Dijkstra算法并非贪心算法：一种全局到局部的正确性证明
Dijkstra Is NOT Greedy: A Global-to-Local Proof of Correctness
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中文总结 AI 辅助
本文挑战Dijkstra算法是贪心算法的传统观点，提出三步排除法从全局到局部证明其正确性，为其提供了更简洁的概念解释。
中文摘要 AI 辅助
Dijkstra算法几乎被普遍归类为经典贪心算法。本文挑战这一传统解读，从不同视角给出简洁直接的正确性证明。与该算法反复做出局部选择从而达到全局最优的普遍直觉相反，我们证明其逻辑方向恰好相反：每次迭代中，算法在所有终点未被求解的路径中识别出一条全局最短路径，该全局最短路径的终点因此被作为“局部”最短路径问题求解。换言之，局部最短路径是全局最小值的直接结果。基于此观察，我们提出三步排除法，将Dijkstra迭代解释为全局路径空间的确定性收缩。所得证明突出了最优子结构、边界状态约简和动态规划结构，为传统贪心解释提供了概念上更简洁的替代方案。
英文摘要
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.
发表机构
- Hangzhou Normal University（杭州师范大学）
机构由 AI 辅助整理，请以论文原文为准。
