Deterministic fast graph coloring algorithm for distributed systems
Distributed $(Δ+1)$-Coloring in $O(\sqrtΔ+\log^* n)$ Rounds
Data Structures and AlgorithmsDistributed, Parallel, and Cluster Computing
Summary
Coloring a network so no connected points share the same color helps organize communication and avoid conflicts. The authors developed a new method that assigns colors quickly even when the network is large and complex. Their approach speeds up this coloring process by combining clever techniques from previous work with new ideas like using special functions for hashing colors. This means distributed systems can coordinate more efficiently using fewer steps.
What this means in practice
- •For network schedulers: Efficiently assign communication slots in large distributed networks by quickly coloring nodes to prevent transmission conflicts.
- •For distributed systems engineers: Implement faster coordination protocols within distributed computing platforms that rely on graph coloring to manage resource allocation.
Authors
Marc Fuchs, Fabian Kuhn
Abstract
We give a deterministic distributed algorithm for $(Δ+1)$-coloring an $n$-node graph of maximum degree $Δ$ in $O(\sqrtΔ+\log^* n)$ rounds in both the LOCAL and CONGEST models. More generally, $(°+1)$-list coloring with lists from a color space of size $C\geqΔ+1$ can be solved in $O(\sqrtΔ\log(1+C/Δ)+\log^* n)$ rounds; in CONGEST, we assume $C\leq n^{O(1)}$. The algorithm combines Barenboim's polynomial coloring procedure with arbdefective decompositions, degree reduction, and a short family of hash functions. Using the polynomial procedure in the degree-reduction framework already gives an $O(\sqrtΔ\logΔ+\log^* n)$-round algorithm when $C=O(Δ)$. Hashing adapts the color space to the current degree and removes the logarithmic factor. We prove the existence of the hash family by the probabilistic method and compute it by deterministic exhaustive search, using unrestricted local computation. We also give an explicit variant based on lossless expanders that avoids this search and achieves $O(\sqrtΔ+\log^* n)$ rounds for $C=O(Δ)$.