318 research outputs found

    Monotone expansion

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    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

    Lossless Dimension Expanders via Linearized Polynomials and Subspace Designs

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    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

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    We show how to construct (1+ε)(1+\varepsilon)-spanner over a set PP of nn points in Rd\mathbb{R}^d that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters ϑ,ε(0,1)\vartheta,\varepsilon \in (0,1), the computed spanner GG has O(εcϑ6nlogn(loglogn)6) O\bigl(\varepsilon^{-c} \vartheta^{-6} n \log n (\log\log n)^6 \bigr) edges, where c=O(d)c= O(d). Furthermore, for any kk, and any deleted set BPB \subseteq P of kk points, the residual graph GBG \setminus B is (1+ε)(1+\varepsilon)-spanner for all the points of PP except for (1+ϑ)k(1+\vartheta)k of them. No previous constructions, beyond the trivial clique with O(n2)O(n^2) edges, were known such that only a tiny additional fraction (i.e., ϑ\vartheta) 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

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    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

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    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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