Distributed $(\Delta+1)$-Coloring via Ultrafast Graph Shattering

Abstract

Vertex coloring is one of the classic symmetry breaking problems studied in distributed computing. In this paper, we present a new algorithm for $(\Delta+1)$-list coloring in the randomized ${LOCAL}$ model running in $O({Det}_{\scriptscriptstyle d}(\operatorname{poly} \log n))=O(\operatorname{poly}(\log\log n))$ time, where ${Det}_{\scriptscriptstyle d}(n')$ is the deterministic complexity of $(\deg+1)$-list coloring on $n'$-vertex graphs. (In this problem, each $v$ has a palette of size $\deg(v)+1$.) This improves upon a previous randomized algorithm of Harris, Schneider, and Su [J. ACM, 65 (2018), 19] with complexity $O(\sqrt{\log \Delta} + \log\log n + {Det}_{\scriptscriptstyle d}(\operatorname{poly}\log n)) = O(\sqrt{\log n})$. Unless $\Delta$ is small, it is also faster than the best known deterministic algorithm of Fraigniaud, Heinrich, and Kosowski [Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2016] and Barenboim, Elkin, and Goldenberg [Proceedings of the 38th Annual ACM Symposium on Principles of Distributed Computing (PODC), 2018], with complexity $O(\sqrt{\Delta\log \Delta}\log^\ast \Delta + \log^* n)$. Our algorithm's running time is syntactically very similar to the $\Omega({Det}(\operatorname{poly}\log n))$ lower bound of Chang, Kopelowitz, and Pettie [SIAM J. Comput., 48 (2019), pp. 122--143], where ${Det}(n')$ is the deterministic complexity of $(\Delta+1)$-list coloring on $n'$-vertex graphs. Although distributed coloring has been actively investigated for 30 years, the best deterministic algorithms for $(\deg+1)$- and $(\Delta+1)$-list coloring (that depend on $n'$ but not $\Delta$) use a black-box application of network decompositions. The recent deterministic network decomposition algorithm of Rozhoň and Ghaffari [Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC), 2020] implies that ${Det}_{\scriptscriptstyle d}(n')$ and ${Det}(n')$ are both $\operatorname{poly}(\log n')$. Whether they are asymptotically equal is an open problem.