Deterministic massively parallel connectivity

We consider the problem of designing fundamental graph algorithms on the model of Massive Parallel Computation (MPC). The input to the problem is an undirected graph G with n vertices and m edges, and with D being the maximum diameter of any connected component in G. We consider the MPC with low local space, allowing each machine to store only Θ(nδ) words for an arbitrary constant δ>0, and with linear global space (which is the number of machines times the local space available), that is, with optimal utilization. In a recent breakthrough, Andoni et al. (FOCS’18) and Behnezhad et al. (FOCS’19) designed parallel randomized algorithms that in O(logD + loglogn) rounds on an MPC with low local space determine all connected components of a graph, improving on the classic bound of O(logn) derived from earlier works on PRAM algorithms. In this paper, we show that asymptotically identical bounds can be also achieved for deterministic algorithms: we present a deterministic MPC low local space algorithm that in O(logD + loglogn) rounds determines connected components of the input graph. Our result matches the complexity of state of the art randomized algorithms for this task. The techniques developed in our paper can be also applied to several related problems, giving new deterministic MPC algorithms for problems like finding a spanning forest, minimum spanning forest, etc. We complement our upper bounds by extending a recent lower bound for connectivity on an MPC conditioned on the 1-vs-2-cycles conjecture (which requires D ≥ log1+Ω(1)n), by showing a related conditional hardness of Ω(logD) MPC rounds for the entire spectrum of D, covering a particularly interesting range when D ≤ O(logn).