Sorting in c log n parallel steps
Combinatorica
Bounded-width polynomial-size branching programs recognize exactly those languages in NC1
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
The probability of generating the symmetric group
Journal of Combinatorial Theory Series A
Local expansion of vertex-transitive graphs and random generation in finite groups
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
On the diameter of a Cayley graph of a simple group of Lie type based on a conjugacy class
Journal of Combinatorial Theory Series A
On sampling generating sets of finite groups and product replacement algorithm: extended abstract
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
The product replacement algorithm is polynomial
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
The Product Replacement Prospector
Journal of Symbolic Computation
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Let G be a finite group. Efficient generation of nearly uniformly distributed random elements in G, starting from a given set of generators of G, is a central problem in computational group theory. In this paper we demonstrate a weakness in the popular "product replacement algorithm," widely used for this purpose. The main results are the following. Let Nk(G) be the set of generating k-tuples of elements of G. Consider the distribution of the first components of the k-tuples in Nk(G) induced by the uniform distribution over Nk(G). We show that there exist infinite sequences of gtoups G such that this distribution is very far from uniform in two different senses: (1) its variation distance from uniform is 1 - ε and (2) there exists a short word (of length (loglog |G|)O(k)) which separates the two distributions with probability 1 - ε. The class of groups we analyze is direct powers of alternating groups. The methods used include statistical analysis of permutation groups, the theory of random walks, the AKS sorting network, and a randomized simulation of monotone Boolean operations by group operations, inspired by Barrington's work on bounded-width branching programs. The problem is motivated by the product replacement algorithm which was introduced in [Comm. Algebra 23 (1995) 4931-4948] and is widely used. Our results show that for certain groups the probability distribution obtained by the product replacement algorithm has a bias which can be detected by a short straight line program.