Case f(x) g(x) 1.1 −1 + bxm/2 + xm 1− bxm/2 + x3m 1.2 1 + bxm/2 + xm b+ xm/2 + x5m/2 1.3 −1 + bxm/2 + xm b− bxm + x5m/2 1.4 −1 + bxm/2 + xm −b− x3m/2 + x5m/2 1.5 1 + bxm/2 + xm b+ bx4m/2 + x5m/2 1.6 1 + bxm/2 + xm 1 + xm + x2m 1.7 −1 + bxm/2 + xm b+ xm + x3m/2 1.8 −1 + bxm/2 + xm −b− bxm + x3m/2 1.9 a− xm/3 + xm −a− xm/3 + x3m 1.10 a− xm/3 + xm 1 + x2m/3 + x8m/3 1.11 a+ xm/3 + xm a+ ax2m/3 + x7...

We consider the task of interconnecting processors to realize efficient parallel algorithms. We propose interconnecting processors using certain graphs called expander graphs, which can provide fast communication from any group of processors to the rest of the network. We show that these interconnections would result in a number of efficient parallel algorithms for sorting, routing, associative...

Abstract The junction-tree representation provides an attractive structural property for organising a decomposable graph . In this study, we present two novel stochastic algorithms, referred to as the expander and collapser , sequential sampling of junction trees graphs. We show that recursive application expander, which expands incrementally underlying with one vertex at time, has full support...

We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results concerning the variation of Galois representations in oneparameter families of abelian varieties.

This paper discusses ‘geometric property (T)’. This is a property of metric spaces introduced in earlier work of the authors for its applications to K-theory. Geometric property (T) is a strong form of ‘expansion property’: in particular for a sequence (Xn) of bounded degree finite graphs, it is strictly stronger than (Xn) being an expander in the sense that the Cheeger constants h(Xn) are boun...

Consider a particle that moves on a connected, undirected graph G with n vertices. At each step the particle goes from the current vertex to one of its neighbors, chosen uniformly at random. The cover time is the rst time when the particle has visited all the vertices in the graph starting from a given vertex. In this paper, we present upper and lower bounds that relate the expected cover time ...

We prove an exponential lower bound on the size of static LovászSchrijver proofs of Tseitin tautologies. We use several techniques, namely, translating static LS+ proof into Positivstellensatz proof of Grigoriev et al., extracting a “good” expander out of a given graph by removing edges and vertices of Alekhnovich et al., and proving linear lower bound on the degree of Positivstellensatz proofs...

We show that the critical probability for percolation on a d-regular nonamenable graph of large girth is close to the critical probability for percolation on an infinite d-regular tree. This is a special case of a conjecture due to O. Schramm on the locality of pc. We also prove a finite analogue of the conjecture for expander graphs.

We show that RL ⊆ L/O(n), i.e., any language computable in randomized logarithmic space can be computed in deterministic logarithmic space with a linear amount of non-uniform advice. To prove our result we use an ultra-low space walk on the Gabber-Galil expander graph due to Gutfreund and Viola. ∗Work done while at TTI-Chicago.

Let G = L ∪ R be a bipartite graph with n left vertices and m right vertices. Then G corresponds to a binary code CG with block length n, and dimension k ≥ n −m, where the left vertices correspond to variables x1, . . . , xn ∈ {0, 1}, while the right vertices c1, . . . , cm correspond to constraints, such that ci is satisfied iff ∑ u↔ci xu = 0. Let SAT be the set of satisfied constraints, and U...