An Efficient Parallel Algorithm for Spectral Sparsification of Laplacian and SDDM Matrix Polynomials

Abstract: For "large" class $\mathcal{C}$ of continuous probability density functions (p.d.f.), we demonstrate that for every $w\in\mathcal{C}$ there is mixture of discrete Binomial distributions (MDBD) with $T\geq N\sqrt{\phi_{w}/\delta}$ distinct Binomial distributions $B(\cdot,N)$ that $\delta$-approximates a discretized p.d.f. $$\widehat{w}(i/N)\triangleq w(i/N)/[\sum_{\ell=0}<sup>{N}w(\ell/N)]$$ for all $i\in[3:N-3]$, where $\phi_{w}\geq\max_{x\in[0,1]}|w(x)|$. Also, we give two efficient parallel algorithms to find such MDBD. Moreover, we propose a sequential algorithm that on input MDBD with $N=2<sup>k$ for $k\in\mathbb{N}<em>{+}$ that induces a discretized p.d.f. $\beta$, $B=D-M$ that is either Laplacian or SDDM matrix and parameter $\epsilon\in(0,1)$, outputs in $$\widehat{O}(\epsilon^{-2}m + \epsilon<sup>{-4}nT)$$ time a spectral sparsifier $$D-\widehat{M}</em>{N} \approx_{\epsilon} D-D\sum_{i=0}^{{N}\beta_{i}(D^{-1}} M)<sup>i$$ of a matrix-polynomial, where $\widehat{O}(\cdot)$ notation hides $\mathrm{poly}(\log n,\log N)$ factors. This improves the Cheng et al.'s [CCLPT15] algorithm whose run time is $$\widehat{O}(\epsilon^{-2} m N² + NT)$$. Furthermore, our algorithm is parallelizable and runs in work $$\widehat{O}(\epsilon^{-2}m + \epsilon<sup>{-4}nT)$$ and depth $O(\log N\cdot\mathrm{poly}(\log n)+\log T)$. Our main algorithmic contribution is to propose the first efficient parallel algorithm that on input continuous p.d.f. $w\in\mathcal{C}$, matrix $B=D-M$ as above, outputs a spectral sparsifier of matrix-polynomial whose coefficients approximate component-wise the discretized p.d.f. $\widehat{w}$. Our results yield the first efficient and parallel algorithm that runs in nearly linear work and poly-logarithmic depth and analyzes the long term behaviour of Markov chains in non-trivial settings. In addition, we strengthen the Spielman and Peng's [PS14] parallel SDD solver.