612 research outputs found
Explicit two-sided unique-neighbor expanders
We study the problem of constructing explicit sparse graphs that exhibit
strong vertex expansion. Our main result is the first two-sided construction of
imbalanced unique-neighbor expanders, meaning bipartite graphs where small sets
contained in both the left and right bipartitions exhibit unique-neighbor
expansion, along with algebraic properties relevant to constructing quantum
codes.
Our constructions are obtained from instantiations of the tripartite line
product of a large tripartite spectral expander and a sufficiently good
constant-sized unique-neighbor expander, a new graph product we defined that
generalizes the line product in the work of Alon and Capalbo and the routed
product in the work of Asherov and Dinur.
To analyze the vertex expansion of graphs arising from the tripartite line
product, we develop a sharp characterization of subgraphs that can arise in
bipartite spectral expanders, generalizing results of Kahale, which may be of
independent interest.
By picking appropriate graphs to apply our product to, we give a strongly
explicit construction of an infinite family of -biregular graphs
(for large enough and ) where all sets with
fewer than a small constant fraction of vertices have
unique-neighbors (assuming ).
Additionally, we can also guarantee that subsets of vertices of size up to
expand losslessly.Comment: New version contains stronger result, and many new technical
ingredients. 45 pages, 2 figure
New Explicit Constant-Degree Lossless Expanders
We present a new explicit construction of onesided bipartite lossless
expanders of constant degree, with arbitrary constant ratio between the sizes
of the two vertex sets. Our construction is simpler to state and analyze than
the only prior construction of Capalbo, Reingold, Vadhan, and Wigderson (2002),
and achieves improvements in some parameters.
We construct our lossless expanders by imposing the structure of a
constant-sized lossless expander "gadget" within the neighborhoods of a large
bipartite spectral expander; similar constructions were previously used to
obtain the weaker notion of unique-neighbor expansion. Our analysis simply
consists of elementary counting arguments and an application of the expander
mixing lemma.Comment: Edits to expositio
Gossip vs. Markov Chains, and Randomness-Efficient Rumor Spreading
We study gossip algorithms for the rumor spreading problem which asks one
node to deliver a rumor to all nodes in an unknown network. We present the
first protocol for any expander graph with nodes such that, the
protocol informs every node in rounds with high probability, and
uses random bits in total. The runtime of our protocol is
tight, and the randomness requirement of random bits almost
matches the lower bound of random bits for dense graphs. We
further show that, for many graph families, polylogarithmic number of random
bits in total suffice to spread the rumor in rounds.
These results together give us an almost complete understanding of the
randomness requirement of this fundamental gossip process.
Our analysis relies on unexpectedly tight connections among gossip processes,
Markov chains, and branching programs. First, we establish a connection between
rumor spreading processes and Markov chains, which is used to approximate the
rumor spreading time by the mixing time of Markov chains. Second, we show a
reduction from rumor spreading processes to branching programs, and this
reduction provides a general framework to derandomize gossip processes. In
addition to designing rumor spreading protocols, these novel techniques may
have applications in studying parallel and multiple random walks, and
randomness complexity of distributed algorithms.Comment: 41 pages, 1 figure. arXiv admin note: substantial text overlap with
arXiv:1304.135
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
- β¦