[Paper Review] Simple and Near-Optimal Distributed Coloring for Sparse Graphs

Mohsen Ghaffari, Christiana Lymouri|arXiv (Cornell University)|Aug 21, 2017

Complexity and Algorithms in Graphs21 references3 citations

TL;DR

This paper presents simple randomized distributed algorithms for coloring sparse graphs using a number of colors that scales near-optimally with arboricity α. By combining H-partitioning with iterative random color selection and deterministic finishing, the algorithms achieve O(α) colors in O(log n · min{log log n, log α}) rounds, nearly matching the theoretical lower bound of Ω(logα n) rounds.

ABSTRACT

Graph coloring is one of the central problems in distributed graph algorithms. Much of the research on this topic has focused on coloring with $Δ+1$ colors, where $Δ$ denotes the maximum degree. Using $Δ+1$ colors may be unsatisfactory in sparse graphs, where not all nodes have such a high degree; it would be more desirable to use a number of colors that improves with sparsity. A standard measure that captures sparsity is arboricity, which is the smallest number of forests into which the edges of the graph can be partitioned. We present simple randomized distributed algorithms that, with high probability, color any $n$-node $α$-arboricity graph: - using $(2+\varepsilon)\cdot α$ colors, for constant $\varepsilon>0$, in $O(\log n)$ rounds, if $α= ildeΩ(\log n)$, or - using $O(α\log α)$ colors, in $O(\log n)$ rounds, or - using $O(α)$ colors, in $O(\log n \cdot \min\{\log\log n,\; \log α\})$ rounds. These algorithms are nearly-optimal, as it is known by results of Linial [FOCS'87] and Barenboim and Elkin [PODC'08] that coloring with $Θ(α)$ colors, or even poly$(α)$ colors, requires $Ω(\log_α n)$ rounds. The previously best-known $O(\log n)$-time result was a deterministic algorithm due to Barenboim and Elkin [PODC'08], which uses $Θ(α^2)$ colors. Barenboim and Elkin stated improving this number of colors as an open problem in their Distributed Graph Coloring Book.

Motivation & Objective

• Address the inefficiency of Δ+1 coloring in sparse graphs where Δ is large but arboricity α is small.
• Develop distributed algorithms that use a number of colors that improves with graph sparsity, measured by arboricity α.
• Achieve near-optimal round complexity and color count, closing the gap between known lower bounds and prior upper bounds.
• Resolve an open problem posed by Barenboim and Elkin on reducing the number of colors from O(α²) to O(α) in O(log n) rounds.

Proposed method

• Construct an H-partition with degree bound d ≤ (2 + ε/2)·α and ℓ = ⌈(1/ε) log n⌉ layers to enable localized coloring.
• Use a randomized iterative procedure where each uncolored node in layer H_j selects a random color from a palette of size (2+ε)·α.
• A node permanently claims a color if no neighbor selects it in the same round, ensuring conflict-free assignment.
• Apply a deterministic finishing phase to the remaining graph, which has no path longer than O(log n) with high probability.
• Leverage the fact that each node has at most d uncolored out-neighbors, ensuring at least one color remains available.
• Combine randomized and deterministic phases to achieve O(log n · min{log log n, log α})-round complexity with O(α) colors.

Experimental results

Research questions

• RQ1 Can we design a distributed algorithm that colors sparse graphs using O(α) colors in near-optimal time?
• RQ2 Is it possible to reduce the number of colors from O(α²) to O(α) in O(log n) rounds for arboricity-α graphs?
• RQ3 How can we achieve a trade-off between color count and round complexity in distributed coloring for sparse graphs?
• RQ4 What is the minimal round complexity required to achieve O(α) coloring, and can we nearly match the known lower bound?
• RQ5 Can a simple randomized approach combined with deterministic finishing yield near-optimal results in distributed coloring?

Key findings

• The algorithm achieves (2+ε)·α colors in O(log n) rounds when α = ~Ω(log n), matching the best-known color count with improved round complexity.
• For general α, the algorithm uses O(α log α) colors in O(log n) rounds, improving over prior deterministic O(α²)-coloring algorithms.
• In the best-case regime, the algorithm uses O(α) colors in O(log n · min{log log n, log α}) rounds, nearly matching the Ω(logα n) lower bound.
• The probability that any node remains uncolored after the randomized phase is at most d⁻², and this is independent across nodes.
• The remaining graph after partial coloring has no path longer than O(log n) with high probability, enabling efficient deterministic finishing.
• The deterministic finishing phase completes in O(log n) rounds, as each node waits at most O(log n) rounds for its neighbors to be colored.