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Hudson River Trading·Machine Learning Engineer·Technical Phone Screen·Senior

Senior
Apr 2026

Summary

HRT ML Engineer interview with a pretty gnarly continuous-time simulation problem. The math was doable but the edge cases kept piling up in ways I didn't fully anticipate going in.

Questions Asked (1)

Q1

A watcher and N people all move along a line at unit speed. The watcher changes direction at known timestamps. Whenever someone falls in the watcher's line of sight, their position is frozen; otherwise they advance normally. Compute each person's final position at time T.

Algorithms & Data StructuresSystem DesignTechnical Trade-offs
Author's notes

The core idea clicked pretty fast: partition the timeline at each direction-change timestamp so you get segments where the watcher's direction is constant, then solve each segment analytically.

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AI HintsAI Generated

Suggested Approach

Model the problem as a sweep over time, tracking the watcher's direction and each person's frozen status. Use a priority queue or sorted events to efficiently determine when each person enters or exits the watcher's line of sight, updating positions accordingly. Finally, compute the total frozen time for each person and subtract from T to get their final position.

Pro tip: Clarify the watcher's line of sight: it likely extends infinitely in the current direction, so only people ahead of the watcher are affected. Also, consider edge cases like simultaneous events and people exactly at the watcher's position.

1. Understand the problem and define states

Clarify that the watcher moves at unit speed and changes direction at given timestamps. Each person moves at unit speed unless frozen. Define the watcher's line of sight as a ray in the current direction.

2. Identify events and conditions for freezing

Determine when a person enters or exits the watcher's line of sight. This depends on relative positions and the watcher's direction. Compute these event times.

3. Simulate or sweep through time

Process events in chronological order, updating the watcher's direction at given timestamps and toggling each person's frozen status. Maintain each person's position and frozen time.

4. Compute final positions

For each person, final position = initial position + (T - total_frozen_time) * speed, where speed is 1. Ensure all events up to time T are processed.

5. Analyze complexity and optimize

Discuss time and space complexity. Use efficient data structures (e.g., priority queue) to handle up to N people and M direction changes. Consider if O((N+M) log N) is achievable.

Key Points to Mention

  • Event-based simulation: treat direction changes and line-of-sight entries/exits as events.
  • Relative motion: transform to watcher's frame to simplify line-of-sight conditions.
  • Frozen time accumulation: track total time each person is frozen, not just final state.
  • Edge cases: simultaneous events, people at the watcher's position, direction changes exactly when someone enters/exits.
  • Complexity analysis: aim for O((N+M) log N) using a priority queue or balanced BST.
  • System design considerations: scalability for large N and M, potential for parallelization or streaming.

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