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View all- Czumaj ADavies-Peck PParter M(2024)Component stability in low-space massively parallel computationDistributed Computing10.1007/s00446-024-00461-937:1(35-64)Online publication date: 8-Feb-2024
Exponentially Faster Massively Parallel Maximal Matching
Abstract
The study of approximate matching in the Massively Parallel Computations (MPC) model has recently seen a burst of breakthroughs. Despite this progress, we still have a limited understanding of maximal matching which is one of the central problems of parallel and distributed computing. All known MPC algorithms for maximal matching either take polylogarithmic time which is considered inefficient, or require a strictly super-linear space of n1+Ω (1) per machine.
In this work, we close this gap by providing a novel analysis of an extremely simple algorithm, which is a variant of an algorithm conjectured to work by Czumaj, Lacki, Madry, Mitrovic, Onak, and Sankowski [15]. The algorithm edge-samples the graph, randomly partitions the vertices, and finds a random greedy maximal matching within each partition. We show that this algorithm drastically reduces the vertex degrees. This, among other results, leads to an O(log log Δ) round algorithm for maximal matching with O(n) space (or even mildly sublinear in n using standard techniques).
As an immediate corollary, we get a 2 approximate minimum vertex cover in essentially the same rounds and space, which is the optimal approximation factor under standard assumptions. We also get an improved O(log log Δ) round algorithm for 1 + ε approximate matching. All these results can also be implemented in the congested clique model in the same number of rounds.
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Published In
October 2023
388 pages
ISSN:0004-5411
EISSN:1557-735X
DOI:10.1145/3627674
- Editor:
- Venkatesan Guruswami
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Association for Computing Machinery
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Publication History
Published: 11 October 2023
Online AM: 25 August 2023
Accepted: 22 June 2023
Revised: 23 May 2023
Received: 05 December 2022
Published in JACM Volume 70, Issue 5
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View all- Czumaj ADavies-Peck PParter M(2024)Component stability in low-space massively parallel computationDistributed Computing10.1007/s00446-024-00461-937:1(35-64)Online publication date: 8-Feb-2024
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