多对数时间上的确定性分布顶点染色
Deterministic Distributed Vertex Coloring in Polylogarithmic Time
摘要：
考虑最大度Delta的n顶点图G =（V，E），并假设每个顶点v \in V托管一个处理器。处理器只允许与G中的邻居进行通信。通信是同步的，即，它以离散的回合进行。在分布式顶点着色问题中，目标是使用尽可能少的通信回合用Delta + 1或略多于Delta + 1的颜色对G着色。(The此后，通信的轮数将被称为运行时间。这个问题的有效随机算法是已知的超过二十年\cite{L86, ABI86}。具体地说，这些算法在O（log n）时间内产生（Delta + 1）-着色，具有很高的概率。另一方面，需要多对数时间的最著名的确定性算法使用O（Delta^2）颜色。该算法是由Linial \cite{L87}在FOCS'87的开创性论文中设计的。它的运行时间是O（log^* n）。在同一篇论文中，利尼亚尔提出了一个问题：在确定性多对数时间内，人们是否可以用明显少于Δ ^2的颜色来着色。到目前为止，Linial的这个问题已经成为这一领域最核心的长期悬而未决的问题之一。在本文中，我们肯定地回答了这个问题，并设计了一个确定性算法，采用Δ^{1 +o（1）}颜色，并在多对数时间内运行。具体地说，我们的算法的运行时间是O（f（Delta）log Delta log n），对于任意缓慢增长的函数f（Delta）= ω（1）。对于任意小的常数η> 0，我们也可以在O（log Delta log n）时间内产生O（Delta^{1 + η}）-着色，对于任意小的常数ε> 0，我们也可以在O（Delta^ε log n）时间内产生O（Delta）-着色。
英文摘要：
Consider an n-vertex graph G = (V,E) of maximum degree Delta, and suppose that each vertex v \in V hosts a processor. The processors are allowed to communicate only with their neighbors in G. The communication is synchronous, i.e., it proceeds in discrete rounds. In the distributed vertex coloring problem the objective is to color G with Delta + 1, or slightly more than Delta + 1, colors using as few rounds of communication as possible. (The number of rounds of communication will be henceforth referred to as running time.) Efficient randomized algorithms for this problem are known for more than twenty years \cite{L86, ABI86}. Specifically, these algorithms produce a (Delta + 1)-coloring within O(log n) time, with high probability. On the other hand, the best known deterministic algorithm that requires polylogarithmic time employs O(Delta^2) colors. This algorithm was devised in a seminal FOCS'87 paper by Linial \cite{L87}. Its running time is O(log^* n). In the same paper Linial asked whether one can color with significantly less than Delta^2 colors in deterministic polylogarithmic time. By now this question of Linial became one of the most central long-standing open questions in this area. In this paper we answer this question in the affirmative, and devise a deterministic algorithm that employs Δ^{1 +o(1)} colors, and runs in polylogarithmic time. Specifically, the running time of our algorithm is O(f(Delta) log Delta log n), for an arbitrarily slow-growing function f(Delta) = ω(1). We can also produce O(Delta^{1 + η})-coloring in O(log Delta log n)-time, for an arbitrarily small constant η> 0, and O(Delta)-coloring in O(Delta^ε log n) time, for an arbitrarily small constant ε> 0.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
