Medians in median graphs and their cube complexes in linear time

Autor: Laurine Bénéteau, Jérémie Chalopin, Victor Chepoi, Yann Vaxès
Přispěvatelé: Algorithmique, Combinatoire et Recherche Opérationnelle (ACRO), Laboratoire d'Informatique et Systèmes (LIS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Algorithmique Distribuée (DALGO), Artur Czumaj and Anuj Dawar and Emanuela Merelli, ANR-17-CE40-0015,DISTANCIA,Théorie métrique des graphes(2017), Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)
Rok vydání: 2019
Předmět:
FOS: Computer and information sciences
General Computer Science
Median Problem
Discrete Mathematics (cs.DM)
Computer Networks and Communications
Mathematics of computing → Graph algorithms
[INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS]
Theory of computation → Computational geometry
0102 computer and information sciences
02 engineering and technology
[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
CAT(0) Cube Complex
Linear Time Algorithm
[INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG]
01 natural sciences
Theoretical Computer Science
TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY
[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
Computer Science - Data Structures and Algorithms
0202 electrical engineering
electronic engineering
information engineering
FOS: Mathematics
Mathematics - Combinatorics
Data Structures and Algorithms (cs.DS)
0101 mathematics
[MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG]
ComputingMilieux_MISCELLANEOUS
Applied Mathematics
010102 general mathematics
Computational Theory and Mathematics
010201 computation theory & mathematics
Median Graph
Event Structure
LexBFS
020201 artificial intelligence & image processing
Combinatorics (math.CO)
Computer Science - Discrete Mathematics
MathematicsofComputing_DISCRETEMATHEMATICS
Zdroj: ICALP 2020 47th International Colloquium on Automata, Languages, and Programming
ICALP 2020 47th International Colloquium on Automata, Languages, and Programming, 2020, Saarbrücken, Germany. pp.10:1--10:17, ⟨10.4230/LIPIcs.ICALP.2020.10⟩
Journal of Computer and System Sciences
Journal of Computer and System Sciences, 2022, 126, pp.80-105. ⟨10.1016/j.jcss.2022.01.001⟩
ISSN: 0022-0000
1090-2724
DOI: 10.48550/arxiv.1907.10398
Popis: International audience; 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 l_1-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, due to their bijections with CAT(0) cube complexes and domains of event structures. Our algorithm is based on the majority rule characterization of medians in median graphs and on a fast computation of parallelism classes of edges (Theta-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 \emph{fellow traveler property} of independent interest: the parents of any two adjacent vertices of G are also adjacent. Using the fast computation of the Theta-classes, we also compute the Wiener index (total distance) of G in linear time and the distance matrix in optimal quadratic time.
Databáze: OpenAIRE
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