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arXiv · 2202.09253

Sketching Distances in Monotone Graph Classes

Louis Esperet·Nathaniel Harms·Andrey Kupavskii
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Abstract

We study the two-player communication problem of determining whether two vertices $x, y$ are nearby in a graph $G$, with the goal of determining the graph structures that allow the problem to be solved with a constant-cost randomized protocol. Equivalently, we consider the problem of assigning constant-size random labels (sketches) to the vertices of a graph, which allow adjacency, exact distance thresholds, or approximate distance thresholds to be computed with high probability from the labels. 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 sketches for exact distance thresholds exist if and only if the class has bounded expansion; constant-size approximate distance threshold (ADT) sketches imply that the class has bounded expansion; any class of constant expansion (i.e. any proper minor closed class) has constant-size ADT sketches; and a class may have arbitrarily small expansion without admitting constant-size ADT sketches.

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BibTeXRIS

Louis Esperet, Nathaniel Harms, Andrey Kupavskii. 2022-02-18. Sketching Distances in Monotone Graph Classes. https://arxiv.org/abs/2202.09253

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