Computational Group Theory Research Articles

Construction of Occurrence Graphs with Permutation Symmetries Aided by the Backtrack Method

<p>This paper recalls the concept of occurrence graphs with permuta- tion symmetries (OS-graphs) for Coloured Petri Nets. It is explained how so-called self-symmetries can help to speed up construction of OS- graphs. The contribution of the paper is to suggest a new method for calculation of self-symmetries, the Backtrack Method. The method is based on the so-called Backtrack Algorithm, which originates in com- putational group theory. The suggestion of the method is justified, both by identifying an important general complexity property and by obtaining encouraging experimental performance measures.</p><p><strong>Topics.</strong> Coloured Petri Nets, reduced state spaces, occurrence graphs with permutation symmetries, self-symmetries, computational group theory, backtrack searches.</p>

The ‘Burnside Process’ Converges Slowly

We consider the problem of sampling ‘unlabelled structures’, i.e., sampling combinatorial structures modulo a group of symmetries. The main tool which has been used for this sampling problem is Burnside’s lemma. In situations where a significant proportion of the structures have no nontrivial symmetries, it is already fairly well understood how to apply this tool. More generally, it is possible to obtain nearly uniform samples by simulating a Markov chain that we call the Burnside process: this is a random walk on a bipartite graph which essentially implements Burnside’s lemma. For this approach to be feasible, the Markov chain ought to be ‘rapidly mixing’, i.e., converge rapidly to equilibrium. The Burnside process was known to be rapidly mixing for some special groups, and it has even been implemented in some computational group theory algorithms. In this paper, we show that the Burnside process is not rapidly mixing in general. In particular, we construct an infinite family of permutation groups for which we show that the mixing time is exponential in the degree of the group.

Cyclic Presentations of the Trivial Group

We report on a computational group theory experiment involving a search for cyclic presentations of the trivial group. The list of such presentations obtained includes counterexamples to a conjecture of M J Dunwoody.

On the Structure and Classification of SOMAs: Generalizations of Mutually Orthogonal Latin Squares

Let $k\ge0$ and $n\ge2$ be integers. A SOMA, or more specifically a SOMA$(k,n)$, is an $n\times n$ array $A$, whose entries are $k$-subsets of a $kn$-set $\Omega$, such that each element of $\Omega$ occurs exactly once in each row and exactly once in each column of $A$, and no 2-subset of $\Omega$ is contained in more than one entry of $A$. A SOMA$(k,n)$ can be constructed by superposing $k$ mutually orthogonal Latin squares of order $n$ with pairwise disjoint symbol-sets, and so a SOMA$(k,n)$ can be seen as a generalization of $k$ mutually orthogonal Latin squares of order $n$. In this paper we first study the structure of SOMAs, concentrating on how SOMAs can decompose. We then report on the use of computational group theory and graph theory in the discovery and classification of SOMAs. In particular, we discover and classify SOMA$(3,10)$s with certain properties, and discover two SOMA$(4,14)$s (SOMAs with these parameters were previously unknown to exist). Some of the newly discovered SOMA$(3,10)$s come from superposing a Latin square of order 10 on a SOMA$(2,10)$.

Computational group theory in St. Petersburg

The paper is devoted to the history of computational group theory in St. Petersburg. Some features of this scientific area are outlined. We mention here some results of A. I. Skopin concerned with computations in Burnside groups of small prime-power exponents, as well as A. V. Rukolaine's work on the computation of ordinary characters of some finite simple groups. The note is dedicated to the 70th anniversary of A. I. Skopin's birth. Bibliography: 24titles.

An Algorithm for Solving the Factorization Problem in Permutation Groups

The factorization problem in permutation groups is to represent an elementgof some permutation groupGas a word over a given setSof generators ofG. For practical purposes, the word should be as short as possible, but must not be minimal. Like many other problems in computational group theory, the problem can be solved from a strong generating set (SGS) and a base ofG. Different algorithms to compute an SGS and a base have been published. The classical algorithm is the Schreier–Sims method. However, for factorization an SGS is needed that has all its elements represented as words overS. The existing methods are not suitable, because they lead to an exponential growth of word lengths. This article presents a simple algorithm to solve the factorization problem. It is based on computing an SGS with elements represented by relatively short words over the generators.

