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DOI: 10.4230/LIPIcs.ICALP.2020.10
URN: urn:nbn:de:0030-drops-124171
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12417/
Bénéteau, Laurine ; Chalopin, Jérémie ; Chepoi, Victor ; Vaxès, Yann
Medians in Median Graphs and Their Cube Complexes in Linear Time
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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.
BibTeX - Entry
@InProceedings{bnteau_et_al:LIPIcs:2020:12417,
author = {Laurine B{\'e}n{\'e}teau and J{\'e}r{\'e}mie Chalopin and Victor Chepoi and Yann Vax{\`e}s},
title = {{Medians in Median Graphs and Their Cube Complexes in Linear Time}},
booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
pages = {10:1--10:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-138-2},
ISSN = {1868-8969},
year = {2020},
volume = {168},
editor = {Artur Czumaj and Anuj Dawar and Emanuela Merelli},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12417},
URN = {urn:nbn:de:0030-drops-124171},
doi = {10.4230/LIPIcs.ICALP.2020.10},
annote = {Keywords: Median Graph, CAT(0) Cube Complex, Median Problem, Linear Time Algorithm, LexBFS}
}
| Keywords: | Median Graph, CAT(0) Cube Complex, Median Problem, Linear Time Algorithm, LexBFS | |
| Seminar: | 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) | |
| Issue date: | 2020 | |
| Date of publication: | 29.06.2020 |
