Dynamic Streaming Spectral Sparsification in Nearly Linear Time and Space
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In this paper we consider the problem of computing spectral approximations to graphs in the single pass dynamic streaming model. We provide a linear sketching based solution that given a stream of edge insertions and deletions to a n-node undirected graph, uses Õ(n) space, processes each update in Õ(1) time, and with high probability recovers a spectral sparsifier in Õ(n) time. Prior to our work, state of the art results either used near optimal Õ(n) space complexity, but brute-force Ω(n^2) recovery time [Kapralov et al.'14], or with subquadratic runtime, but polynomially suboptimal space complexity [Ahn et al.'14, Kapralov et al.'19]. Our main technical contribution is a novel method for `bucketing' vertices of the input graph into clusters that allows fast recovery of edges of sufficiently large effective resistance. Our algorithm first buckets vertices of the graph by performing ball-carving using (an approximation to) its effective resistance metric, and then recovers the high effective resistance edges from a sketched version of an electrical flow between vertices in a bucket, taking nearly linear time in the number of vertices overall. This process is performed at different geometric scales to recover a sample of edges with probabilities proportional to effective resistances and obtain an actual sparsifier of the input graph. This work provides both the first efficient ℓ_2-sparse recovery algorithm for graphs and new primitives for manipulating the effective resistance embedding of a graph, both of which we hope have further applications.
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