Distributed testing of graph isomorphism in the CONGEST model

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, GU, is isomorphic to GK 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 GU and GK are isomorphic from the case where GU and GK are not isomorphic. Specifically, if GU and GK 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 GK is given only to a single node. We prove that for this setting the number of rounds of any deterministic algorithm is Ω(~ 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 GU and GK are isomorphic from the case that GU and GK 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 GK 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.