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Distributed Testing of Graph Isomorphism in the CONGEST Model

Authors Reut Levi , Moti Medina

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Author Details

Reut Levi
  • Efi Arazi School of Computer Science, The Interdisciplinary Center, Herzliya, Israel
Moti Medina
  • School of Electrical & Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva, Israel

Funding

  • Levi, Reut: This research was supported by the ERC advanced grant 834735.
  • Medina, Moti: This research was supported by the Israel Science Foundation under Grant 867/19.

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Reut Levi and Moti Medina. Distributed Testing of Graph Isomorphism in the CONGEST Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 19:1-19:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.19

Abstract

In this paper we study the problem of testing graph isomorphism (GI) in the CONGEST distributed model. In this setting we test whether the distributive network, G_U, is isomorphic to G_K which is given as an input to all the nodes in the network, or alternatively, only to a single node.
We first consider the decision variant of the problem in which the algorithm should distinguish the case where G_U and G_K are isomorphic from the case where G_U and G_K are not isomorphic. Specifically, if G_U and G_K are not isomorphic then w.h.p. at least one node should output reject and otherwise all nodes should output accept . We provide a randomized algorithm with O(n) rounds for the setting in which G_K is given only to a single node. We prove that for this setting the number of rounds of any deterministic algorithm is Ω̃(n²) rounds, where n denotes the number of nodes, which implies a separation between the randomized and the deterministic complexities of deciding GI . Our algorithm can be adapted to the semi-streaming model, where a single pass is performed and Õ(n) bits of space are used.
We then consider the property testing variant of the problem, where the algorithm is only required to distinguish the case that G_U and G_K are isomorphic from the case that G_U and G_K are far from being isomorphic (according to some predetermined distance measure). We show that every (possibly randomized) algorithm, requires Ω(D) rounds, where D denotes the diameter of the network. This lower bound holds even if all the nodes are given G_K as an input, and even if the message size is unbounded. We provide a randomized algorithm with an almost matching round complexity of O(D+(ε^{-1}log n)²) rounds that is suitable for dense graphs (namely, graphs with Ω(n²) edges).
We also show that with the same number of rounds it is possible that each node outputs its mapping according to a bijection which is an approximate isomorphism.
We conclude with simple simulation arguments that allow us to adapt centralized property testing algorithms and obtain essentially tight algorithms with round complexity Õ(D) for special families of sparse graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • the CONGEST model
  • graph isomorphism
  • distributed property testing
  • distributed decision
  • graph algorithms

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