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Sketching Distances in Monotone Graph Classes

Authors Louis Esperet , Nathaniel Harms , Andrey Kupavskii



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Author Details

Louis Esperet
  • Univ. Grenoble Alpes, CNRS, Laboratoire G-SCOP, Grenoble, France
Nathaniel Harms
  • University of Waterloo, Canada
Andrey Kupavskii
  • Univ. Grenoble Alpes, CNRS, Laboratoire G-SCOP, Grenoble, France
  • Moscow Institute of Physics and Technology, Russia
  • Huawei R&D Moscow, Russia

Acknowledgements

We thank Gwenaël Joret for many helpful discussions. We thank Viktor Zamaraev for leading us to Corollary 3.6 and carefully proofreading our manuscript. We thank Alexandr Andoni for a helpful discussion and for sharing with us the manuscript [Andoni and Krauthgamer, 2008]. We thank Renato Ferreira Pinto Jr. and Sebastian Wild for comments on the presentation of this article.

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Louis Esperet, Nathaniel Harms, and Andrey Kupavskii. Sketching Distances in Monotone Graph Classes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 18:1-18:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.18

Abstract

We study the problems of adjacency sketching, small-distance sketching, and approximate distance threshold (ADT) sketching for monotone classes of graphs. The algorithmic problem is to assign random sketches to the vertices of any graph G in the class, so that adjacency, exact distance thresholds, or approximate distance thresholds of two vertices u,v can be decided (with probability at least 2/3) from the sketches of u and v, by a decoder that does not know the graph. The goal is to determine when sketches of constant size exist. Our main results are that, for monotone classes of graphs: constant-size adjacency sketches exist if and only if the class has bounded arboricity; constant-size small-distance sketches exist if and only if the class has bounded expansion; constant-size ADT sketches imply that the class has bounded expansion; any class of constant expansion (i.e. any proper minor closed class) has a constant-size ADT sketch; and a class may have arbitrarily small expansion without admitting a constant-size ADT sketch.

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Distributed algorithms
  • Theory of computation → Sketching and sampling
Keywords
  • adjacency labelling
  • informative labelling
  • distance sketching
  • adjacency sketching
  • communication complexity

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