(2Δ - l)-Edge-Coloring is Much Easier than Maximal Matching in the Distributed Setting
(2Δ - l)-Edge-Coloring is Much Easier than Maximal Matching in the Distributed Setting
复制标题
(2Δ - l)-边缘着色比分布式设置中的最大匹配容易得多
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Hsin
中科院分区：
文献类型：
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作者：
Michael Elkin;Seth Pettie;Hsin
图形是分布式计算中的一个中心问题。 n对于任何E> 0，特别是在EO（[等式] log n）回合中。需要ω（[等式] log n）时间[15]。
我们为任意较小的常数e> 0设计A（1 + E）δ-边缘色的算法。 *δ+[方程]）。 ）我们算法的运行时间仅是o（log* n）。
我们对（2δ-1） - 边缘色的结果也遵循我们有关（1- e） - 局部稀疏图的更一般的结果。 - 在o（log*δ + log（1/e））中，对于任何e> 0的宽稀疏图，前提是eδ=（log n）1 +ω（1）。 ） - 可以解决（1- e）的vertex着色问题 - 可以在稀疏图中解决o（log（1/e）） + eo（[等式] log log n）时间。 + 1） - 原始图的线图的vertex颜色，并且由于线图为（1/2 + O（1）） - 局部稀疏。
Graph coloring is a central problem in distributed computing. Both vertex- and edge-coloring problems have been extensively studied in this context. In this paper we show that a (2Δ − 1)-edge-coloring can be computed in time smaller than loge n for any e > 0, specifically, in eO([EQUATION]log log n) rounds. This establishes a separation between the (2Δ − 1)-edge-coloring and Maximal Matching problems, as the latter is known to require Ω([EQUATION]log n) time [15]. No such separation is currently known between the (Δ + 1)-vertex-coloring and Maximal Independent Set problems.
We devise a (1 + e)Δ-edge-coloring algorithm for an arbitrarily small constant e > 0. This result applies whenever Δ ≥ Δe, for some constant Δe which depends on e. The running time of this algorithm is O(log* Δ + [EQUATION]). A much earlier logarithmic-time algorithm by Dubhashi, Grable and Panconesi [11] assumed Δ ≥ (log n)1+Ω(1). For Δ = (log n)1+Ω(1) the running time of our algorithm is only O(log* n). This constitutes a drastic improvement of the previous logarithmic bound [11, 9].
Our results for (2Δ − 1)-edge-coloring also follows from our more general results concerning (1 − e)-locally sparse graphs. Specifically, we devise a (Δ + 1)-vertex coloring algorithm for (1 − e)-locally sparse graphs that runs in O(log* Δ + log(1/e)) rounds for any e > 0, provided that eΔ = (log n)1+Ω(1). We conclude that the (Δ + 1)-vertex coloring problem for (1 − e)-locally sparse graphs can be solved in O(log(1/e)) + eO([EQUATION]log log n) time. This imply our result about (2Δ − 1)-edge-coloring, because (2Δ − 1)-edge-coloring reduces to (Δ + 1)-vertex-coloring of the line graph of the original graph, and because line graphs are (1/2 + o(1))-locally sparse.
影响因子：
1.3
作者：
Kai-Min Chung;Seth Pettie;Hsin-Hao Su
通讯作者：
Kai-Min Chung;Seth Pettie;Hsin-Hao Su