Deterministic 3-Coloring of Trees in the Sublinear MPC model

We present deterministic $O(\log^2 \log n)$ time sublinear Massively Parallel Computation (MPC) algorithms for 3-coloring, maximal independent set and maximal matching in trees with $n$ nodes. In accordance with the sublinear MPC regime, our algorithms run on machines that have memory as little as $O(n^\delta)$ for any arbitrary constant $0<\delta<1$. Furthermore, our algorithms use only $O(n)$ global memory. Our main result is the 3-coloring algorithm, which contrasts the probabilistic 4-coloring algorithm of Ghaffari, Grunau and Jin [DISC'20]. The maximal independent set and maximal matching algorithms follow in $O(1)$ time after obtaining the coloring. The key ingredient of our 3-coloring algorithm is an $O(\log^2 \log n)$ time MPC implementation of a variant of the rake-and-compress tree decomposition used by Chang and Pettie [FOCS'17], which is closely related to the $H$-partition by Barenboim and Elkin [PODC'08]. When restricting to trees of constant maximum degree, we bring the runtime down to $O(\log \log n)$.