Distributed (Δ + 1)-coloring in linear (in Δ) time

Leonid Barenboim, Michael Elkin

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


The distributed (Δ + 1)-coloring problem is one of most fundamental and well-studied problems in Distributed Algorithms. Starting with the work of Cole and Vishkin in 86, there was a long line of gradually improving algorithms published. The current state-of-the-art running time is O(Δ log Δ+logz.ast; n), due to Kuhn andWattenhofer, PODC'06. Linial (FOCS'87) has proved a lower bound of 1/2 log* n for the problem, and Szegedy and Vishwanathan (STOC'93) provided a heuristic argument that shows that algorithms from a wide family of locally iterative algorithms are un-likely to achieve running time smaller than Θ(Δlog Δ). We present a deterministic (Δ+1)-coloring distributed algorithm with running time O(Δ)+ 1/2 log* n. We also present a tradeoff between the running time and the number of colors, and devise an O(Δ t)-coloring algorithm with running time O(Δ/t+log* n), for any parameter t, 1 < t ≤ Δ1-e, for an arbitrarily small constant e, 0 < e < 1. Our algorithm breaks the heuristic barrier of Szegedy and Vishwanathan, and achieves running time which is linear in the maximum degree Δ. On the other hand, the conjecture of Szegedy and Vishwanathan may still be true, as our algorithm is not from the family of locally iterative algorithms. On the way to this result we study a generalization of the notion of graph coloring, which is called defective coloring. In an m-defective p-coloring the vertices are colored with p colors so that each vertex has up to m neighbors with the same color. We show that an m-defective p-coloring with reasonably small m and p can be computed very efficiently. We also develop a technique to employ multiple defective colorings of various subgraphs of the original graph G for computing a (Δ + 1)-coloring of G. We believe that these techniques are of independent interest.

Original languageEnglish
Title of host publicationSTOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing
Number of pages10
StatePublished - 2009
Externally publishedYes
Event41st Annual ACM Symposium on Theory of Computing, STOC '09 - Bethesda, MD, United States
Duration: 31 May 20092 Jun 2009

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Conference41st Annual ACM Symposium on Theory of Computing, STOC '09
Country/TerritoryUnited States
CityBethesda, MD


  • Coloring
  • Defective-coloring
  • Distributed algorithms


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