Efficient Deterministic Distributed Coloring with Small Bandwidth

We show that the $(degree+1)$-list coloring problem can be solved deterministically in $O(D \cdot \log n \cdot\log^2Δ)$ rounds in the \CONGEST model, where $D$ is the diameter of the graph, $n$ the number of nodes, and $Δ$ the maximum degree. Using the recent polylogarithmic-time deterministic network decomposition algorithm by Rozhoň and Ghaffari [STOC 2020], this implies the first efficient (i.e., $\poly\log n$-time) deterministic \CONGEST algorithm for the $(Δ+1)$-coloring and the $(\mathit{degree}+1)$-list coloring problem. Previously the best known algorithm required $2^{O(\sqrt{\log n})}$ rounds and was not based on network decompositions. Our techniques also lead to deterministic $(\mathit{degree}+1)$-list coloring algorithms for the congested clique and the massively parallel computation (MPC) model. For the congested clique, we obtain an algorithm with time complexity $O(\logΔ\cdot\log\logΔ)$, for the MPC model, we obtain algorithms with round complexity $O(\log^2Δ)$ for the linear-memory regime and $O(\log^2Δ+ \log n)$ for the sublinear memory regime.

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