Space-Round Tradeoffs for MapReduce Computations

TL;DR

This paper introduces a formal MapReduce model capturing data-centric constraints, and analyzes fundamental tradeoffs between space and round complexity in matrix multiplication, inversion, and matching tasks.

Contribution

It provides a new formal model for MapReduce and establishes space-round tradeoffs for key matrix computations, advancing understanding of computational limits.

Findings

Upper and lower bounds for dense matrix multiplication

Tradeoffs between space and rounds in sparse matrix multiplication

Derived space-round tradeoffs for matrix inversion and matching

Abstract

This work explores fundamental modeling and algorithmic issues arising in the well-established MapReduce framework. First, we formally specify a computational model for MapReduce which captures the functional flavor of the paradigm by allowing for a flexible use of parallelism. Indeed, the model diverges from a traditional processor-centric view by featuring parameters which embody only global and local memory constraints, thus favoring a more data-centric view. Second, we apply the model to the fundamental computation task of matrix multiplication presenting upper and lower bounds for both dense and sparse matrix multiplication, which highlight interesting tradeoffs between space and round complexity. Finally, building on the matrix multiplication results, we derive further space-round tradeoffs on matrix inversion and matching.