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22
7

Ultrafast Distributed Coloring of High Degree Graphs

10 May 2021
Magnús M. Halldórsson
Alexandre Nolin
Tigran Tonoyan
ArXiv (abs)PDFHTML
Abstract

We give a new randomized distributed algorithm for the Δ+1\Delta+1Δ+1-list coloring problem. The algorithm and its analysis dramatically simplify the previous best result known of Chang, Li, and Pettie [SICOMP 2020]. This allows for numerous refinements, and in particular, we can color all nnn-node graphs of maximum degree Δ≥log⁡2+Ω(1)n\Delta \ge \log^{2+\Omega(1)} nΔ≥log2+Ω(1)n in O(log⁡∗n)O(\log^* n)O(log∗n) rounds. The algorithm works in the CONGEST model, i.e., it uses only O(log⁡n)O(\log n)O(logn) bits per message for communication. On low-degree graphs, the algorithm shatters the graph into components of size poly⁡(log⁡n)\operatorname{poly}(\log n)poly(logn) in O(log⁡∗Δ)O(\log^* \Delta)O(log∗Δ) rounds, showing that the randomized complexity of Δ+1\Delta+1Δ+1-list coloring in CONGEST depends inherently on the deterministic complexity of related coloring problems.

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