Exponential Speedup Over Locality in MPC with Optimal Memory

Abstract

Locally Checkable Labeling ( LCL ) 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 coloring problems. A successful line of research has been studying the complexities of LCL problems on paths/cycles, trees, and general graphs, providing many interesting results for the LOCAL model of distributed computing. In this work, we initiate the study of LCL problems in the low-space Massively Parallel Computation ( MPC ) model. In particular, on forests, we provide a method that, given the complexity of an LCL problem in the LOCAL model, automatically provides an exponentially faster algorithm for the low-space MPC setting that uses optimal global memory, that is, truly linear. While restricting to forests may seem to weaken the result, we emphasize that all known (conditional) lower bounds for the MPC setting are obtained by lifting lower bounds obtained in the distributed setting in tree-like networks (either forests or high girth graphs), and hence the problems that we study are challenging already on forests. Moreover, the most important technical feature of our algorithms is that they use optimal global memory, that is, 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, because the given graph is already as sparse as it can be, and using optimal memory requires new solutions. Graph Exponentiation. A reoccurring challenge for all regimes lies in respecting the linear global memory, which roughly means that on average, every node can use only a constant amount of memory. This is particularly unfortunate because almost all recent MPC results – and in particular all that achieve exponential speedups – rely on the memory-intense graph exponentiation technique [45]. Informally, this technique enables a node to gather its 2 k -hop neighborhood in k communication rounds. Doing this in parallel for every node in the graph results in a ∆ 2 k overhead in global memory. For this technique to be useful, k has to be ω (1), yielding a non-constant multiplicative increase in the global memory requirement. In order to use this technique but not violate linear global memory, we develop new solutions that are discussed in the following paragraphs. ▶ Lemma 9. The distance- k O (∆ 2 k ) -coloring problem on general graphs can be solved in the low-space MPC model with a O (log log ∗ n + log k ) -time deterministic algorithm, as long as ∆ k < n δ . The algorithm requires O (∆ k ) words of local and O ( m + n · ∆ k ) words of global memory. If k and ∆ are constants, the runtime reduces to O (log log ∗ n ) and we require O (1) words of local and O ( m + n ) words of global memory.