The Product Replacement Algorithm is a practical algorithm for generating random elements of a finite group. The algorithm can be described as a random walk on a graph whose vertices are the generating k-tuples of the group (for a fixed k). We show that there is a function c(r) such that for any finite simple group of Lie type, with Lie rank r, the Product Replacement Graph of the generating k-...

The product replacement algorithm is a commonly used heuristic to generate random group elements in a finite group G, by running a random walk on generating k-tuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an extensive review of both positive and negative theoretical results in the analysis of the algorithm.

A problem of great importance in computational group theory is to generate (nearly) uniformly distributed random elements in a finite groupG. A good example of such an algorithm should start at any given set of generators, use no prior knowledge of the structure of G, and in a polynomial number of group operations return the desired random group elements (see [Bb2]). Then these random elements ...

We present new variations on the product replacement method for generating random elements in a black box group, together with some basic analysis and conjectures.

The \product replacement algorithm" is a commonly used heuristic to generate random group elements in a nite group G, by running a random walk on generating k-tuples of G. While experiments showed outstanding performance, the theoretical explanation remained mysterious. In this paper we propose a new approach to study of the algorithm, by using Kazhdan's property (T) from representation theory ...

We prove that the product replacement graph on generating 3-tuples of An is connected for n ≤ 11. We employ an efficient heuristic based on [P1] which works significantly faster than brute force. The heuristic works for any group. Our tests were confined to An due to the interest in Wiegold’s Conjecture, usually stated in terms of T -systems (see [P2]). Our results confirm Wiegold’s Conjecture ...

The product replacement algorithm is a commonly used heuristic to generate random group elements in a nite group G, by running a random walk on generating k-tuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an extensive review of both positive and negative theoretical results in the analysis of the algorithm.

Warranty is a powerful marketing tool for the manufacturer and a good protection for both the manufacturer and the customer. However, warranty always involves additional costs to the manufacturer, which depend on product reliability characteristics and warranty parameters. This paper presents an approach to optimisation of warranty parameters for known product failure distribution to reduce the...