Locally-Iterative Distributed (Delta + 1)-Coloring below Szegedy-Vishwanathan Barrier, and Applications to Self-Stabilization and to Restricted-Bandwidth Models
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We consider graph coloring and related problems in the distributed message-passing model. 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 (Delta+1)-coloring algorithm requires Omega(Delta log Delta + log^* n) rounds, unless there is "a very special type of coloring that can be very efficiently reduced" SV93. In this paper we obtain this special type of coloring. Specifically, we devise a locally-iterative (Delta+1)-coloring algorithm with running time O(Delta + log^* n), i.e., 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: - We obtain self-stabilizing distributed algorithms for (Delta+1)-vertex-coloring, (2Delta-1)-edge-coloring, maximal independent set and maximal matching with O(Delta+log^* n) time. This significantly improves previously-known results that have O(n) or larger running times GK10. - We devise a (2Delta-1)-edge-coloring algorithm in the CONGEST model with O(Delta + log^* n) time and in the Bit-Round model with O(Delta + log n) time. Previously-known algorithms had superlinear dependency on Delta for (2Delta-1)-edge-coloring in these models. - We obtain an arbdefective coloring algorithm with running time O(√(D)elta + log^* n). We employ it in order to compute proper colorings that improve the recent state-of-the-art bounds of Barenboim from PODC'15 B15 and Fraigniaud et al. from FOCS'16 FHK16 by polylogarithmic factors. - Our algorithms are applicable to the SET-LOCAL model of HKMS15.
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