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 analysis, 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(α log n) space and approximates the size of the maximum matching in graphs with arboricity α within a factor of O(α). This improves significantly upon the state-of-The-Art Õ(α n2/3)-space streaming algorithms, and is the first poly-logarithmic space algorithm for this problem. (Dynamic Streams) Given a dynamic graph stream (i.e., inserts and deletes) of edges of an underlying α -bounded arboricity graph, we present an one-pass algorithm that uses space Õ(α 10/3n2/3) and returns an O(α)-estimator for the size of the maximum matching on the condition that the number edge deletions in the stream is bounded by O(α n). For this class of inputs, our algorithm improves the state-of-The-Art O (α n4/5)-space algorithms, where the O (.) notation hides logarithmic in n dependencies. In contrast to prior work, 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.