Cover or Pack: New Upper and Lower Bounds for Massively Parallel Joins

This paper considers the worst-case complexity of multi-round join evaluation in the Massively Parallel Computation (MPC) model. Unlike the sequential RAM model, in which there is a unified optimal algorithm based on the AGM bound for all join queries, worst-case optimal algorithms have been achieved on a very restrictive class of joins in the MPC model. The only known lower bound is still derived from the AGM bound, in terms of the optimal fractional edge covering number of the query. In this work, we make efforts towards bridging this gap. We design an instance-dependent algorithm for the class of α-acyclic join queries. In particular, when the maximum size of input relations is bounded, this complexity has a closed form in terms of the optimal fractional edge covering number of the query, which is worst-case optimal. Beyond acyclic joins, we surprisingly find that the optimal fractional edge covering number does not lead to a tight lower bound. More specifically, we prove for a class of cyclic joins a higher lower bound in terms of the optimal fractional edge packing number of the query, which is matched by existing algorithms, thus optimal. This new result displays a significant distinction for join evaluation, not only between acyclic and cyclic joins, but also between the fine-grained RAM and coarse-grained MPC model.