eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-29
10:1
10:17
10.4230/LIPIcs.ICALP.2020.10
article
Medians in Median Graphs and Their Cube Complexes in Linear Time
Bénéteau, Laurine
1
Chalopin, Jérémie
1
Chepoi, Victor
1
Vaxès, Yann
1
Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol168-icalp2020/LIPIcs.ICALP.2020.10/LIPIcs.ICALP.2020.10.pdf
Median Graph
CAT(0) Cube Complex
Median Problem
Linear Time Algorithm
LexBFS