All you need are random walks: fast and simple distributed conductance testing

Batu, Tugkan ORCID: 0000-0003-3914-4645, Trehan, Amitabh and Trehan, Chhaya ORCID: 0000-0002-3249-3212 (2024) All you need are random walks: fast and simple distributed conductance testing. In: Emek, Yuval, (ed.) Structural Information and Communication Complexity: 31st International Colloquium, SIROCCO 2024, Vietri sul Mare, Italy, May 27–29, 2024, Proceedings. Lecture Notes in Computer Science. Springer, Cham, CH, 64 - 82. ISBN 9783031606021

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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 (preprocessing), running O(log n) random walks to generate statistics (Phase 1), and then convergecasting to the root to make the decision (Phase 2). The whole of our algorithm is, however, similar to their Phase 1 running only O(m2) = O(n4) 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.

Item Type:	Book Section
Additional Information:	© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
Divisions:	Mathematics LSE
Subjects:	Q Science > QA Mathematics
Date Deposited:	01 Feb 2024 00:01
Last Modified:	18 Nov 2024 17:21
URI:	http://eprints.lse.ac.uk/id/eprint/121617

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