Locally-Iterative Distributed (Δ+ 1): -Coloring below Szegedy-Vishwanathan Barrier, and Applications to Self-Stabilization and to Restricted-Bandwidth Models

We consider graph coloring and related problems in the distributed message-passing model. \em Locally-iterative algorithms are especially important in this setting. These are algorithms in which each vertex decides about its next color only as a function of the current colors in its 1 - hop-neighborhood. In STOC'93 Szegedy and Vishwanathan showed that any locally-iterative (Δ + 1)-coloring algorithm requires Ω(Δ log Δ + log^* n) rounds, unless there exists "a very special type of coloring that can be very efficiently reduced" \citeSV93. No such special coloring has been found since then. This led researchers to believe that Szegedy-Vishwanathan barrier is an inherent limitation for locally-iterative algorithms, and to explore other approaches to the coloring problem \citeBE09,K09,B15,FHK16. The latter gave rise to faster algorithms, but their heavy machinery which is of non-locally-iterative nature made them far less suitable to various settings. In this paper we obtain the aforementioned special type of coloring. Specifically, we devise a locally-iterative (Δ + 1)-coloring algorithm with running time O(Δ + log^* n), i.e., \em below Szegedy-Vishwanathan barrier. This demonstrates that this barrier is not an inherent limitation for locally-iterative algorithms. As a result, we also achieve significant improvements for dynamic, self-stabilizing and bandwidth-restricted settings. This includes the following results. \beginitemize ıtem We obtain self-stabilizing distributed algorithms for (Δ + 1)-vertex-coloring, (2Δ - 1)-edge-coloring, maximal independent set and maximal matching with O(Δ + log^* n) time. This significantly improves previously-known results that have O(n) or larger running times \citeGK10. ıtem We devise a (2Δ - 1)-edge-coloring algorithm in the CONGEST model with O(Δ + log^* n) time and O(Δ)-edge-coloring in the Bit-Round model with O(Δ + log n) time. The factors of log^* n and log n are unavoidable in the CONGEST and Bit-Round models, respectively. Previously-known algorithms had superlinear dependency on Δ for (2Δ - 1)-edge-coloring in these models. ıtem We obtain an arbdefective coloring algorithm with running time O(\sqrt Δ + log^* n). Such a coloring is not necessarily proper, but has certain helpful properties. We employ it in order to compute a proper (1 + ε)Δ-coloring within O(√ Δ + log^* n) time, and √(Δ + 1)√-coloring within √O(√ Δ log Δ log^* Δ + log^* n)√ time. This improves the recent state-of-the-art bounds of Barenboim from PODC'15 \citeB15 and Fraigniaud et al. from FOCS'16 \citeFHK16 by polylogarithmic factors. ıtem Our algorithms are applicable to the SET-LOCAL model \citeHKMS15 (also known as the weak LOCAL model). In this model a relatively strong lower bound of √Ω(Δ^1/3 )√ is known for √(Δ + 1)√-coloring. However, most of the coloring algorithms do not work in this model. (In \citeHKMS15 only Linial's √O(Δ^2)√-time algorithm and Kuhn-Wattenhofer √O(Δ log Δ)√-time algorithms are shown to work in it.) We obtain the first linear-in-Δ algorithms that work also in this model. \enditemize