Matching and MIS for Uniformly Sparse Graphs in the Low-Memory MPC Model
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The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale parallel computation frameworks and has recently gained a lot of importance, especially in the context of classic graph problems. Unsatisfactorily, all current poly ( n)-round MPC algorithms seem to get fundamentally stuck at the linear-memory barrier: their efficiency crucially relies on each machine having space at least linear in the number n of nodes. As this might not only be prohibitively large, but also allows for easy if not trivial solutions for sparse graphs, we are interested in the low-memory MPC model, where the space per machine is restricted to be strongly sublinear, that is, n^δ for any 0<δ<1. We devise a degree reduction technique that reduces maximal matching and maximal independent set in graphs with arboricity λ to the corresponding problems in graphs with maximum degree poly(λ) in O(^2 n) rounds. This gives rise to O(^2 n + T(polyλ))-round algorithms, where T(Δ) is the Δ-dependency in the round complexity of maximal matching and maximal independent set in graphs with maximum degree Δ. A concurrent work by Ghaffari and Uitto shows that T(Δ)=O(√(Δ)). For graphs with arboricity λ=poly( n), this almost exponentially improves over Luby's O( n)-round PRAM algorithm [STOC'85, JALG'86], and constitutes the first poly ( n)-round maximal matching algorithm in the low-memory MPC model, thus breaking the linear-memory barrier. Previously, the only known subpolylogarithmic algorithm, due to Lattanzi et al. [SPAA'11], required strongly superlinear, that is, n^1+Ω(1), memory per machine.
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