This is basically a connected components count dressed up in a physics metaphor.
Model the problem as a graph where each ball is a node and edges connect balls with Euclidean distance < d. The minimum number of triggers equals the number of connected components, which can be found using Union-Find (DSU) or BFS/DFS. For efficiency, avoid checking all pairs by using a spatial grid or sweep-line to find nearby balls.
Pro tip: Mention that the strict inequality (< d) means you must handle floating-point precision carefully, and that using squared distances avoids square roots. Also, discuss the trade-off between Union-Find with path compression (near O(n α(n))) and BFS/DFS (O(n + m)), noting that building the graph can be the bottleneck.
Confirm the problem: given points and threshold d, find minimum triggers to absorb all balls, where connections are transitive. Clarify that a trigger absorbs a connected component, so the answer is the number of components.
Represent each ball as a node. Add an edge between two balls if their Euclidean distance is strictly less than d. The problem reduces to counting connected components in this graph.
Use Union-Find (DSU) to efficiently merge connected balls and count components, or BFS/DFS if the graph is sparse. Discuss time and space complexity trade-offs.
Naively checking all pairs is O(n^2). Use a spatial grid with cell size d or a sweep-line to find neighbors within distance d, reducing time for large n.
Address empty list, single ball, and floating-point precision by comparing squared distances. Ensure strict inequality is correctly implemented.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.