We study property testing in the distributed model and extend its setting from testing with one-sided 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) / (epsilon Phi^2)) in the CONGEST model, which accepts graphs with conductance Phi and rejects graphs that are epsilon-far from having conductance at least Phi^2 / 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 Omega(log n) 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.
@InProceedings{fichtenberger_et_al:LIPIcs.MFCS.2018.19, author = {Fichtenberger, Hendrik and Vasudev, Yadu}, title = {{A Two-Sided Error Distributed Property Tester For Conductance}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {19:1--19:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.19}, URN = {urn:nbn:de:0030-drops-96019}, doi = {10.4230/LIPIcs.MFCS.2018.19}, annote = {Keywords: property testing, distributed algorithms, conductance} }
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