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Dijkstra’s algorithm and L-concave function maximization

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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.

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Notes

  1. A polyhedral \(L^{\natural }\)-concave function satisfying the condition (12) is called a polyhedral L-concave function (see, e.g., [12]).

  2. The value of \(\hat{\rho }^{\prime }(p;\chi _X)\) admits an explicit formula, which is omitted here for simplicity of the description.

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Acknowledgments

This research is partially supported by KAKENHI (21360045, 21740060) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Global COE program “The Research and Training Center for New Development in Mathematics,” and the Aihara Project, the FIRST program from the Japan Society for the Promotion of Science (JSPS).

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Authors and Affiliations

  1. Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656, Japan

    Kazuo Murota

  2. Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

    Akiyoshi Shioura

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  1. Kazuo Murota
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  2. Akiyoshi Shioura
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Correspondence to Akiyoshi Shioura.

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Murota, K., Shioura, A. Dijkstra’s algorithm and L-concave function maximization. Math. Program. 145, 163–177 (2014). https://doi.org/10.1007/s10107-013-0643-2

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  • DOI: https://doi.org/10.1007/s10107-013-0643-2

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