Efficient $\widetilde{O}(n/\epsilon)$ Spectral Sketches for the Laplacian and its Pseudoinverse
نویسندگان
چکیده
In this paper we consider the problem of efficiently computing ǫ-sketches for the Laplacian and its pseudoinverse. Given a Laplacian and an error tolerance ǫ, we seek to construct a function f such that for any vector x (chosen obliviously from f), with high probability (1 − ǫ)xAx ≤ f(x) ≤ (1+ǫ)xAx where A is either the Laplacian or its pseudoinverse. Our goal is to construct such a sketch f efficiently and to store it in the least space possible. We provide nearly-linear time algorithms that, when given a Laplacian matrix L ∈ R and an error tolerance ǫ, produce Õ(n/ǫ)-size sketches of both L and its pseudoinverse. Our algorithms improve upon the previous best sketch size of Õ(n/ǫ) for sketching the Laplacian form by [1] and O(n/ǫ) for sketching the Laplacian pseudoinverse by [2]. Furthermore we show how to compute all-pairs effective resistances from our Õ(n/ǫ) size sketch in Õ(n/ǫ) time. This improves upon the previous best running time of Õ(n/ǫ) by [3]. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-114747.
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