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What Do We Know About The Product Replacement Algorithm?
 in: Groups ann Computation III
, 2000
"... . 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 ktuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an exten ..."
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Cited by 30 (7 self)
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. 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 ktuples 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. Introduction In the past few decades the study of groups by means of computations has become a wonderful success story. The whole new field, Computational Group Theory, was developed out of needs to discover and prove new results on finite groups. More recently, the probabilistic method became an important tool for creating faster and better algorithms. A number of applications were developed which assume a fast access to (nearly) uniform group elements. This led to a development of the so called "product replacement algorithm", which is a commonly used heuristic to generate random group elemen...
The Product Replacement Algorithm is Polynomial
 In Proc. 41 st IEEE Symposium on Foundations of Computer Science (FOCS
, 2000
"... The product replacement algorithm is a heuristic designed to generate random group elements. The idea is to run a random walk on generating ktuples of the group, and then output a random component. The algorithm was designed by LeedhamGreen and Soicher ([31]), and further investigated in [12]. It ..."
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Cited by 21 (4 self)
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The product replacement algorithm is a heuristic designed to generate random group elements. The idea is to run a random walk on generating ktuples of the group, and then output a random component. The algorithm was designed by LeedhamGreen and Soicher ([31]), and further investigated in [12]. It was found to have an outstanding performance, much better than the the previously known algorithms (see [12, 22, 26]). The algorithm is now included in two major group algebra packages GAP [42] and MAGMA [10]. In spite of the many serious attempts and partial results, (see [6, 14, 15, 21, 22, 32, 39, 40]), the analysis of the algorithm remains difficult at best. For small values of k even graph connectivity becomes a serious obstacle (see [19, 37, 39, 40]). The most general results are due to Diaconis and SaloffCoste [22], who used a state of the art analytic technique to obtain polynomial bounds in special cases, and (sub)exponential bounds in general case. The main result of this pape...
Expansion of Product Replacement Graphs
 Combinatorica
, 2001
"... . We establish a connection between the expansion coefficient of the product replacement graph \Gamma k (G) and the minimal expansion coefficient of a Cayley graph of G with k generators. In particular, we show that the product replacement graphs \Gamma k \Gamma PSL(2; p) \Delta form an expander ..."
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Cited by 9 (1 self)
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. We establish a connection between the expansion coefficient of the product replacement graph \Gamma k (G) and the minimal expansion coefficient of a Cayley graph of G with k generators. In particular, we show that the product replacement graphs \Gamma k \Gamma PSL(2; p) \Delta form an expander family, under assumption that all Cayley graphs of PSL(2; p), with at most k generators are expanders. This gives a new explanation of the outstanding performance of the product replacement algorithm and supports the speculation that all product replacement graphs are expanders [LP,P3].
On sampling generating sets of finite groups and product replacement algorithm. (Extended Abstract)
 Proceedings of ISSAC'99, 9196
, 1999
"... ) IGOR PAK , SERGEY BRATUS y 1 Introduction Let G be a finite group. A sequence of k group elements (g1 ; : : : ; gk ) is called a generating ktuple of G if the elements generate G (we write hg1 ; : : : ; gk i = G). Let Nk (G) be the set of all generating ktuples of G, and let Nk (G) = jNk ( ..."
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Cited by 8 (8 self)
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) IGOR PAK , SERGEY BRATUS y 1 Introduction Let G be a finite group. A sequence of k group elements (g1 ; : : : ; gk ) is called a generating ktuple of G if the elements generate G (we write hg1 ; : : : ; gk i = G). Let Nk (G) be the set of all generating ktuples of G, and let Nk (G) = jNk (G)j. We consider two related problems on generating ktuples. Given G and k ? 0, 1) Determine Nk (G) 2) Generate random element of Nk (G), each with probability 1=Nk (G) The problem of determining the structure of Nk (G) is of interest in several contexts. The counting problem goes back to Philip Hall, who expressed Nk (G) as a Mobius type summation of Nk (H) over all maximal subgroups H ae G (see [23]). Recently the counting problem has been studied for large simple groups where remarkable progress has been made (see [25, 27]). In this paper we analyze Nk for solvable groups and products of simple groups. The sampling problem, while often used in theory as a tool for approximate counting...
Black box exceptional groups of Lie type
 In preparation
, 2002
"... If a black box group is known to be isomorphic to an exceptional simple group of Lie type of rank> 1, other than any 2 F4(q), over a field of known size, a Las Vegas algorithm is used to produce a constructive isomorphism. This yields an upgrade of all known nearly linear time Monte Carlo permutatio ..."
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Cited by 5 (2 self)
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If a black box group is known to be isomorphic to an exceptional simple group of Lie type of rank> 1, other than any 2 F4(q), over a field of known size, a Las Vegas algorithm is used to produce a constructive isomorphism. This yields an upgrade of all known nearly linear time Monte Carlo permutation group algorithms to Las Vegas algorithms when the input group has no composition factor isomorphic to a rank 1 group or to any 2 F4(q). 1
Computing with Matrix Groups
 GROUPS, COMBINATORICS AND GEOMETRY
, 2001
"... A group is usually input into a computer by specifying the group either using a presentation or using a generating set of permutations or matrices. Here we will emphasize the latter approach, referring to [Si3, Si4, Ser1] for details of the other situations. Thus, the basic computational setting dis ..."
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Cited by 4 (4 self)
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A group is usually input into a computer by specifying the group either using a presentation or using a generating set of permutations or matrices. Here we will emphasize the latter approach, referring to [Si3, Si4, Ser1] for details of the other situations. Thus, the basic computational setting discussed here is as follows: a group is given, speciﬁed as G = X in terms of some generating set X of its elements, where X is an arbitrary subset of either Sn or GL(d, q ) (a familiar example is the group of Rubik’s cube). The goal is then to ﬁnd properties of G eﬃciently, such as G, the derived series, a composition series, Sylow subgroups, and so on.
Polynomialtime computation in matrix groups
, 1999
"... This dissertation investigates deterministic polynomialtime computation in matrix groups over finite fields. Of particular interest are matrixgroup problems that resemble testing graph isomorphism. The main results are instances where the problems admit polynomialtime solutions and methods that e ..."
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Cited by 4 (3 self)
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This dissertation investigates deterministic polynomialtime computation in matrix groups over finite fields. Of particular interest are matrixgroup problems that resemble testing graph isomorphism. The main results are instances where the problems admit polynomialtime solutions and methods that enable such efficiency.