Distributed (Delta+1)-Coloring Via Ultrafast Graph Shattering

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(d)(polylogn)) time, where Det(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(root log Delta+log log n+Det(d)(poly log n)) = O(root 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(root Delta log log* Delta + log* n). Our algorithm's running time is syntactically very similar to the Omega(Det(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 determin- istic 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 Rozhoil and Ghaffari [Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC), 2020] implies that Det(d)(n') and Det(n') are both poly(log n'). Whether they are asymptotically equal is an open problem.