Graph-Based Optimization Problem (Observation-Based)

Problem: A coding problem discussed in the context of graphs; a straightforward graph-based solution is extremely suboptimal, and the intended solution requires spotting a key observation that avoids the graph traversal altogether.

Approach: Start with the naive graph-traversal solution to establish correctness, then look for structural patterns in the input that let you shortcut the traversal (e.g. the answer depends only on some aggregate/frequency property rather than actual path-finding). Once the observation is found, the optimized solution typically drops to linear or near-linear time. A harder follow-up variation asks for an even more optimized solution.