Faster Deterministic Distributed Coloring Through Recursive List Coloring

Abstract

We provide novel deterministic distributed vertex coloring algorithms. As our main result, we give a deterministic distributed algorithm to compute a (∆ + 1)-coloring of an n-node graph with maximum degree ∆ in 2 O( √ log ∆) · log n rounds. This improves on the best previously known time complexity of min O √ ∆ log ∆ log * ∆ + log * n , 2 O( √ log n) for a large range of values of ∆. For graphs with arboricity a, we obtain a deterministic distributed algorithm to compute a (2+o(1))acoloring in time 2 O( √ log a) · log 2 n. Further, for graphs with bounded neighborhood independence, we show that a (∆ + 1)-coloring can be computed more efficiently in time 2 O(This in particular implies that also a (2∆ − 1)-edge coloring can be computed deterministically in 2 O( √ log ∆) + O(log * n) rounds, which improves the best known time bound for small values of ∆. All results even hold for the list coloring variants of the problems. As a consequence, we also obtain an improved deterministic 2 O( √ log ∆) · log 3 n-round algorithm for ∆-coloring non-complete graphs with maximum degree ∆ ≥ 3. Most of our algorithms only require messages of O(log n) bits (including the (∆ + 1)-vertex coloring algorithms).Our main technical contribution is a recursive deterministic distributed list coloring algorithm to solve list coloring problems with lists of size ∆ 1+o(1) . Given some list coloring problem and an orientation of the edges, we show how to recursively divide the global color space into smaller subspaces, assign one of the subspaces to each node of the graph, and compute a new edge orientation such that for each node, the list size to out-degree ratio degrades at most by a constant factor on each recursion level.

Improved distributed $$\Delta $$-coloring

“…As one of many implications the runtime of our deterministic -coloring algorithm (for unbounded degree) drops to poly logn and our randomized algorithm for non-clique graphs with (unbounded) maximum degree ≥ 4 drops to O(log ) + poly log log n (Theorem 2). Further, [24] provides an improved list coloring algorithm. Combining it with the techniques in our paper [24] gives a log 3 n • 2 O( √ log ) round algorithm for -coloring.…”

“…Further, [24] provides an improved list coloring algorithm. Combining it with the techniques in our paper [24] gives a log 3 n • 2 O( √ log ) round algorithm for -coloring. Also [28] provides improved list coloring algorithms, i.e., it shaves of the O(log * ) term in Theorem 6 if the color space is at most exponential in the maximum degree of the uncolored graph.…”

On the Locality of Nash-Williams Forest Decomposition and Star-Forest Decomposition

“…The forest decomposition problem was first studied in the LOCAL model by Barenboim and Elkin [7], who developed the -partition algorithm to compute a (2 + ) -forest decomposition in (log / ) rounds. This has been a building block in many other distributed and parallel algorithms [7,8,10,41,54]. Barenboim and Elkin [7] also showed that an ( )-FD would require Ω( log log − log * ) rounds.…”

Distributed Backup Placement in One Round and its Applications to Maximum Matching Approximation and Self-Stabilization

“…This includes unit disk graphs, unit balls graphs, line graphs, line graphs of hypergraphs, claw-free graphs, graphs of bounded diversity, and many more. Consequently, this graph family and its subtypes have been very widely studied, especially in the distributed computing setting [7,14,2,3,4,1,9]. For example, unit disc graphs can model certain types of wireless networks.…”

“…A notable example of the benefit of analyzing such graphs is the very recent breakthrough of Kuhn [9]. Kuhn obtained a (2∆−1)-edge-coloring of general graphs by analyzing graphs with constant neighborhood independence.…”