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Two Sigma·Machine Learning Engineer·Technical Phone Screen·Senior

Senior
Jul 2026

Summary

Two Sigma MLE interview with two algorithmic problems that felt more like competitive programming than anything ML-specific. The questions were genuinely hard and required careful thought on complexity, not just getting a working solution.

Questions Asked (2)

Q1

Given a rooted tree represented as a parent array and a weight array, cut exactly one edge to split the tree into two connected components. Return the minimum absolute difference between the sum of weights in each component.

Algorithms & Data Structures
Author's notes

My first instinct was to compute subtree sums via a post-order DFS, which is the right move.

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AI HintsAI Generated

Suggested Approach

Compute the total sum of all node weights, then perform a post-order traversal to compute subtree sums. For each edge (except the root edge), consider cutting it, which yields component sums equal to the subtree sum and total minus subtree sum; track the minimum absolute difference.

Pro tip: Clarify that the parent array defines a rooted tree and that cutting an edge disconnects a subtree from the rest. Emphasize that you only need to consider edges where the child is not the root, and that the total sum can be computed in the same traversal to avoid a second pass.

1. Understand the problem and clarify assumptions

Confirm that the tree is rooted, the parent array defines edges from parent to child, and that cutting an edge splits the tree into a subtree and the remaining tree. Ask if weights can be negative (though typically they are non-negative).

2. Compute total sum and subtree sums

Calculate the total sum of all weights. Then, using a post-order DFS or BFS, compute the sum of weights for each subtree rooted at every node.

3. Iterate over possible cuts

For each node except the root, consider cutting the edge between it and its parent. The two component sums are the subtree sum of that node and total sum minus that subtree sum. Compute the absolute difference.

4. Track the minimum difference

Keep a running minimum of the absolute differences encountered. Return this minimum after checking all edges.

5. Analyze complexity and edge cases

Discuss time and space complexity (O(n) time, O(n) space). Handle edge cases like a tree with only one node (no edge to cut) or all weights equal.

Key Points to Mention

  • Tree representation: parent array and weight array; root identification (parent[i] == -1 or i == 0).
  • Subtree sum computation via post-order traversal (DFS or BFS).
  • Total sum and the relationship: component sums are subtree_sum and total_sum - subtree_sum.
  • Minimizing absolute difference by iterating over all valid edges (all nodes except root).
  • Time complexity O(n) and space complexity O(n) due to recursion/stack or adjacency list.
  • Edge cases: single node tree (no edge to cut), negative weights (if allowed), and large trees requiring iterative traversal to avoid stack overflow.

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

Q2

Design an IPO share allocation system: given a total share count and a list of bids with bidder ID, shares requested, price, and timestamp, allocate shares by processing price tiers in descending order. Within a tier, use round-robin ordered by ascending timestamp, removing bidders once fulfilled. Return the per-bid allocation in input order.

Algorithms & Data StructuresSystem Design
Author's notes

This one took me a while to even parse.

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AI HintsAI Generated

Suggested Approach

Clarify the allocation rules and edge cases, then propose an efficient algorithm using sorting and a queue-based round-robin within each price tier. Walk through a small example to validate the approach, and analyze time and space complexity.

Pro tip: Emphasize that within a price tier, the round-robin order is based on ascending timestamp, and bidders are removed once fully allocated; this ensures fairness and prevents over-allocation. Also, discuss how to handle ties in timestamp (e.g., by bidder ID) for deterministic output.

1. Clarify Requirements and Edge Cases

Ask questions to confirm allocation rules: e.g., what if total demand is less than supply? How to handle ties in price or timestamp? Should partial allocations be allowed? Confirm that output must be in input order.

2. Design the Algorithm

Sort bids by price descending, then group by price tier. For each tier, sort bidders by timestamp ascending and use a queue to allocate shares in round-robin, removing bidders when fulfilled. Continue until all shares are allocated or all bids processed.

3. Walk Through an Example

Use a small example with multiple price tiers and bidders to demonstrate the round-robin allocation, showing how shares are distributed and bidders removed. Verify that the total allocated equals the total shares (if demand suffices).

4. Analyze Complexity and Optimize

Discuss time complexity: sorting O(n log n), allocation O(n + S) where S is total shares (or O(n) if using efficient queue operations). Space complexity O(n). Mention potential optimizations for large S.

5. Handle Edge Cases and Return Result

Address edge cases: all bids same price, insufficient demand, zero shares, etc. Map allocations back to original input order and return the list.

Key Points to Mention

  • Sorting bids by price descending and grouping into tiers.
  • Within a tier, round-robin allocation ordered by ascending timestamp.
  • Removing bidders from the round-robin once their requested shares are fully allocated.
  • Handling ties in price and timestamp deterministically (e.g., by bidder ID).
  • Time and space complexity analysis, including the impact of total share count.
  • Returning allocations in the original input order.

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