We present the first single pass algorithm for computing spectral sparsifiers of graphs in the dynamic semi-streaming model. Given a single pass over a stream containing insertions and deletions of edges to a graph G, our algorithm maintains a randomized linear sketch of the incidence matrix of G into dimension O( 1 2 n polylog(n)). Using this sketch, at any point, the algorithm can output a (1 ± ) spectral sparsifier for G with high probability. While O( 1 2 n polylog(n)) space algorithms are

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... nown for computing cut sparsifiers in dynamic streams [AGM12b, GKP12] and spectral sparsifiers in insertion-only streams [KL11], prior to our work, the best known single pass algorithm for maintaining spectral sparsifiers in dynamic streams required sketches of dimension Ω( 1 2 n 5/3 ) [AGM14]. To achieve our result, we show that, using a coarse sparsifier of G and a linear sketch of G's incidence matrix, it is possible to sample edges by effective resistance, obtaining a spectral sparsifier of arbitrary precision. Sampling from the sketch requires a novel application of 2 / 2 sparse recovery, a natural extension of the 0 methods used for cut sparsifiers in [AGM12b] . Recent work of [MP12] on row sampling for matrix approximation gives a recursive approach for obtaining the required coarse sparsifiers. Under certain restrictions, our approach also extends to the problem of maintaining a spectral approximation for a general matrix A A given a stream of updates to rows in A.