Distributed graph coloring in a few rounds
Kishore Kothapalli, Sriram V. Pemmaraju · 2011
This paper considers the question of how many colors a distributed graph coloring algorithm would need to use if it had only k rounds available, for any positive integer k. In our main result, we present an algorithm that runs in O(k) rounds for any k bounded below by ©(log log n) and bounded above by O(√log n), and uses O(a Å n1/k) colors to color a graph with arboricity a. This result is optimal since the palette size matches the lower bound of Barenboim and Elkin (PODC 2008). This result is achieved via the use of several new results developed in this paper on coloring graphs whose edges have been acyclically oriented. For example, suppose that G is an n-vertex, acyclically oriented graph with maximum out-degree Δo. We present an algorithm that, for any k ≥ 2 loglog n, runs in O(k) rounds on G to produce an (i) O(Δo)-coloring when Δo Δ ©(maxkn2/k2log1+1/k n, 2k) and an (ii) O(Δo Å n2/k2)-coloring when Δo ∈ Ω(maxk log1+1/k n, 2k). These results are useful in any setting where it is possible to efficiently compute acyclic orientations of a graph with Δo << Δ. We derive non-trivial bounds on the palette size even when k < 2 loglog n.