Mathematical Programming

, Volume 145, Issue 1–2, pp 163–177 | Cite as

Dijkstra’s algorithm and L-concave function maximization

Full Length Paper Series A

Abstract

Dijkstra’s algorithm is a well-known algorithm for the single-source shortest path problem in a directed graph with nonnegative edge length. We discuss Dijkstra’s algorithm from the viewpoint of discrete convex analysis, where the concept of discrete convexity called L-convexity plays a central role. We observe first that the dual of the linear programming (LP) formulation of the shortest path problem can be seen as a special case of L-concave function maximization. We then point out that the steepest ascent algorithm for L-concave function maximization, when applied to the LP dual of the shortest path problem and implemented with some auxiliary variables, coincides exactly with Dijkstra’s algorithm.

Keywords

Shortest path problem Dijkstra’s algorithm Discrete concave function Steepest ascent algorithm 

Mathematics Subject Classification (2010)

90C27 68Q25 

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan
  2. 2.Graduate School of Information SciencesTohoku UniversitySendaiJapan

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