On Estimating Path Aggregates over Streaming Graphs

Abstract

We consider the updatable streaming graph model, where edges of a graph arrive or depart in arbitrary sequence and are processed in an online fashion using sub-linear space and time. We study the problem of estimating aggregate path metrics P _k defined as the number of pairs of vertices that have a simple path between them of length k. For a streaming undirected graph with n vertices, m edges and r components, we present an $\tilde{O}(m(m-r)^{-1/4})$ space algorithm for estimating P ₂ and an $\Omega(\sqrt{m})$ space lower bound. We show that estimating P ₂ over directed streaming graphs, and estimating P _k over streaming graphs (whether directed or undirected), for any k ≥3 requires Ω(n ²) space. We also present a space lower bound of Ω(n ²) for the problems of (a) deterministically testing the connectivity, and, (b) estimating the size of transitive closure, of undirected streaming graphs that allow both edge-insertions and deletions.