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In this paper we consider the problem of efficiently computing ϵ-sketches for the Laplacian and its pseudoinverse. Given a Laplacian and an error tolerance ϵ, we seek to construct a function f such that for any vector x (chosen obliviously from f), with high probability (1−ϵ)x┬ Ax≤f(x)≤(1+ϵ)x┬ Ax 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 [EQUATION] ∈ Rn×n and an error tolerance ϵ, produce O(n/ϵ)-size sketches of both [EQUATION] and its pseudoinverse. Our algorithms improve upon the previous best sketch size of O(n/ϵ1.6) for sketching the Laplacian form by [1] and O(n/ϵ2) for sketching the Laplacian pseudoinverse by [2]. Furthermore we show how to compute all-pairs effective resistances from our O(n/ϵ) size sketch in O(n2/ϵ) time. This improves upon the previous best running time of O(n2/ϵ2) by [3].

Efficient Õ(n/∊) Spectral Sketches for the Laplacian and its Pseudoinverse