Processor interconnection networks from Cayley graphs

Cayley graphs of groups are presently being considered by the computer science community as models of architectures for large scale parallel processor computers. In the first section of this paper we discuss Cayley graphs and show how they may be used as a tool for the design and analysis of network architectures for these types of computers. Observing that routing on a Cayley graph is equivalent to a certain factoring problem in the associated group, we have been able to use a known powerful factoring technique in computational group theory to produce a fast efficient routing algorithm on the associated Cayley graph. In the second section of this paper we present this work. This research can be regarded as a first attempt to find general purpose routing algorithms for interconnection networks. Believing that average diameter of a network for a large scale MIMD machine is the predominant factor in determining network performance, we designed Cayley graphs to be used in a special study performed at the Supercomputing Research Center (SRC). The importance of the average diameter in determining network performance was supported by the fact that the graphs found by us had the smallest average diameter and outperformed all other graphs evaluated in the study. In fact, before being driven into saturation, one of our graphs sustained 9.4% more network traffic than the next best candidate, a butterfly architecture, and 74.3% better than the bench mark 2-d mesh. The last section of our paper is devoted to this work. This paper is divided into three sections. In the first section we discuss Cayley graphs and show how they may be used as a tool for the design and analysis of network architectures for parallel computers. In the second section we present our research on the routing problem. This research can be regarded as a first attempt to find general purpose routing algorithms for interconnection networks. In the last section we present some evidence that average diameter of a network for a large scale MIMD machine is the predominant factor in determining network performance.

New methods for using Cayley graphs in interconnection networks

A number of researchers have proposed Cayley graphs and Schreier coset graphs as models for interconnection networks. New algorithms are presented for generating Cayley graphs in a more time-efficient manner than was previously possible. Alternatively, a second algorithm is provided for storing Cayley graphs in a space-efficient manner (log 2(3) bits per node), so that copies could be cheaply stored at each node of an interconnection network. The second algorithm is especially useful for providing a compact encoding of an optimal routing table (for example, a 13 kilobyte optimal table for 64,000 nodes). The algorithm relies on using a compact encoding of group elements known from computational group theory. Generalizations of all of the above are presented for Schreier coset graphs.

On doing Todd-Coxeter coset enumeration in parallel

The Todd-Coxeter coset enumeration method is without any doubt the best known and most used method in computational group theory. Surprisingly, however, it seems that no attempt has yet been made at producing a parallel version of it. This paper reports on such a parallel formulation.

Permutation group algorithms based on partitions, I: Theory and algorithms

A technique for computing in permutation groups of high degree is developed. The technique uses the idea of successive refinement of ordered partitions, introduced by B. McKay in connection with the graph isomorphism problem, to supplement the techniques of base and strong generating set developed earlier by Sims. Applications to a number of specific problems in computational group theory are presented.

A strong generating test and short presentations for permutation groups

The group membership problem for permutation groups is one of the most important problems of computational group theory. Solution of this problem seems to depend intrinsically on constructing a strong generating set. Until now, recognizing if a set of generators is strong has been thought to be as hard as constructing a strong generating set from an arbitrary generating set. This paper shows how to verify a strong generating set in O(n 4) time, where n is the size of the set on which the group acts. This is faster than the best known algorithms in the literature. The work also leads to related algorithms for discovering all orbit information contained in an arbitrary set of generators S in O( n|S| + n log n) time, and, if S is strong, for finding a presentation with no more than | S|( n −1) relations. Refinements in the analysis are given for the case in which a small base exists.

Constructing matrix representations of finitely presented groups

The Todd-Coxeter coset enumeration algorithm is one of the most important tools of computational group theory. It may be viewed as a means of constructing permutation representations of finitely presented groups. In this paper we present an analogous algorithm for directly constructing matrix representations over various fields. In fact the algorithm is more general than this, and can be used to construct matrix representations of finitely generated algebras. The algorithm (with some restrictions) has been implemented as a C program and some results obtained with this implementation are described.

