SIAM Journal on Computing, Volume 49, Issue 3, Page 497-539, January 2020.
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.

中文翻译：

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通过超快图形粉碎进行分布式$（\ Delta + 1）$着色

SIAM计算杂志，第49卷，第3期，第497-539页，2020年1月。
顶点着色是在分布式计算中研究的经典对称破坏问题之一。在本文中，我们为在$ O（{Det} _ {\ scriptscriptstyle d}（\ operatorname {poly}）中运行的随机$ {LOCAL} $模型中的$（\ Delta + 1）$列表着色提供了一种新算法\ log n））= O（\ operatorname {poly}（\ log \ log n））$时间，其中$ {Det} _ {\ scriptscriptstyle d}（n'）$是$（\ deg +的确定性复杂度1）$ n'$-顶点图上的$ -list着色。（在这个问题中，每个$ v $都有一个大小为\\ deg（v）+ 1 $的调色板。）这是对Harris，Schneider和Su [J. ACM，65（2018），19]，复杂度$ O（\ sqrt {\ log \ Delta} + \ log \ log n + {Det} _ {\ scriptscriptstyle d}（\ operatorname {poly} \ log n））= O（\ sqrt {\ log n}）$。除非$ \ Delta $很小，否则它也比最著名的Fraigniaud确定性算法快，Heinrich和Kosowski [第57届IEEE年度计算机科学基础学术研讨会（FOCS），2016年]和Barenboim，Elkin和Goldenberg [第38届ACM分布式计算原理学术年会（PODC），2018年]，复杂度为$ O（\ sqrt {\ Delta \ log \ Delta} \ log ^ \ ast \ Delta + \ log ^ * n）$。我们的算法的运行时间在语法上与Chang，Kopelowitz和Pettie的$ \ Omega（{Det}（\ operatorname {poly} \ log n））$下限非常相似[SIAM J. Comput。，48（2019）， pp。122--143]，其中$ {Det}（n'）$是$ n'$顶点图上$（\ Delta + 1）$-list着色的确定性复杂度。尽管已经对分布式着色进行了30年的积极研究，但用于$（\ deg + 1）$-和$（\ Delta + 1）$列表着色的最佳确定性算法（取决于$ n' $，而不是$ \ Delta $）使用网络分解的黑盒应用程序。Rozhoň和Ghaffari的最新确定性网络分解算法[第52届ACM SIGACT年度计算机理论研讨会论文集（STOC），2020年]暗示$ {Det} _ {\ scriptscriptstyle d}（n'）$和$ {Det }（n'）$均为$ \ operatorname {poly}（\ log n'）$。它们是否渐近相等是一个开放的问题。