The Sparse Awakens: Streaming Algorithms for Matching Size Estimation in Sparse Graphs

Estimating the size of the maximum matching is a canonical problem in graph algorithms, and one that has attracted extensive study over a range of different computational models. We present improved streaming algorithms for approximating the size of maximum matching with sparse (bounded arboricity) graphs. • (Insert-Only Streams) We present a one-pass algorithm that takes O(c log n) space and approximates the size of the maximum matching in graphs with arboricity c within a factor of O(c). This improves significantly upon the state-of-the-art Õ(cn)-space streaming algorithms. • (Dynamic Streams) Given a dynamic graph stream (i.e., inserts and deletes) of edges of an underlying c-bounded arboricity graph, we present an one-pass algorithm that uses space Õ(cn) and returns an O(c)-estimator for the size of the maximum matching. This algorithm improves the state-of-the-art Õ(cn)-space algorithms, where the Õ(.) notation hides logarithmic in n dependencies. In contrast to the previous works, our results take more advantage of the streaming access to the input and characterize the matching size based on the ordering of the edges in the stream in addition to the degree distributions and structural properties of the sparse graphs. Department of Computer Science, University of Warwick, UK. Supported in part by European Research Council grant ERC2014-CoG 647557 and a Royal Society Wolfson Research Merit Award. [email protected]. Department of Computer Science, University of Warwick, UK. Supported by European Research Council grant ERC-2014CoG 647557. [email protected] . Rutgers University, Piscataway, NJ, USA. [email protected]. Rutgers University, Piscataway, NJ, USA. [email protected].