Secure multiparty computation (MPC) on incomplete communication networks has been studied within two primary models: (1) Where a partial network is fixed priori, and thus corruptions can occur dependent its structure, (2) edges in the graph are determined dynamically as part of protocol. Whereas rich literature succeeded mapping out feasibility limitations structures supporting secure fixed-gra...

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 [a] 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 much...

We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is. As an application, we show that if G is a...

This paper shows how to compute O( √ logn)-approximations to the Sparsest Cut and Balanced Separator problems in Õ(n2) time, thus improving upon the recent algorithm of Arora, Rao, and Vazirani [Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 222–231]. Their algorithm uses semidefinite programming and requires Õ(n9.5) time. Our algorithm relies on efficiently find...

We consider dictionaries that perform lookups by probing a single word of memory, knowing only the size of the data structure. We describe a randomized dictionary where a lookup returns the correct answer with probability 1− , and otherwise returns “don’t know”. The lookup procedure uses an expander graph to select the memory location to probe. Recent explicit expander constructions are shown t...

Now, it is easy to see from the definition that a complete graph is definitely an edge expander as per the above definition. For our applications, we will be mostly interested in expander graphs which have a much smaller number of edges. For example, we would be interested in d-regular graphs which are expanders with parameters d = O(1) and α = θ(1). Clearly, α cannot exceed d. The existence of...

A matrix-weighted graph is an undirected with a k×k symmetric positive semidefinite matrix assigned to each edge. Such graphs admit natural generalizations of the Laplacian and adjacency matrices, leading generalized notion expansion. Extensions some theorems about expansion hold for graphs—in particular, analogue expander mixing lemma one half Cheeger-type inequality. These results lead defini...

Here we describe at greater length the mathematical basis and practical implementation of the poolMC smart pooling strategy. Also included are complete versions of the figures used in the paper to analyze the performance of poolMC in a pooled microarray experiment. Pooling strategy Several pooling methods have been discussed in the literature [1, 2] (Note: The reference numbers pertain to those...

We construct a deterministic approximation algorithm for computing a permanent of a 0, 1 n by n matrix to within a multiplicative factor (1 + ǫ)n, for arbitrary ǫ > 0. When the graph underlying the matrix is a constant degree expander our algorithm runs in polynomial time (PTAS). In the general case the running time of the algorithm is exp(O(n 2 3 log n)). For the class of graphs which are cons...

We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a (1 + ε)-resistance sparsifier of size Õ(n/ε), and conjecture this bound holds for all graphs on n nodes. In comparison, spectral sparsification i...