Calculating characters of p-groups

A central problem in computational group theory is that of getting the table of ordinary characters for a finite group. The current technique (due to Dixon and Schneider) is based on an old idea of Burnside and takes no account of any special properties which the group may have. Over the last few years a number of very poverful techniques have been developed for computing with p -groups defined by a power-commutator presentation. In this paper a character table algorithm for such groups is presented. A trial implementation of this new algorithm indicates its superiority over the generalDixon-Schneider algorithm for p -groups having order greater than p 12 .

A surprising isomorphism

The group 4 +( a 12), as described by N. Biggs in “Computational Group Theory” (M. Atkinson, Ed., pp. 57–63, Academic Press, San Diego, CA/London, 1984) is a finitely presented group which arises naturally from the study of trivalent symmetric graphs containing a circuit of length 12. In this paper it is shown that 4 + ( a 12) is isomorphic to SL(3, Z ) extended by the automorphism which takes a matrix to the transpose of its inverse. Other finitely presented groups associated with this one are also discussed.

A New Base Change Algorithm for Permutation Groups

The computation of a strong generating set for a permutation group acting on a set $\Omega $ of n points is the fundamental operation that underlies most of the algorithms in computational group theory. Sims gave a change of basis algorithm that transforms a strong generating set relative to one ordering of $\Omega $ into a strong generating set relative to a different ordering. Base change is crucial for many of the important algorithms that have been implemented in the Cayley system, and is also important for many applications of computational group theory to combinatorial and search problems. Sims’s base change has worst-case time $O(n^5 )$. The main result of this paper is a new change of basis algorithm that has worst-case time $O(n^3 )$.

An application of quantum chemical methodology to liquid phase studies: Monte Carlo simulation of nonrigid acetone dissolved in carbon disulfide

Abstract This paper is concerned with the application of Quantum Chemistry to the study of liquid phase. The way we follow is Semi-classical: translation and overall rotation of molecules are treated classically (e.g. : Monte Carlo Computer Simulation): internal degrees of freedom and intermolecular interactions in the liquid are taken into account with the aid of Quantum Chemical tools (e.g.: Semiempirical Methods and Perturbation Theory). In this article we study a dilute solution of acetone (with motion of its two rotors) in carbon disulfide by means of the Monte Carlo method (MC). The structural information about the way in which solvent molecules surround acetone is given through atom-atom radial distributions functions (rdfs). The internal distribution function (idf) for the two-rotor molecule has been computed too. In order to save computational effort Group Theory of Nonrigid Molecules has been employed along the MC run.

A General backtrack algorithm for the isomorphism problem of combinatorial objects

Our aim is to present a practical algorithm for the isomorphism problem that can be easily adapted to any class of combinatorial objects. We investigate the underlying principles of backtrack algorithms that determine a canonical representative of a combinatorial object. We identify the parts of the algorithm that are dependent on the class of combinatorial objects and those parts that are independent of the class. An interface between the two parts is developed to provide a general backtrack algorithm for the isomorphism problem of combinatorial objects that incorporates the technique of branch-and-bound, and that also uses the automorphisms of the combinatorial object to prune the search tree. Our general algorithm incorporates from computational group theory an algorithm known as the base change algorithm. The base change algorithm allows one to recover as much information as possible about the automorphism group when a new branch of the search tree is processed. Thus, it can lead to greater pruning of the search tree. This work is intended to lead to a better understanding of the practical isomorphism algorithms. It is not intended as a contribution to the theoretical study of the complexity of the isomorphism problem.

Computational group theory in crystal physics: A review

A brief account is given of the rôle played by three-dimensional crystallographic point and space groups and their representations in computer programs in crystal physics. It covers their use in specific applications, for example energy band programs and lattice dynamics programs, together with their use in those programs written to calculate, for example, irreducible characters and matrix representatives, selection rules, Clebsch-Gordan coefficients, etc., which have general application in crystal physics.

Saving space in coset enumeration

Possibly the most frequently used algorithm in computational group theory is the Todd-Coxeter process for enumerating the cosets of a subgroup of a finitely presented group. In all implementations the power of this algorithm is limited much more by its storage requirements than by its execution time. This note outlines a technique which can often be used to economise on storage at the expense of an increased execution time. Before giving details it will be helpful to review some background.