Motivated by recent progress on symmetry breaking problems such as maximal independent set (MIS) and maximal matching in the low-memory Massively Parallel Computation (MPC) model (e.g., Behnezhad et al. PODC 2019; Ghaffari-Uitto SODA 2019), we investigate the complexity of ruling set problems in this model. The MPC model has become very popular as a model for large-scale distributed computing and it comes with the constraint that the memory-per-machine is strongly sublinear in the input size. For graph problems, extremely fast MPC algorithms have been designed assuming Ω̃(n) memory-per-machine, where n is the number of nodes in the graph (e.g., the O(log log n) MIS algorithm of Ghaffari et al., PODC 2018). However, it has proven much more difficult to design fast MPC algorithms for graph problems in the low-memory MPC model, where the memory-per-machine is restricted to being strongly sublinear in the number of nodes, i.e., O(n^ε) for constant 0 < ε < 1.

In this paper, we present an algorithm for the 2-ruling set problem, running in Õ(log^{1/6} Δ) rounds whp, in the low-memory MPC model. Here Δ is the maximum degree of the graph. We then extend this result to β-ruling sets for any integer β > 1. Specifically, we show that a β-ruling set can be computed in the low-memory MPC model with O(n^ε) memory-per-machine in Õ(β ⋅ log^{1/(2^{β+1}-2)} Δ) rounds, whp. From this it immediately follows that a β-ruling set for β = Ω(log log log Δ)-ruling set can be computed in in just O(β log log n) rounds whp. The above results assume a total memory of Õ(m + n^{1+ε}). We also present algorithms for β-ruling sets in the low-memory MPC model assuming that the total memory over all machines is restricted to Õ(m). For β > 1, these algorithms are all substantially faster than the Ghaffari-Uitto Õ(√{log Δ})-round MIS algorithm in the low-memory MPC model.

All our results follow from a Sample-and-Gather Simulation Theorem that shows how random-sampling-based Congest algorithms can be efficiently simulated in the low-memory MPC model. We expect this simulation theorem to be of independent interest beyond the ruling set algorithms derived here.