My first instinct was to just simulate it step by step, which is probably the right call, but I spent too long second-guessing whether there was some closed-form trick.
Model the problem as a simulation over time, tracking each person's position and the watcher's state (position and direction). At each time step, update the watcher's direction based on the schedule, then move each person right by one unit unless they are strictly on the watcher's facing side. Count how many reach or pass L after T steps.
Pro tip: Clarify edge cases upfront: people exactly at W are unaffected, and the watcher's direction changes at specified times—likely before movement in that step. Also, consider if multiple people can occupy the same position; typically they can, so no collision handling is needed.
Read N, L, T, initial positions, W, initial direction, and reversal times. Initialize each person's position and the watcher's current direction.
For t from 1 to T: update watcher's direction if t is a reversal time; then for each person, if they are not strictly on the watcher's facing side, move them right by 1.
After each move, check if a person's position is >= L. If so, mark them as arrived (or remove them from further movement) and increment a counter.
After T steps, output the number of people who have reached or passed L.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the new problem: the watcher's direction changes only at discrete switch times, but movement is continuous, so the watcher's position evolves linearly between switches. Then, adapt the step-based solution by replacing discrete position updates with continuous-time integration, while keeping the same decision points for direction changes. Finally, discuss the algorithmic implications, such as event-driven simulation or closed-form position calculations, and any trade-offs in complexity or precision.
Pro tip: Emphasize that the core insight—the watcher only makes decisions at discrete times—remains unchanged, so you can reuse the same decision logic and simply interpolate positions between switch times. This shows you can separate the problem's invariant structure from its implementation details.
Confirm that the watcher moves continuously at a constant speed (or known velocity) and only changes direction at discrete switch times. Clarify whether switch times are predetermined or adaptive.
In the step-based version, positions update at each step; now, positions must be computed as continuous functions of time between switches. The decision logic at switch times remains the same.
Use event-driven simulation: maintain the watcher's position and direction, and jump from one switch time to the next, updating position via linear interpolation. Alternatively, derive closed-form expressions for position at any time.
Replace discrete step loops with a priority queue of switch times or a sorted list of events. For each interval, compute the watcher's trajectory and check for interactions (e.g., with targets) using continuous collision detection.
Discuss complexity: event-driven may be O(k log k) for k switches vs. O(n) for n steps. Address precision issues, simultaneous events, and whether the watcher's speed is constant or variable.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.