Optimal Deterministic Massively Parallel Connectivity on Forests

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We show fast deterministic algorithms for fundamental problems on forests in the challenging low-space regime of the well-known Massive Parallel Computation (MPC) model. A recent breakthrough result by Coy and Czumaj [STOC'22] shows that, in this setting, it is possible to deterministically identify connected components on graphs in O(\log D + \log\log n) rounds, where D is the diameter of the graph and n the number of nodes. The authors left open a major question: is it possible to get rid of the additive \log\log n factor and deterministically identify connected components in a runtime that is completely independent of n?
We answer the above question in the affirmative in the case of forests. We give an algorithm that identifies connected components in O(\log D) deterministic rounds. The total memory required is O(n+m) words, where m is the number of edges in the input graph, which is optimal as it is only enough to store the input graph. We complement our upper bound results by showing that \Omega(\log D) time is necessary even for component-unstable algorithms, conditioned on the widely believed 1 vs. 2 cycles conjecture. Our techniques also yield a deterministic forest-rooting algorithm with the same runtime and memory bounds.
Furthermore, we consider Locally Checkable Labeling problems (LCLs), whose solution can be verified by checking the O(1)-radius neighborhood of each node. We show that any LCL problem on forests can be solved in O(\log D) rounds with a canonical deterministic algorithm, improving over the O(\log n) runtime of Brandt, Latypov and Uitto [DISC'21]. We also show that there is no algorithm that solves all LCL problems on trees asymptotically faster.