159 research outputs found
Testing Small Set Expansion in General Graphs
We consider the problem of testing small set expansion for general graphs. A
graph is a -expander if every subset of volume at most has
conductance at least . Small set expansion has recently received
significant attention due to its close connection to the unique games
conjecture, the local graph partitioning algorithms and locally testable codes.
We give testers with two-sided error and one-sided error in the adjacency
list model that allows degree and neighbor queries to the oracle of the input
graph. The testers take as input an -vertex graph , a volume bound ,
an expansion bound and a distance parameter . For the
two-sided error tester, with probability at least , it accepts the graph
if it is a -expander and rejects the graph if it is -far
from any -expander, where and
. The
query complexity and running time of the tester are
, where is the number of
edges of the graph. For the one-sided error tester, it accepts every
-expander, and with probability at least , rejects every graph
that is -far from -expander, where
and for any . The query
complexity and running time of this tester are
.
We also give a two-sided error tester with smaller gap between and
in the rotation map model that allows (neighbor, index) queries and
degree queries.Comment: 23 pages; STACS 201
Bidirectional PageRank Estimation: From Average-Case to Worst-Case
We present a new algorithm for estimating the Personalized PageRank (PPR)
between a source and target node on undirected graphs, with sublinear
running-time guarantees over the worst-case choice of source and target nodes.
Our work builds on a recent line of work on bidirectional estimators for PPR,
which obtained sublinear running-time guarantees but in an average-case sense,
for a uniformly random choice of target node. Crucially, we show how the
reversibility of random walks on undirected networks can be exploited to
convert average-case to worst-case guarantees. While past bidirectional methods
combine forward random walks with reverse local pushes, our algorithm combines
forward local pushes with reverse random walks. We also discuss how to modify
our methods to estimate random-walk probabilities for any length distribution,
thereby obtaining fast algorithms for estimating general graph diffusions,
including the heat kernel, on undirected networks.Comment: Workshop on Algorithms and Models for the Web-Graph (WAW) 201
AMS Without 4-Wise Independence on Product Domains
In their seminal work, Alon, Matias, and Szegedy introduced several sketching
techniques, including showing that 4-wise independence is sufficient to obtain
good approximations of the second frequency moment. In this work, we show that
their sketching technique can be extended to product domains by using
the product of 4-wise independent functions on . Our work extends that of
Indyk and McGregor, who showed the result for . Their primary motivation
was the problem of identifying correlations in data streams. In their model, a
stream of pairs arrive, giving a joint distribution ,
and they find approximation algorithms for how close the joint distribution is
to the product of the marginal distributions under various metrics, which
naturally corresponds to how close and are to being independent. By
using our technique, we obtain a new result for the problem of approximating
the distance between the joint distribution and the product of the
marginal distributions for -ary vectors, instead of just pairs, in a single
pass. Our analysis gives a randomized algorithm that is a
approximation (with probability ) that requires space logarithmic in
and and proportional to
- …