An Asymptotically Optimal Algorithm for Maximum Matching in Dynamic Streams

Sepehr Assadi, Vihan Shah

Research output: Chapter in Book/Report/Conference proceedingConference contribution

7 Scopus citations

Abstract

We present an algorithm for the maximum matching problem in dynamic (insertion-deletions) streams with asymptotically optimal space: for any n-vertex graph, our algorithm with high probability outputs an α-approximate matching in a single pass using O(n23) bits of space. A long line of work on the dynamic streaming matching problem has reduced the gap between space upper and lower bounds first to no(1) factors [Assadi-Khanna-Li-Yaroslavtsev; SODA 2016] and subsequently to polylog(n) factors [Dark-Konrad; CCC 2020]. Our upper bound now matches the Dark-Konrad lower bound up to O(1) factors, thus completing this research direction. Our approach consists of two main steps: we first (provably) identify a family of graphs, similar to the instances used in prior work to establish the lower bounds for this problem, as the only “hard” instances to focus on. These graphs include an induced subgraph which is both sparse and contains a large matching. We then design a dynamic streaming algorithm for this family of graphs which is more efficient than prior work. The key to this efficiency is a novel sketching method, which bypasses the typical loss of polylog (n)-factors in space compared to standard L0-sampling primitives, and can be of independent interest in designing optimal algorithms for other streaming problems.

Original languageEnglish (US)
Title of host publication13th Innovations in Theoretical Computer Science Conference, ITCS 2022
EditorsMark Braverman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772174
DOIs
StatePublished - Jan 1 2022
Event13th Innovations in Theoretical Computer Science Conference, ITCS 2022 - Berkeley, United States
Duration: Jan 31 2022Feb 3 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume215
ISSN (Print)1868-8969

Conference

Conference13th Innovations in Theoretical Computer Science Conference, ITCS 2022
Country/TerritoryUnited States
CityBerkeley
Period1/31/222/3/22

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Graph streaming algorithms
  • Maximum matching
  • Sketching

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