For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work Ω(t). We desire local access algorithms supporting positionG(t) queries, which return the position of a random walk from some fixed start vertex s at time t, where the joint distribution of returned positions is 1/ poly(n) close to those of a uniformly random walk in ℓ1 distance.
We first give an algorithm for local access to random walks on a given undirected dregular graph with eO( 1 1−λ √ n) runtime per query, where λ is the secondlargest eigenvalue of the random walk matrix of the graph in absolute value. Since random dregular graphs G(n, d) are expanders with high probability, this gives an eO(√ n) algorithm for a graph drawn from G(n, d) whp, which improves on the naive method for small numbers of queries. We then prove that no algorithm with subconstant error given probe access to an input dregular graph can have runtime better than Ω(√ n/ log(n)) per query in expectation when the input graph is drawn from G(n, d), obtaining a nearly matching lower bound. We further show an Ω(n1/4) runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance). We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on smalldegree abelian Cayley graphs, including cycles and hypercubes, with runtime polylog(n) per query. This also allows for efficient
local access to walks on polylog degree expanders. We show that our techniques apply to graphs with high degree by extending or results to graphs constructed using the tensor product (giving fast local access to walks on degree nϵ graphs for any ϵ ∈ (0, 1]) and Cartesian product.
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On the trace of random walks on random graphs
We study graphtheoretic properties of the trace of a random walk on a
random graph. We show that for any $\varepsilon>0$ there exists $C>1$ such
that the trace of the simple random walk of length $(1+\varepsilon)n\ln{n}$
on the random graph $G\sim\gnp$ for $p>C\ln{n}/n$ is, with high
probability, Hamiltonian and $\Theta(\ln{n})$connected. In the special
case $p=1$ (i.e.\ when $G=K_n$), we show a hitting time result according to
which, with high probability, exactly one step after the last vertex has
been visited, the trace becomes Hamiltonian, and one step after the last
vertex has been visited for the $k$'th time, the trace becomes
$2k$connected.
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 Award ID(s):
 1661063
 NSFPAR ID:
 10054179
 Date Published:
 Journal Name:
 Journal of the London Mathematical Society
 ISSN:
 14697750
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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