An optimal distributed (Δ+1)-coloring algorithm?

Abstract

Vertex coloring is one of the classic symmetry breaking problems studied in distributed computing. In this paper we present a new algorithm for (∆ + 1)-list coloring in the randomized LOCAL model running in O(Detd(poly log n)) time, where Detd(n ) is the deterministic complexity of (deg +1)-list coloring on n -vertex graphs. (In this problem, each v has a palette of size deg(v) + 1.) This improves upon a previous randomized algorithm of Harris, Schneider, and Su [STOC 2016, J. ACM 2018] with complexity O( √ log ∆ + log log n + Detd(poly log n)), and, for some range of ∆, is much faster than the best known deterministic algorithm of Fraigniaud, Heinrich, and Kosowski [FOCS 2016] and Barenboim, Elkin, and Goldenberg [PODC 2018], with complexity O( √ ∆ log ∆ log * ∆ + log * n). Our algorithm appears to be optimal, in view of the Ω(Det(poly log n)) randomized lower bound due to Chang, Kopelowitz, and Pettie [FOCS 2016, SIAM J. Comput. 2019], where Det is the deterministic complexity of (∆ + 1)-list coloring. At present, the best upper bounds on Detd(n ) and Det(n ) are both 2 O √ log n and use a black box application of network decompositions due to Panconesi and Srinivasan [J.Algorithms 1996]. It is quite possible that the true deterministic complexities of both problems are the same, asymptotically, which would imply the randomized optimality of our (∆ + 1)-list coloring algorithm.1 In the case of MIS, the subproblems actually have size poly(∆) log n, but satisfy the additional property that they contain distance-5 dominating sets of size O(log n), which is often just as good as having poly log(n) size. See [9, §3] or [21, §4] for more discussion of this.2 See [33,14,12] for the formal definition of the class of locally checkable labeling (LCL) problems.

The Complexity of (Δ+1) Coloring in Congested Clique, Massively Parallel Computation, and Centralized Local Computation

“…This notion of sparsification is a key idea behind some recent algorithms [Gha17,GU19]. In the present paper, a key technical ingredient is providing such a sparsification for the (∆ + 1) coloring algorithm of CLP [CLP18]. The pre-shattering phase of the CLP algorithm [CLP18] consists of three parts: (i) initial coloring, (ii) dense coloring, and (iii) color bidding.…”

“…Given ID(v), the algorithm A returns A(v) = the output of v, after making a small number of queries. It is required that the output of A at different vertices are consistent with one legal solution of P. [CLP18] can be implemented in the centralized local computation model with query complexity ∆ O(log * ∆) ·O(log n).…”

“…In the present paper, a key technical ingredient is providing such a sparsification for the (∆ + 1) coloring algorithm of CLP [CLP18]. The pre-shattering phase of the CLP algorithm [CLP18] consists of three parts: (i) initial coloring, (ii) dense coloring, and (iii) color bidding. Parts (i) and (ii) take O(1) rounds; 3 part (iii) takes τ = O(log * ∆) rounds.…”

“…The algorithm of [CLP18] is based on the graph shattering framework. Each vertex successfully colors itself with probability 1 − 1/poly(∆) during the pre-shattering phase of [CLP18], and so by the shattering lemma (Lemma 2.2), the remaining uncolored vertices V Bad form connected components of size ∆ O(1) O(poly log n). 9 The post-shattering phase then applies a deterministic (deg +1)-list coloring algorithm to color them.…”

A Topological Perspective on Distributed Network Algorithms

“…This domain of distributed computing is extremely active and productive this last decade, analyzing a large variety of network problems in the so-called LOCAL model [37], capturing the ability to solve task locally in networks 1 . We refer to [4,5,8,13,18,20,21,24,42] for a non exhaustive list of achievements in context. However, all these achievements were based on an operational approach, using sophisticated algorithmic techniques and tools solely from graph theory.…”