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Reports tagged with Expander Graphs:
TR97-020 | 15th May 1997
Oded Goldreich

A Sample of Samplers -- A Computational Perspective on Sampling (survey).

We consider the problem of estimating the average of a huge set of values.
That is,
given oracle access to an arbitrary function $f:\{0,1\}^n\mapsto[0,1]$,
we need to estimate $2^{-n} \sum_{x\in\{0,1\}^n} f(x)$
upto an additive error of $\epsilon$.
We are allowed to employ a randomized algorithm which may ... more >>>

TR98-072 | 14th December 1998
Ziv Bar-Yossef, Oded Goldreich, Avi Wigderson

Deterministic Amplification of Space Bounded Probabilistic Algorithms.

This paper initiates the study of deterministic amplification of space
bounded probabilistic algorithms. The straightforward implementations of
known amplification methods cannot be used for such algorithms, since they
consume too much space. We present a new implementation of the
Ajtai-Koml\'{o}s-Szemer\'{e}di method, that enables to amplify an $S$ ... more >>>

TR99-046 | 17th November 1999
Ran Raz, Omer Reingold, Salil Vadhan

Extracting All the Randomness and Reducing the Error in Trevisan's Extractors

We give explicit constructions of extractors which work for a source of
any min-entropy on strings of length n. These extractors can extract any
constant fraction of the min-entropy using O(log^2 n) additional random
bits, and can extract all the min-entropy using O(log^3 n) additional
random bits. Both of these ... more >>>

TR00-090 | 3rd December 2000
Oded Goldreich

Candidate One-Way Functions Based on Expander Graphs

We suggest a candidate one-way function using combinatorial
constructs such as expander graphs. These graphs are used to
determine a sequence of small overlapping subsets of input bits,
to which a hard-wired random predicate is applied.
Thus, the function is extremely easy to evaluate:
all that is needed ... more >>>

TR01-018 | 23rd February 2001
Omer Reingold, Salil Vadhan, Avi Wigderson

Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders and Extractors

The main contribution of this work is a new type of graph product, which we call the zig-zag
product. Taking a product of a large graph with a small graph, the resulting graph inherits
(roughly) its size from the large one, its degree from the small one, and ... more >>>

TR05-012 | 17th January 2005
Luca Trevisan, Salil Vadhan, David Zuckerman

Compression of Samplable Sources

We study the compression of polynomially samplable sources. In particular, we give efficient prefix-free compression and decompression algorithms for three classes of such sources (whose support is a subset of {0,1}^n).

1. We show how to compress sources X samplable by logspace machines to expected length H(X)+O(1).

Our next ... more >>>

TR05-022 | 19th February 2005
Omer Reingold, Luca Trevisan, Salil Vadhan

Pseudorandom Walks in Biregular Graphs and the RL vs. L Problem

Motivated by Reingold's recent deterministic log-space algorithm for Undirected S-T Connectivity (ECCC TR 04-94), we revisit the general RL vs. L question, obtaining the following results.

1. We exhibit a new complete problem for RL: S-T Connectivity restricted to directed graphs for which the random walk is promised to have ... more >>>

TR05-061 | 15th June 2005
Ronen Gradwohl, Guy Kindler, Omer Reingold, Amnon Ta-Shma

On the Error Parameter of Dispersers

Optimal dispersers have better dependence on the error than
optimal extractors. In this paper we give explicit disperser
constructions that beat the best possible extractors in some
parameters. Our constructions are not strong, but we show that
having such explicit strong constructions implies a solution
to the Ramsey graph construction ... more >>>

TR05-092 | 23rd August 2005
Eyal Rozenman, Salil Vadhan

Derandomized Squaring of Graphs

We introduce a "derandomized" analogue of graph squaring. This
operation increases the connectivity of the graph (as measured by the
second eigenvalue) almost as well as squaring the graph does, yet only
increases the degree of the graph by a constant factor, instead of
squaring the degree.

One application of ... more >>>

TR05-098 | 4th September 2005
Oded Goldreich

Bravely, Moderately: A Common Theme in Four Recent Results

We highlight a common theme in four relatively recent works
that establish remarkable results by an iterative approach.
Starting from a trivial construct,
each of these works applies an ingeniously designed
sequence of iterations that yields the desired result,
which is highly non-trivial. Furthermore, in each iteration,
more >>>

TR06-058 | 25th April 2006
Alexander Healy

Randomness-Efficient Sampling within NC^1

Revisions: 1

We construct a randomness-efficient averaging sampler that is computable by uniform constant-depth circuits with parity gates (i.e., in AC^0[mod 2]). Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within NC^1. For example, we obtain the following results:

... more >>>

TR09-057 | 23rd June 2009
Yonatan Bilu, Nathan Linial

Are stable instances easy?

We introduce the notion of a stable instance for a discrete
optimization problem, and argue that in many practical situations
only sufficiently stable instances are of interest. The question
then arises whether stable instances of NP--hard problems are
easier to solve. In particular, whether there exist algorithms
that solve correctly ... more >>>

TR09-105 | 27th October 2009
Vikraman Arvind, Srikanth Srinivasan

The Remote Point Problem, Small Bias Spaces, and Expanding Generator Sets

Using $\epsilon$-bias spaces over F_2 , we show that the Remote Point Problem (RPP), introduced by Alon et al [APY09], has an $NC^2$ algorithm (achieving the same parameters as [APY09]). We study a generalization of the Remote Point Problem to groups: we replace F_n by G^n for an arbitrary fixed ... more >>>

TR09-135 | 10th December 2009
Zeev Dvir, Avi Wigderson

Monotone expanders - constructions and applications

The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to:
(1) Constant degree dimension expanders in finite ... more >>>

TR16-152 | 27th September 2016
Oded Goldreich

Deconstructing 1-local expanders

Contemplating the recently announced 1-local expanders of Viola and Wigderson (ECCC, TR16-129, 2016), one may observe that weaker constructs are well know. For example, one may easily obtain a 4-regular $N$-vertex graph with spectral gap that is $\Omega(1/\log^2 N)$, and similarly a $O(1)$-regular $N$-vertex graph with spectral gap $1/\tildeO(\log N)$.
more >>>

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