Abstract
We present a randomized distributed algorithm that computes a Delta -coloring in any non-complete graph with maximum degree Delta ge 4 in O(log Delta ) + 2^{O(sqrt{log log n})} rounds, as well as a randomized algorithm that computes a Delta -coloring in O((log log n)^2) rounds when Delta in [3, O(1)]. Both these algorithms improve on an O(log ^3 n / log Delta )-round algorithm of Panconesi and Srinivasan (STOC’93), which has remained the state of the art for the past 25 years. Moreover, the latter algorithm gets (exponentially) closer to an Omega (log log n) round lower bound of Brandt et al. (STOC’16).
Highlights
Introduction and related work1.1 Background and state of the artThis paper presents faster distributed algorithms, in the LOCAL model, for computing a -coloring of any nonclique graph with maximum degree ≥ 3
We use the Distributed Brooks Theorem to provide two deterministic algorithms for -coloring. These algorithms already contain much of the high level structure of our randomized algorithms that we present in Sect
We have provided several structural results for the -coloring (Sect. 2) that hopefully will be of use for future algorithmic improvements to the problem
Summary
This paper presents faster distributed algorithms, in the LOCAL model, for computing a -coloring of any nonclique graph with maximum degree ≥ 3. In the context of lower bounds for the Lovász Local Lemma problem, Brandt et al [9] proved that (log log n) rounds are required by any randomized -coloring algorithm, even in constant-degree graphs. These results led to two problems which exhibit an exponential separation between their randomized and deterministic complexity. In the special case of trees of large enough maximum degree, Chang et al [12] give an O(log log n)-round randomized algorithm for computing a -coloring This, combined with their deterministic lower bound (log n) [12], gives an exponential separation on trees. Our algorithms establish this separation in the general bounded-degree case
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