31 research outputs found
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
Simple random walk on distance-regular graphs
A survey is presented of known results concerning simple random walk on the
class of distance-regular graphs. One of the highlights is that electric
resistance and hitting times between points can be explicitly calculated and
given strong bounds for, which leads in turn to bounds on cover times, mixing
times, etc. Also discussed are harmonic functions, moments of hitting and cover
times, the Green's function, and the cutoff phenomenon. The main goal of the
paper is to present these graphs as a natural setting in which to study simple
random walk, and to stimulate further research in the field
Mixing and relaxation time for Random Walk on Wreath Product Graphs
Suppose that G and H are finite, connected graphs, G regular, X is a lazy
random walk on G and Z is a reversible ergodic Markov chain on H. The
generalized lamplighter chain X* associated with X and Z is the random walk on
the wreath product H\wr G, the graph whose vertices consist of pairs (f,x)
where f=(f_v)_{v\in V(G)} is a labeling of the vertices of G by elements of H
and x is a vertex in G. In each step, X* moves from a configuration (f,x) by
updating x to y using the transition rule of X and then independently updating
both f_x and f_y according to the transition probabilities on H; f_z for z
different of x,y remains unchanged. We estimate the mixing time of X* in terms
of the parameters of H and G. Further, we show that the relaxation time of X*
is the same order as the maximal expected hitting time of G plus |G| times the
relaxation time of the chain on H.Comment: 30 pages, 1 figur