Citadel·Data Scientist·Technical Phone Screen·Senior

SeniorPrefer not to say

Apr 2026

Summary

Citadel Data Scientist interview with a pretty brutal dynamic programming question that had multiple layers stacked on top of each other. The kind of problem where you think you're done and then they ask you to prove it.

Questions Asked (1)

Given an array of daily stock prices and an integer k, implement a bottom-up dynamic programming solution to find the maximum profit from at most k buy-sell transactions with no overlapping positions. Return both the profit and the actual list of buy/sell day pairs, breaking ties by lexicographically smallest sequence. Achieve O(nk) time and O(k) space, and handle edge cases like k=0, fewer than 2 prices, and plateau prices. Also prove correctness using optimal substructure and explain how you avoid the naive O(n^2 k) approach.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

This wrecked me a little.

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AI HintsAI Generated

Suggested Approach

Start by clarifying the problem and edge cases, then derive the DP recurrence that tracks the best profit up to each day for each transaction count while maintaining the minimum effective buy price. Explain how to reconstruct the transaction sequence using parent pointers or by storing decisions, and prove correctness via optimal substructure and induction. Finally, analyze time and space complexity and contrast with the naive O(n^2 k) approach.

Pro tip: Emphasize that the O(k) space optimization requires careful handling of state dependencies and that tie-breaking for lexicographically smallest sequence can be resolved by consistently preferring earlier buy/sell days when profits are equal.

1. Clarify problem and edge cases

Restate the problem, confirm constraints (e.g., k can be 0, n < 2, plateau prices), and discuss expected output format for the transaction list.

2. Define DP state and recurrence

Define dp[t][i] as max profit using at most t transactions up to day i, and derive the recurrence dp[t][i] = max(dp[t][i-1], max_{j<i}(prices[i] - prices[j] + dp[t-1][j])). Explain how to maintain the max term efficiently.

3. Optimize to O(nk) time and O(k) space

Show how to compute the inner max in O(1) by keeping a running maximum of dp[t-1][j] - prices[j], and reduce space by only storing the previous transaction row and current row.

4. Reconstruct transaction sequence with tie-breaking

Describe how to store parent pointers or decisions during DP to backtrack and build the list of buy/sell pairs, ensuring lexicographically smallest sequence by preferring earlier transactions when profits tie.

5. Prove correctness and analyze complexity

Prove optimal substructure and that the DP considers all valid transaction sequences, then state time O(nk) and space O(k), and contrast with naive O(n^2 k).

Key Points to Mention

Optimal substructure: the optimal solution for t transactions up to day i either doesn't use day i or uses it as a sell day after some buy day j, reducing to t-1 transactions up to j.
State transition: dp[t][i] = max(dp[t][i-1], prices[i] + max_{j<i}(dp[t-1][j] - prices[j])).
Space optimization: only keep two rows (previous and current) of size n+1, and use a running maximum to avoid O(n) inner loop.
Tie-breaking: when multiple optimal sequences exist, choose the one with lexicographically smallest list of (buy, sell) pairs, which can be achieved by preferring earlier buy days during reconstruction.
Edge cases: k=0 returns 0 and empty list; n<2 returns 0 and empty list; plateau prices may yield zero-profit transactions but should be handled without infinite loops.
Complexity: O(nk) time and O(k) space, compared to naive O(n^2 k) which recomputes max for each j.

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