Distributed Computing

, Volume 30, Issue 5, pp 325–338 | Cite as

Linear-in-\(\varDelta \) lower bounds in the LOCAL model

Article

Abstract

By prior work, there is a distributed graph algorithm that finds a maximal fractional matching (maximal edge packing) in \(O(\varDelta )\) rounds, independently of \(n\); here \(\varDelta \) is the maximum degree of the graph and \(n\) is the number of nodes in the graph. We show that this is optimal: there is no distributed algorithm that finds a maximal fractional matching in \(o(\varDelta )\) rounds, independently of \(n\). Our work gives the first linear-in-\(\varDelta \) lower bound for a natural graph problem in the standard \(\mathsf{LOCAL }\) model of distributed computing—prior lower bounds for a wide range of graph problems have been at best logarithmic in \(\varDelta \).

Keywords

Local distributed algorithms Lower bounds Maximal edge packing Maximal fractional matching 

Notes

Acknowledgments

We thank the anonymous reviewers for their helpful feedback. The combinatorial proof in “Appendix B” is joint work with Christoph Lenzen and Roger Wattenhofer. This work is supported in part by the Academy of Finland, Grants 132380 and 252018, and by the Research Funds of the University of Helsinki. Much of this research was done while the authors were affiliated with the Department of Computer Science, University of Helsinki.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada
  2. 2.Helsinki Institute for Information Technology HIIT, Department of Computer ScienceAalto UniversityEspooFinland

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