Citadel·Software Engineer·Technical Phone Screen·Senior

SeniorPrefer not to say

May 2026

Summary

Citadel SWE interview with a pretty gnarly data structures design problem. One question, but it had enough depth to keep me busy for a while.

Questions Asked (1)

Design a data structure that supports inserting items with weights, deleting items, and sampling a random item with probability proportional to its weight.

Algorithms & Data StructuresSystem DesignTechnical Trade-offs

Author's notes

I started with a naive approach, basically just keeping a list and recomputing cumulative sums on each sample.

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AI HintsAI Generated

Suggested Approach

Start by clarifying the requirements: operations needed (insert, delete, sample), expected time complexities, and whether weights can change. Then propose a solution using a Fenwick tree (Binary Indexed Tree) or segment tree to store cumulative weights, enabling O(log n) insert, delete, and sample operations. Explain the sampling algorithm: generate a random number between 0 and total weight, then binary search the cumulative array to find the corresponding item.

Pro tip: Mention that a naive array with linear scan gives O(n) sampling, which is inefficient; the Fenwick tree approach achieves O(log n) for all operations, demonstrating strong algorithmic maturity. Also, discuss handling edge cases like zero weights and deletions.

1. Clarify requirements and constraints

Ask about expected operation frequencies, whether weights can be updated, and if the data structure needs to be thread-safe. This shows you consider practical aspects before diving into design.

2. Propose a naive solution and its limitations

Describe a simple array of items with weights and linear scan for sampling, noting O(n) time for sampling and O(1) for insert/delete (if using a map). Highlight the inefficiency for large n.

3. Introduce an efficient data structure

Propose using a Fenwick tree (BIT) or segment tree to maintain cumulative weights. Explain that each node stores the sum of weights in its range, allowing O(log n) updates and prefix sum queries.

4. Detail the sampling algorithm

Generate a random number r in [0, total_weight). Use binary search on the Fenwick tree to find the smallest index i such that prefix_sum(i) > r. Return the item at index i.

5. Discuss trade-offs and extensions

Compare Fenwick tree vs. segment tree (Fenwick is simpler and faster for prefix sums, segment tree supports more complex queries). Mention handling deletions by setting weight to 0 and updating the tree, and potential need for coordinate compression if items are not indexed 0..n-1.

Key Points to Mention

Time complexity: O(log n) for insert, delete, and sample with Fenwick tree; O(n) for naive array.
Space complexity: O(n) for the tree and item storage.
Fenwick tree (Binary Indexed Tree) implementation details: 1-indexed array, update and query operations.
Sampling algorithm: random number generation and binary search on cumulative weights.
Handling deletions: set weight to 0 and update tree, or use a map to track active items.
Edge cases: zero weights, empty data structure, and large weights requiring 64-bit integers.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.