Dijkstra Is NOT Greedy: A Global-to-Local Proof of Correctness

Abstract

Dijkstra's algorithm is almost universally classified as a canonical greedy algorithm. This paper challenges that conventional interpretation and presents a simple, direct proof of correctness from a different viewpoint. Contrary to the widespread intuition that the algorithm repeatedly makes a local choice and thereby reaches a global optimum, we show that the logical direction can be read in exactly the opposite way: at each iteration, the algorithm identifies a globally shortest path among all paths whose destinations remain unsolved, and the endpoint of that globally shortest path is therefore solved as a ``local''shortest-path problem. In other words, the local shortest path is obtained as an immediate consequence of a global minimum. Based on this observation, we formulate a Three-Step Exclusion Method that interprets Dijkstra's iteration as deterministic contraction of the global path space. The resulting proof highlights optimal substructure, boundary-state reduction, and dynamic-programming structure, and offers a conceptually simple alternative to the usual greedy explanation.