The greedy algorithm is optimal for on-line edge coloring
Information Processing Letters
Randomized algorithms
Randomized Distributed Edge Coloring via an Extension of the Chernoff--Hoeffding Bounds
SIAM Journal on Computing
Nearly optimal distributed edge coloring in O(log log n) rounds
Random Structures & Algorithms
Near-optimal, distributed edge colouring via the nibble method
ESA '95 Selected papers from the third European symposium on Algorithms
Switch Scheduling via Randomized Edge Coloring
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Concentration of Measure for the Analysis of Randomized Algorithms
Concentration of Measure for the Analysis of Randomized Algorithms
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A classic theorem by Vizing proves that if the maximum degree of a graph is Δ then it is possible to color its edges, in polynomial time, using at most Δ+1 colors. However, this algorithm is offline, i.e., it assumes the whole graph is known in advance. A natural question then is how well we can do in the online setting, where the edges of the graph are revealed one by one, and we need to color each edge as soon as it is added to the graph. Online edge coloring has an important application in fast switch scheduling. Here, a natural model is that edges arrive online, but in a random permutation. Even in the random permutations model, the best analysis for any algorithm is factor 2, which comes from the simple greedy algorithm (which is factor 2 even in the worst case online model). The algorithm of Aggarwal et al. [1] provides a 1+o(1) factor algorithm, but for the case of multigraphs, when Δ = ω(n2), where n is the number of vertices. In this paper, we show that for graphs with Δ = ω(log n), it is possible to color the graph with 1.43Delta; + o(Δ) colors in the online random order model. Our algorithm is inspired by a 1.6 factor distributed offline algorithm of Panconesi and Srinivasan [9], which we extend by reusing colors online in multiple rounds.