23 research outputs found
A polynomial time approximation scheme for computing the supremum of Gaussian processes
We give a polynomial time approximation scheme (PTAS) for computing the
supremum of a Gaussian process. That is, given a finite set of vectors
, we compute a -factor approximation
to deterministically in time . Previously, only a constant factor
deterministic polynomial time approximation algorithm was known due to the work
of Ding, Lee and Peres [Ann. of Math. (2) 175 (2012) 1409-1471]. This answers
an open question of Lee (2010) and Ding [Ann. Probab. 42 (2014) 464-496]. The
study of supremum of Gaussian processes is of considerable importance in
probability with applications in functional analysis, convex geometry, and in
light of the recent breakthrough work of Ding, Lee and Peres [Ann. of Math. (2)
175 (2012) 1409-1471], to random walks on finite graphs. As such our result
could be of use elsewhere. In particular, combining with the work of Ding [Ann.
Probab. 42 (2014) 464-496], our result yields a PTAS for computing the cover
time of bounded-degree graphs. Previously, such algorithms were known only for
trees. Along the way, we also give an explicit oblivious estimator for
semi-norms in Gaussian space with optimal query complexity. Our algorithm and
its analysis are elementary in nature, using two classical comparison
inequalities, Slepian's lemma and Kanter's lemma.Comment: Published in at http://dx.doi.org/10.1214/13-AAP997 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A sharp estimate for cover times on binary trees
We compute the second order correction for the cover time of the binary tree
of depth by (continuous-time) random walk, and show that with probability
approaching 1 as increases,
\sqrt{\tau_{\mathrm{cov}}}=\sqrt{|E|}[\sqrt{2\log 2}\cdot n - {\log
n}/{\sqrt{2\log 2}} + O((\log\logn)^8], thus showing that the second order
correction differs from the corresponding one for the maximum of the Gaussian
free field on the tree.Comment: 14 pages, no figur
The evolution of the cover time
The cover time of a graph is a celebrated example of a parameter that is easy
to approximate using a randomized algorithm, but for which no constant factor
deterministic polynomial time approximation is known. A breakthrough due to
Kahn, Kim, Lovasz and Vu yielded a (log log n)^2 polynomial time approximation.
We refine this upper bound, and show that the resulting bound is sharp and
explicitly computable in random graphs. Cooper and Frieze showed that the cover
time of the largest component of the Erdos-Renyi random graph G(n,c/n) in the
supercritical regime with c>1 fixed, is asymptotic to f(c) n \log^2 n, where
f(c) tends to 1 as c tends to 1. However, our new bound implies that the cover
time for the critical Erdos-Renyi random graph G(n,1/n) has order n, and shows
how the cover time evolves from the critical window to the supercritical phase.
Our general estimate also yields the order of the cover time for a variety of
other concrete graphs, including critical percolation clusters on the Hamming
hypercube {0,1}^n, on high-girth expanders, and on tori Z_n^d for fixed large
d. For the graphs we consider, our results show that the blanket time,
introduced by Winkler and Zuckerman, is within a constant factor of the cover
time. Finally, we prove that for any connected graph, adding an edge can
increase the cover time by at most a factor of 4.Comment: 14 pages, to appear in CP