5. Discuss complexity analysis, emphasizing space trade-offs, and mention handling potential floating-point issues by using squared Euclidean distance to avoid precision errors.
Key Points to Cover
- Recognize that squared Euclidean distance prevents floating-point precision errors
- Apply the permutation formula n * (n-1) rather than combinations when order matters
- Demonstrate clear transition from O(n^3) brute force to O(n^2) hash map optimization
- Explicitly state that the center point must be distinct from the two outer points
- Explain space complexity trade-offs between the input array and the hash map
Sample Answer
To solve the Number of Boomerangs problem efficiently, I would first clarify that we are looking for ordered triplets (i, j, k) where the distance from i to j equals the distance from i to k. A naive solution checking al…
Common Mistakes to Avoid
- Using standard combinations instead of permutations, ignoring that (j,k) and (k,j) are distinct boomerangs
- Calculating actual square roots for distances, introducing unnecessary floating-point rounding errors
- Failing to reset the hash map for each new center point, leading to incorrect aggregated counts
- Overlooking edge cases where multiple points share the exact same coordinates
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.