We propose a problem concerning the determination of the threshold function for the edge probability that guarantees, almost surely, the existence of various sparse spanning subgraphs in random graphs. We prove some bounds and demonstrate them in the cases of a d-cube and a two dimensional lattice. B. Bollobás (cf. e.g., [3]) raised the following problem: Let G be a random graph with n = 2d ver...

We present an all-pairs shortest path algorithm whose running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from [0, 1] is O(n), in expectation and with high probability. This resolves a long standing open problem. The algorithm is a variant of the dynamic all-pairs shortest paths algorithm of Demetrescu and Italiano. The ana...

We consider the problem of testing graph expansion (either vertex or edge) in the bounded degree model (Goldreich & Ron, ECCC 2000). We give a property tester that takes as input a graph with degree bound d, an expansion bound α, and a parameter ε > 0. The tester accepts the graph with high probability if its expansion is more than α, and rejects it with high probability if it is ε-far from any...

Finding the largest clique is a notoriously hard problem, even on random graphs. It is known that the clique number of a random graph G(n, 1/2) is almost surely either k or k + 1, where k = ⌈2 log n − 2 log log n − 1⌉ (Section 4.5 in [1], also [2]). However, a simple greedy algorithm finds a clique of size only log n (1 + o(1)), with high probability, and finding larger cliques – that of size e...

We show that for random bit strings, Up(n), with probability, p = 12 , the firstorder quantifier depth D(Up(n)) needed to distinguish non-isomorphic structures is Θ(lg lg n), with high probability. Further, we show that, with high probability, for random ordered graphs, G≤,p(n) with edge probabiltiy p = 12 , D(G≤,p(n)) = Θ(log∗ n), contrasting with the results of random (non-ordered) graphs, Gp...

Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are the classical results of Tutte and Nash-Williams from 1961 which show that a λ...

Our interest is in the cumulative probabilities Pr(L(t) ≤ l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry r...

The single source shortest path (SSSP) problem lacks parallel solutions which are fast and simultaneously work-eecient. We propose simple criteria which divide Dijkstra's sequential SSSP algorithm into a number of phases, such that the operations within a phase can be done in parallel. We give a PRAM algorithm based on these criteria and analyze its performance on random digraphs with random ed...

The single source shortest path (SSSP) problem lacks parallel solutions which are fast and simultaneously work-eecient. We propose simple criteria which divide Dijkstra's sequential SSSP algorithm into a number of phases, such that the operations within a phase can be done in parallel. We give a PRAM algorithm based on these criteria and analyze its performance on random digraphs with random ed...

We present a method for dynamically gen­ erating Bayesian networks from knowledge bases consisting of first-order probability logic sentences. We present a subset of proba­ bility logic sufficient for representing the class of Bayesian networks with discrete-valued nodes. We impose constraints on the form of the sentences that guarantee that the knowl­ edge base contains all the probabilistic i...