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arXiv · 2002.05121

An Optimal Decentralized $(\Delta + 1)$-Coloring Algorithm

Daniel Bertschinger·Johannes Lengler·Anders Martinsson·Robert Meier·Angelika Steger·Miloš Trujić·Emo Welzl
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Abstract

Consider the following simple coloring algorithm for a graph on $n$ vertices. Each vertex chooses a color from $\{1, \dotsc, \Delta(G) + 1\}$ uniformly at random. While there exists a conflicted vertex choose one such vertex uniformly at random and recolor it with a randomly chosen color. This algorithm was introduced by Bhartia et al. [MOBIHOC'16] for channel selection in WIFI-networks. We show that this algorithm always converges to a proper coloring in expected $O(n \log \Delta)$ steps, which is optimal and proves a conjecture of Chakrabarty and Supinski [SOSA'20].

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Daniel Bertschinger, Johannes Lengler, Anders Martinsson, Robert Meier, Angelika Steger, Miloš Trujić, Emo Welzl. 2020-02-12. An Optimal Decentralized $(\Delta + 1)$-Coloring Algorithm. https://arxiv.org/abs/2002.05121

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