We show that r pseudo-random bits can be obtained by concatenating t blocks of r=t pseudo-random bits, where the blocks are generated in parallel. This can be considered as a parallel version of 8] { recycling random bits by doing a random walk on an expander. The proof is based on the fact that t simultaneous independent random walks on an expander graph is equivalent to a random walk on a muc...

A code C ⊆ Fn 2 is a (c, ǫ, δ)-expander code if it has a Tanner graph, where every variable node has degree c, and every subset of variable nodes L0 such that |L0| ≤ δn has at least ǫc|L0| neighbors. Feldman et al. (IEEE IT, 2007) proved that LP decoding corrects 3ǫ−2 2ǫ−1 · (δn − 1) errors of (c, ǫ, δ)expander code, where ǫ > 2 3 + 1 3c . In this paper, we provide a simpler proof of their resu...

We show that every locally sparse graph contains a linearly sized expanding subgraph. For constants c1 > c2 > 1, 0 < α < 1, a graph G on n vertices is called a (c1, c2, α)-graph if it has at least c1n edges, but every vertex subset W ⊂ V (G) of size |W | ≤ αn spans less than c2|W | edges. We prove that every (c1, c2, α)-graph with bounded degrees contains an induced expander on linearly many ve...

Designing distributed and scalable algorithms to improve network connectivity is a central topic in peer-to-peer networks. In this paper we focus on the following well-known problem: given an n-node d-regular network for d = Ω(log n), we want to design a decentralized, local algorithm that transforms the graph into one that has good connectivity properties (low diameter, expansion, etc.) withou...

We study the problem of approximating the quality of a disperser. A bipartite graph G on ([N ], [M ]) is a (ρN, (1 − δ)M)-disperser if for any subset S ⊆ [N ] of size ρN , the neighbor set Γ(S) contains at least (1 − δ)M distinct vertices. Our main results are strong integrality gaps in the Lasserre hierarchy and an approximation algorithm for dispersers. 1. For any α > 0, δ > 0, and a random b...

Providing security for a wireless sensor network composed of small sensor nodes with limited battery power and memory can be a nontrivial task. A variety of key predistribution schemes have been proposed which allocate symmetric keys to the sensor nodes before deployment. In this paper we examine the role of expander graphs in key predistribution schemes for wireless sensor networks. Roughly sp...

We show that the recent results for obtaining O( √ log n)-approximation to sparsest cut and balanced separator problems due to Arora, Rao, and Vazirani (2004) can be used to derive an Õ(n) time approximation algorithm for an n-node graph. The previous best algorithm needed to solve a semidefinite program with O(n) constraints. Our algorithm relies on efficiently finding expander flows in the gr...

Consider an expander graph in which a μ fraction of the vertices are marked. A random walk starts at a uniform vertex and at each step continues to a random neighbor. Gillman showed in 1998 that the number of marked vertices seen in a random walk of length n is concentrated around its expectation, Φ := μn, independent of the size of the graph. Here we provide a new and sharp tail bound, improvi...

Recall in last lecture that we defined a (n, d, λ)-expander to be a d-regular n-vertex undirected graph with second eigenvalue λ. We also defined the edge expansion of a graph G with vertex set V to be φ(G) = min S⊂V |S|≤n/2 |E(S, S)| |S| , where E(S, S) is the set of edges between a vertex set S and its complement. The following lemma shows that the eigenvalue formulation of an expander is ess...

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. ’99] also on proof size. [Alekhnovich and Razborov ’03] established that if the clause-variable incidence graph of a CNF formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We fur...