🤖 AI Summary
This paper challenges the conventional classification of Dijkstra’s algorithm as a greedy method, rectifying the widespread misconception that local choices inherently lead to global optimality. It proposes a three-step elimination method and a deterministic path-space contraction strategy, which are formally proven in conjunction with optimal substructure and boundary state reduction theory. These analyses reveal that the algorithm fundamentally constitutes a dynamic programming process deriving local solutions from the global shortest path. By establishing this paradigm-shifting "global-to-local" perspective, the study constructs a more concise correctness proof and clarifies the algorithm's essential nature. Ultimately, this work provides a novel theoretical interpretive framework for shortest-path problems in graph theory.
📝 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.