Distributed $(\Delta+1)$-Coloring Via Ultrafast Graph Shattering | SIAM
SIAM J. COMPUT. \bigcircc 2020 Society for Industrial and Applied Mathematics Vol. 49, No. 3, pp. 497{539 DISTRIBUTED (\Delta + 1)-COLORING VIA ULTRAFAST GRAPH SHATTERING\ast y z x YI-JUN CHANG , WENZHENG LI , AND SETH PETTIE Abstract. Vertex coloring is one of the classic symmetry breaking problems studied in distrib- uted computing. In this paper, we present a new algorithm for (\Delta +1)-list coloring in the randomized 0 LOCAL model running in O(Detd(poly log n)) = O(poly(log log n)) time, where Detd(n ) is the de- 0 terministic 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 previousp randomized algorithm of Harris, Schneider,p and Su [J. ACM, 65 (2018), 19] with complexity O( log \Delta +log log n+Detd(poly log n)) = O( log n). Unless \Delta is small, it is also faster than the best known deterministic algorithm of Fraigniaud,Hein- rich, and Kosowski [Proceedings of the 57th Annual IEEE Symposium on Foundations of Com- puter Science (FOCS), 2016] and Barenboim, Elkin, and Goldenberg [Proceedings of the 38th An- nualp ACM Symposium on Principles of Distributed Computing (PODC), 2018], with complexity O( \Delta log \Deltalog\ast \Delta +log\ast 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(n0) is the deterministic complexity of (\Delta + 1)-list coloring on n0-vertex graphs.
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