How to construct random functions
Journal of the ACM (JACM)
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FoCM '97 Selected papers of a conference on Foundations of computational mathematics
A Pseudorandom Generator from any One-way Function
SIAM Journal on Computing
On the Implementation of Huge Random Objects
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
The Strange Logic of Random Graphs
The Strange Logic of Random Graphs
Deterministic Extractors for Affine Sources over Large Fields
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Implementing Huge Sparse Random Graphs
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
On the Implementation of Huge Random Objects
SIAM Journal on Computing
On the complexity of powering in finite fields
Proceedings of the forty-third annual ACM symposium on Theory of computing
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We construct efficiently computable sequences of random-looking graphs that preserve properties of the canonical random graphs G (2n,p(n)). We focus on first-order graph properties, namely properties that can be expressed by a formula φ in the language where variables stand for vertices and the only relations are equality and adjacency (e.g. having an isolated vertex is a first-order property ∃x∀y(¬EDGE(x, y))). Random graphs are known to have remarkable structure w.r.t. first order properties, as indicated by the following 0/1 law: for a variety of choices of p(n), any fixed first-order property φ holds for G (2n,p(n)) with probability tending either to 0 or to 1 as n grows to infinity. We first observe that similar 0/1 laws are satisfied by G (2n,p(n)) even w.r.t. sequences of formulas {φn}n∈ℕ with bounded quantifier depth, ${\it depth}(\phi_{n}) \leq {\frac {1}{{\rm lg} (1/p(n))}}$. We also demonstrate that 0/1 laws do not hold for random graphs w.r.t. properties of significantly larger quantifier depth. For most choices of p(n), we present efficient constructions of huge graphs with edge density nearly p(n) that emulate G (2n,p(n)) by satisfying ${\it \Theta} ({\frac {1}{{\rm lg} (1/p(n))}})-0/1$ laws. We show both probabilistic constructions (which also have other properties such as K-wise independence and being computationally indistinguishable from G (N,p(n)) ), and deterministic constructions where for each graph size we provide a specific graph that captures the properties of G (2n,p(n)) for slightly smaller quantifier depths.