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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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. 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...
On the Diameter of Finite Groups
 SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 1990
"... The diameter of a group G with respect to a set S of generators is the maximum over g 2 G of the length of the shortest word in S [ S 1 representing g. This concept arises in the contexts of efficient communication networks and Rubik's cube type puzzles. "Best" generators (giving mini ..."
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Cited by 29 (4 self)
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The diameter of a group G with respect to a set S of generators is the maximum over g 2 G of the length of the shortest word in S [ S 1 representing g. This concept arises in the contexts of efficient communication networks and Rubik's cube type puzzles. "Best" generators (giving minimum diameter while keeping the number of generators limited) are pertinent to networks, "worst" and "average" generators seem a more adequate model for puzzles. We survey a substantial body of recent work by the authors on these subjects. Regarding the "best" case, we show that while the structure of the group is essentially irrelevant if S is allowed to exceed (log G) 1+c (c > 0), it plays a heavy role when jSj = O(1). In particular, every nonabelian nite simple group has a set of 7 generators giving logarithmic diameter. This cannot happen for groups with an abelian subgroup of bounded index. { Regarding the worst case, we are concerned primarily with permutation groups of degree n and obtain a tight exp((n ln n) 1=2 (1 + o(1))) upper bound. In the average case, the upper bound improves to exp((ln n) 2 (1 + o(1))). As a rst step toward extending this result to simple groups other than An , we establish that almost every pair of elements of a classical simple group G generates G, a result previously proved by J. Dixon for An . In the limited space of this article, we try to illuminate some of the basic underlying techniques.
Random Walk on Finite Groups
 In Probability on Discrete Structures
, 2003
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Simple groups, permutation groups, and probability
 In Proceedings of the International Congress of Mathematicians
, 1999
"... In recent years probabilistic methods have proved useful in the solution of several problems concerning finite groups, mainly involving simple groups and permutation groups. In some cases the probabilistic nature of the problem is apparent from its very formulation (see [KL], [GKS], [LiSh1]); but in ..."
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Cited by 21 (1 self)
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In recent years probabilistic methods have proved useful in the solution of several problems concerning finite groups, mainly involving simple groups and permutation groups. In some cases the probabilistic nature of the problem is apparent from its very formulation (see [KL], [GKS], [LiSh1]); but in other cases the use of probability,
Random permutations: some grouptheoretic aspects
 257–262. SIZE AND METRIC DIMENSION Page 33 of 34
, 1993
"... The study of asymptotics of random permutations was initiated by Erdos and Tunto. in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group theory make it reasonable to ask questions with a more grouptheoretic flavour. Two examples considered ..."
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Cited by 12 (2 self)
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The study of asymptotics of random permutations was initiated by Erdos and Tunto. in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group theory make it reasonable to ask questions with a more grouptheoretic flavour. Two examples considered here are membership in a proper transitive subgroup, and the intersection of a subgroup with a random conjugate. These both arise from other topics (quasigroups, bases for permutation groups, and design constructions). 1. Permutations lying in a transitive subgroup Sn and An denote the symmetric and alternating groups on the set X = {I,.... n}. A subgroup G of S " is transitive if, for all i, j E X, there exists g E G with ig ~ j. In a preliminary version of this paper, we asked the following question: Question 1.1. Is it true that,/or almost all permutations g E Sn. the only transitive subgroups containing g are Sn and (possihly) An? Here, of course, 'almost all g E S " have property P ' means 'the proportion of elements of S " not having property P tends to 0 as n> ex'. An affirmative answer to this question was given by Luczak and Pyber, in [15]. We will
Residual properties of free groups and probabilistic methods, submitted for publication
 J. reine angew. Math. (Crelle’s
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Expander graphs in pure and applied mathematics
 Bull. Amer. Math. Soc. (N.S
"... Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number th ..."
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Cited by 10 (0 self)
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Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics. This paper is based on notes prepared for the Colloquium Lectures at the
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].
The probability of generating the symmetric group
 Bull. London Math. Soc
, 1978
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