[Paper Review] Distributed $(Δ+1)$-Coloring in Sublogarithmic Rounds

David G. Harris, Johannes Schneider+1 more|arXiv (Cornell University)|Mar 4, 2016

Complexity and Algorithms in Graphs48 references3 citations

TL;DR

This paper presents a randomized distributed algorithm for $(\Delta+1)$-coloring in the LOCAL model that runs in $O(\sqrt{\log\Delta}) + 2^{O(\sqrt{\log\log n})}$ rounds with high probability, marking the first sublogarithmic-round algorithm for this problem. The approach uses a novel decomposition of graphs into dense and sparse parts, enabling efficient symmetry breaking and establishing a separation between $(\Delta+1)$-coloring and the maximal independent set problem.

ABSTRACT

We give a new randomized distributed algorithm for $(Δ+1)$-coloring in the LOCAL model, running in $O(\sqrt{\log Δ})+ 2^{O(\sqrt{\log \log n})}$ rounds in a graph of maximum degree~$Δ$. This implies that the $(Δ+1)$-coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds of $Ω\left( \min \left( \sqrt{\frac{\log n}{\log \log n}}, \frac{\log Δ}{\log \log Δ} ight) ight)$ by Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Our algorithm also extends to list-coloring where the palette of each node contains $Δ+1$ colors. We extend the set of distributed symmetry-breaking techniques by performing a decomposition of graphs into dense and sparse parts.

Motivation & Objective

• To develop a faster distributed algorithm for $(\Delta+1)$-coloring in the LOCAL model, overcoming the $\Omega(\log n)$ round barrier of prior randomized approaches.
• To demonstrate that $(\Delta+1)$-coloring is strictly easier than the maximal independent set (MIS) problem by showing a separation in round complexity.
• To extend the algorithm to list-coloring in locally sparse graphs, achieving improved bounds for graphs with $(1-\delta)$-local sparsity.
• To introduce a new graph decomposition technique that separates vertices into dense and sparse components based on neighborhood density, enabling tailored coloring strategies.

Proposed method

• The algorithm decomposes the graph into dense and sparse parts using a parameterized threshold based on neighborhood density, where vertices with many neighbors are classified as dense.
• For sparse vertices, the algorithm uses a randomized coloring procedure based on a $\epsilon$-friend definition, ensuring sufficient independence in neighbor choices.
• The dense part is handled via a recursive coloring process that reduces the problem size by eliminating vertices with high external degree, leveraging concentration bounds.
• The algorithm applies a network decomposition technique to partition the graph into components where coloring can be performed efficiently in parallel.
• It uses probabilistic analysis to bound the number of external and internal neighbors per vertex after each round, ensuring that the number of conflicting neighbors remains sublinear.
• The final coloring is achieved by combining results from both sparse and dense components, with residual conflicts resolved using known subroutines.

Experimental results

Research questions

• RQ1 Can a randomized distributed algorithm for $(\Delta+1)$-coloring achieve sublogarithmic round complexity in the LOCAL model?
• RQ2 Does $(\Delta+1)$-coloring have a strictly lower round complexity than the maximal independent set problem?
• RQ3 Can the algorithm be extended to list-coloring in graphs with local sparsity properties?
• RQ4 What is the optimal trade-off between dependence on $\Delta$ and $n$ in distributed coloring algorithms?
• RQ5 Can a decomposition into dense and sparse components lead to improved bounds for symmetry-breaking problems?

Key findings

• The algorithm runs in $O(\sqrt{\log\Delta}) + 2^{O(\sqrt{\log\log n})}$ rounds with high probability, which is $o(\log n)$ for all graphs.
• This result establishes a strict separation between $(\Delta+1)$-coloring and the maximal independent set problem, as the latter has a lower bound of $\Omega(\min(\sqrt{\log n / \log\log n}, \log\Delta / \log\log\Delta))$ rounds.
• For $(1-\delta)$-locally sparse graphs, the algorithm achieves list-coloring in $O(\log(1/\delta)) + 2^{O(\sqrt{\log\log n})}$ rounds with high probability.
• The number of external neighbors per vertex after the initial coloring phase is bounded by $O(\log n) \cdot 2^{O(\sqrt{\log\Delta})}$, ensuring efficient conflict resolution.
• The method introduces a new graph decomposition strategy that separates vertices based on neighborhood density, enabling tailored coloring procedures for each component.
• The algorithm improves upon prior randomized bounds, which required $O(\log n)$ or $O(\log\Delta + \sqrt{\log n})$ rounds, by achieving sublogarithmic complexity in both $\Delta$ and $n$.