For "large" class C of continuous probability density functions
(p.d.f.), we demonstrate that for every w∈C there is mixture of
discrete Binomial distributions (MDBD) with T≥Nϕw/δ
distinct Binomial distributions B(⋅,N) that δ-approximates a
discretized p.d.f. w(i/N)≜w(i/N)/[∑ℓ=0Nw(ℓ/N)] for all i∈[3:N−3], where
ϕw≥maxx∈[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=2k
for k∈N+ that induces a discretized p.d.f. β, B=D−M
that is either Laplacian or SDDM matrix and parameter ϵ∈(0,1),
outputs in O(ϵ−2m+ϵ−4nT) time a spectral
sparsifier D−MN≈ϵD−D∑i=0Nβi(D−1M)i of a matrix-polynomial, where
O(⋅) notation hides poly(logn,logN) factors.
This improves the Cheng et al.'s [CCLPT15] algorithm whose run time is
O(ϵ−2mN2+NT).
Furthermore, our algorithm is parallelizable and runs in work
O(ϵ−2m+ϵ−4nT) and depth O(logN⋅poly(logn)+logT). Our main algorithmic contribution is to
propose the first efficient parallel algorithm that on input continuous p.d.f.
w∈C, matrix B=D−M as above, outputs a spectral sparsifier of
matrix-polynomial whose coefficients approximate component-wise the discretized
p.d.f. 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