Exponential speedup over locality in MPC with optimal memory

Abstract Locally Checkable Labeling () problems are graph problems in which a solution is correct if it satisfies some given constraints in the local neighborhood of each node. Example problems in this class include maximal matching, maximal independent set, and colorings. A successful line of research has been studying the complexities of s on paths/cycles, trees, and general graphs, providing many interesting results for the model of distributed computing. In this work, we initiate the study of problems in the low-space Massively Parallel Computation () model. In particular, on forests, we provide a method that, given the complexity of an problem in the model, automatically provides an exponentially faster algorithm for the low-space setting that uses optimal global memory, that is, truly linear. While restricting to forests may seem to weaken the results, we emphasize that all known (conditional) lower bounds for the setting are obtained through lower bounds for problems in the distributed setting in tree-like networks (either trees or high-girth graphs), and hence the problems that we study are challenging already on trees. Moreover, our algorithms use optimal global memory, i.e., memory linear in the number of edges of the graph. In contrast, most of the state-of-the-art algorithms use more than linear global memory. Further, they typically start with a dense graph, sparsify it, and then solve the problem on the residual graph, exploiting the relative increase in global memory. On forests this is not possible, hence using optimal memory requires new solutions.