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• W2129021807 abstract "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 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 - e and (2) there exists a short word (of length (loglog |G|)O(k)) which separates the two distributions with probability 1 - e. 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." @default.
• W2129021807 created "2016-06-24" @default.
• W2129021807 creator A5014978503 @default.
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• W2129021807 date "2000-02-01" @default.
• W2129021807 modified "2023-09-26" @default.
• W2129021807 title "Strong bias of group generators: an obstacle to the “product replacement algorithm”" @default.
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• W2129021807 doi "https://doi.org/10.5555/338219.338616" @default.
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