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Abstract

We propose a simple and time-optimal algorithm for property testing a graph for its conductance in the CONGEST model. Our algorithm takes only \(O(\log n)\) rounds of communication (which is known to be optimal), and consists of simply running multiple random walks of \(O(\log n)\) length from a certain number of random sources, at the end of which nodes can decide if the underlying network is a good conductor or far from it. Unlike previous algorithms, no aggregation is required even with a smaller number of walks. Our main technical contribution involves a tight analysis of this process for which we use spectral graph theory. We introduce and leverage the concept of sticky vertices which are vertices in a graph with low conductance such that short random walks originating from these vertices end in a region around them. The present state-of-the-art distributed CONGEST algorithm for the problem by Fichtenberger and Vasudev [MFCS 2018], runs in \(O(\log n)\) rounds using three distinct phases : building a rooted spanning tree (\emph{preprocessing}), running \(O(n^{100})\) random walks to generate statistics (\emph{Phase~1}), and then convergecasting to the root to make the decision (\emph{Phase~2}). The whole of our algorithm is, however, similar to their Phase~1 running only \(O(m^2) = O(n^4)\) walks. Note that aggregation (using spanning trees) is a popular technique but spanning tree(s) are sensitive to node/edge/root failures, hence, we hope our work points to other more distributed, efficient and robust solutions for suitable problems.

Details

1009240
Title
A Distributed Conductance Tester Without Global Information Collection
Publication title
arXiv.org; Ithaca
Publication year
2024
Publication date
Mar 11, 2024
Section
Computer Science
Publisher
Cornell University Library, arXiv.org
Source
arXiv.org
Place of publication
Ithaca
Country of publication
United States
University/institution
Cornell University Library arXiv.org
e-ISSN
2331-8422
Source type
Working Paper
Language of publication
English
Document type
Working Paper
Publication history
 
 
Online publication date
2024-03-12
Milestone dates
2023-05-23 (Submission v1); 2024-03-11 (Submission v2)
Publication history
 
 
   First posting date
12 Mar 2024
ProQuest document ID
2818541250
Document URL
https://www.proquest.com/working-papers/distributed-conductance-tester-without-global/docview/2818541250/se-2?accountid=208611
Full text outside of ProQuest
Copyright
© 2024. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
Last updated
2024-03-13
Database
ProQuest One Academic