Efficient Deterministic Distributed Coloring with Small Bandwidth

Abstract

We show that the $(degree+1)$-list coloring problem can be solved deterministically in $O(D \cdot \log n \cdot\log^2\Delta)$ rounds in the \CONGEST model, where $D$ is the diameter of the graph, $n$ the number of nodes, and $\Delta$ the maximum degree. Using the recent polylogarithmic-time deterministic network decomposition algorithm by Rozho\v{n} and Ghaffari [STOC 2020], this implies the first efficient (i.e., $\poly\log n$-time) deterministic \CONGEST algorithm for the $(\Delta+1)$-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 $O(\log\Delta\cdot\log\log\Delta)$, for the MPC model, we obtain algorithms with round complexity $O(\log^2\Delta)$ for the linear-memory regime and $O(\log^2\Delta + \log n)$ for the sublinear memory regime.