First thing I did was ask whether the pickup locations had to sit on integer coordinates or be chosen from existing people's positions, which felt like the right move.
Start by clarifying the problem constraints (N, K, coordinate ranges) and discussing the NP-hardness of the continuous K-median problem. Then present a practical solution using Lloyd's algorithm with Manhattan distance (K-medians), and optionally mention approximation algorithms or heuristics for large-scale settings.
Pro tip: Acknowledge that Manhattan distance makes the problem separable in x and y, so you can optimize each dimension independently using 1D K-medians, which is a key insight for efficient solutions.
Ask about N, K, coordinate ranges, and whether pickup points can be placed anywhere (continuous) or at discrete locations. Note that the problem is NP-hard for general K.
For small N and K, mention dynamic programming or exhaustive search. For large-scale, propose approximation algorithms like K-medians clustering or local search.
Explain that Manhattan distance decomposes into independent x and y components, so the problem reduces to two 1D K-median problems, which can be solved optimally in polynomial time using dynamic programming.
Describe Lloyd's algorithm adapted for Manhattan distance (K-medians): initialize centroids, assign points to nearest centroid, update centroids to median of assigned points, and iterate until convergence.
Compare time complexity, convergence guarantees, and quality of solutions. Discuss how to handle large N with sampling or streaming approaches, and mention evaluation metrics.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.