A Two-Sided Error Distributed Property Tester For Conductance

We study property testing in the distributed model and extend its setting from testing with onesided error to testing with two-sided error. In particular, we develop a two-sided error property tester for general graphs with round complexity O(log(n)/( Φ2)) in the CONGEST model, which accepts graphs with conductance Φ and rejects graphs that are -far from having conductance at least Φ2/1000 with constant probability. Our main insight is that one can start poly(n) random walks from a few random vertices without violating the congestion and unite the results to obtain a consistent answer from all vertices. For connected graphs, this is even possible when the number of vertices is unknown. We also obtain a matching Ω(logn) lower bound for the LOCAL and CONGEST models by an indistinguishability argument. Although the power of vertex labels that arises from two-sided error might seem to be much stronger than in the sequential query model, we can show that this is not the case. 2012 ACM Subject Classification Theory of computation → Distributed algorithms

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