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Medians in Median Graphs and Their Cube Complexes in Linear Time

Authors Laurine Bénéteau, Jérémie Chalopin, Victor Chepoi, Yann Vaxès

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Author Details

Laurine Bénéteau
  • Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France
Jérémie Chalopin
  • Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France
Victor Chepoi
  • Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France
Yann Vaxès
  • Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France

Funding

The research on this paper was supported by the ANR project DISTANCIA (ANR-17-CE40-0015).

Acknowledgements

The authors are grateful to the anonymous referees for useful remarks that helped improving the presentation of this work.

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Laurine Bénéteau, Jérémie Chalopin, Victor Chepoi, and Yann Vaxès. Medians in Median Graphs and Their Cube Complexes in Linear Time. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.10

Abstract

The median of a set of vertices P of a graph G is the set of all vertices x of G minimizing the sum of distances from x to all vertices of P. In this paper, we present a linear time algorithm to compute medians in median graphs, improving over the existing quadratic time algorithm. We also present a linear time algorithm to compute medians in the 𝓁₁-cube complexes associated with median graphs. Median graphs constitute the principal class of graphs investigated in metric graph theory and have a rich geometric and combinatorial structure. Our algorithm is based on the majority rule characterization of medians in median graphs and on a fast computation of parallelism classes of edges (Θ-classes or hyperplanes) via Lexicographic Breadth First Search (LexBFS). To prove the correctness of our algorithm, we show that any LexBFS ordering of the vertices of G satisfies the following fellow traveler property of independent interest: the parents of any two adjacent vertices of G are also adjacent.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Computational geometry
Keywords
  • Median Graph
  • CAT(0) Cube Complex
  • Median Problem
  • Linear Time Algorithm
  • LexBFS

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