Dijkstra's Algorithm

Bellman-Ford handles negative weights but pays $O(VE)$ for that generality. When every edge weight is guaranteed non-negative — true for almost every genuinely physical or economic "cost" (time, distance, a bid-ask spread, a transaction cost) — Dijkstra's algorithm exploits that guarantee with a greedy relaxation order, driven by a min-priority-queue, achieving $O((V+E)\log V)$ with a binary heap. This note derives the algorithm, proves its correctness rests entirely on non-negativity, and shows precisely why negative weights break it.