318 research outputs found
Monotone expansion
This work, following the outline set in [B2], presents an explicit
construction of a family of monotone expanders. The family is essentially
defined by the Mobius action of SL_2(R) on the real line. For the proof, we
show a product-growth theorem for SL_2(R).Comment: 37 page
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Expanders with Symmetry: Constructions and Applications
Expanders are sparse yet well-connected graphs with numerous theoretical and practical uses. Symmetry is a valuable structure for expanders as it enables efficient algorithms and a richer set of applications. This thesis studies expanders with symmetry, giving new constructions and applications. We extend expander construction techniques to work with symmetry and give explicit constructions of expanders with varying quality of expansion and symmetries of various groups. In particular, we construct graphs with large Abelian group symmetries via the technique of \textit{graph lifts}. We also give a generic amplification procedure that converts a weak expander to an almost optimal one while preserving symmetries. This procedure is obtained by generalizing prior amplification techniques that work for Cayley graphs over Abelian groups to Cayley graphs over any finite group. In particular, we obtain almost-Ramanujan expanders over every non-abelian finite simple group. We then explore the utility of having both symmetry and expansion simultaneously. We obtain explicit quantum LDPC codes of almost linear distance and \textit{good} classical quasi-cyclic codes with varying circulant sizes using prior results and our constructions of graphs with Abelian symmetries. We show how our generic amplification machinery boosts various structured expander-like objects: \textit{quantum expanders}, \textit{dimension expanders}, and \textit{monotone expanders}. Finally, we prove a structural result about expanding Cayley graphs, showing that they satisfy a \enquote{degree-2} variant of the \textit{expander mixing lemma}. As an application of this, we give a randomness-efficient query algorithm for \textit{homomorphism testing} of unitary-valued functions on finite groups and a derandomized version of the celebrated Babai--Nikolov--Pyber (BNP) lemma
Lossless Dimension Expanders via Linearized Polynomials and Subspace Designs
For a vector space F^n over a field F, an (eta,beta)-dimension expander of degree d is a collection of d linear maps Gamma_j : F^n -> F^n such that for every subspace U of F^n of dimension at most eta n, the image of U under all the maps, sum_{j=1}^d Gamma_j(U), has dimension at least beta dim(U). Over a finite field, a random collection of d = O(1) maps Gamma_j offers excellent "lossless" expansion whp: beta ~~ d for eta >= Omega(1/d). When it comes to a family of explicit constructions (for growing n), however, achieving even modest expansion factor beta = 1+epsilon with constant degree is a non-trivial goal.
We present an explicit construction of dimension expanders over finite fields based on linearized polynomials and subspace designs, drawing inspiration from recent progress on list-decoding in the rank-metric. Our approach yields the following:
- Lossless expansion over large fields; more precisely beta >= (1-epsilon)d and eta >= (1-epsilon)/d with d = O_epsilon(1), when |F| >= Omega(n).
- Optimal up to constant factors expansion over fields of arbitrarily small polynomial size; more precisely beta >= Omega(delta d) and eta >= Omega(1/(delta d)) with d=O_delta(1), when |F| >= n^{delta}. Previously, an approach reducing to monotone expanders (a form of vertex expansion that is highly non-trivial to establish) gave (Omega(1),1+Omega(1))-dimension expanders of constant degree over all fields. An approach based on "rank condensing via subspace designs" led to dimension expanders with beta >rsim sqrt{d} over large fields. Ours is the first construction to achieve lossless dimension expansion, or even expansion proportional to the degree
A Spanner for the Day After
We show how to construct -spanner over a set of
points in that is resilient to a catastrophic failure of nodes.
Specifically, for prescribed parameters , the
computed spanner has edges, where . Furthermore, for any , and
any deleted set of points, the residual graph is -spanner for all the points of except for
of them. No previous constructions, beyond the trivial clique
with edges, were known such that only a tiny additional fraction
(i.e., ) lose their distance preserving connectivity.
Our construction works by first solving the exact problem in one dimension,
and then showing a surprisingly simple and elegant construction in higher
dimensions, that uses the one-dimensional construction in a black box fashion
Explicit expanders with cutoff phenomena
The cutoff phenomenon describes a sharp transition in the convergence of an
ergodic finite Markov chain to equilibrium. Of particular interest is
understanding this convergence for the simple random walk on a bounded-degree
expander graph. The first example of a family of bounded-degree graphs where
the random walk exhibits cutoff in total-variation was provided only very
recently, when the authors showed this for a typical random regular graph.
However, no example was known for an explicit (deterministic) family of
expanders with this phenomenon. Here we construct a family of cubic expanders
where the random walk from a worst case initial position exhibits
total-variation cutoff. Variants of this construction give cubic expanders
without cutoff, as well as cubic graphs with cutoff at any prescribed
time-point.Comment: 17 pages, 2 figure
Expander Construction in VNC1
We give a combinatorial analysis (using edge expansion) of a variant of the iterative expander construction due to Reingold, Vadhan, and Wigderson (2002), and show that this analysis can be formalized in the bounded arithmetic system VNC^1 (corresponding to the "NC^1 reasoning"). As a corollary, we prove the assumption made by Jerabek (2011) that a construction of certain bipartite expander graphs can be formalized in VNC^1. This in turn implies that every proof in Gentzen\u27s sequent calculus LK of a monotone sequent can be simulated in the monotone version of LK (MLK) with only polynomial blowup in proof size, strengthening the quasipolynomial simulation result of Atserias, Galesi, and Pudlak (2002)
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