Abstract. A variety of elementary combinatorial techniques for permutation groups are reviewed. It is shown how to apply these techniques to yield faster and/or more space-efficient algorithms for problems including group membership, normal closure, center, base change and Cayley graphs. Emphasis is placed on randomized techniques and new data structures. The paper includes both a survey of recent algorithms with which the authors have been associated, and some new algorithms in the same spirit that have not previously appeared in print. Many of the results include both complexity bounds and pseudo-code, along with comments for faster software implementations.

Many users of symbolic algebra systems have felt the need for greater CPU power. Yet few of them have ventured into parallel programming due to the steep learning curve and the unfamiliar programming environment entailed by such an effort. In an attempt to remedy that situation, the parallel library MPI has been integrated into both GCL (GNU Common LISP) and GAP [14] (a general purpose language for mathematical group theory). These implementations are examples that extend bindings of MPI to interactive languages. (MPI already has bindings to the compiled languages C and FORTRAN.) Further, this binding to an interactive language retains the interactive environment during execution. Further, STAR/MPI represents a blueprint for binding MPI to other interactive languages besides GCL and GAP, from which comes the name STAR/MPI, or /MPI. STAR/MPI includes a simple SPMD architecture on top of this MPI binding. An important class of sequential algorithms is described that can be parallelized with little effort using STAR/MPI architecture. Since GAP is representative of systems for discrete mathematics and LISP is the basis for several symbolic algebra systems with strengths in nondiscrete mathematics, it is hoped to gain broad feedback on the issues involved. Although vendor-specific, interactive, parallel languages exist, this appears to be the first attempt at defining a binding of a vendor-independent, portable, parallel library to arbitrary interactive languages.

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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.

l over the generators grows as c l for some constant c>1 depending on G. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals and Babai, who give a Las Vegas algorithm for the case of finite groups, as well as recent work of Babai, Beals, Cai, Ivanyos, and Luks, who give a deterministic algorithm for the case of abelian groups. # 1999 Academic Press Article ID jcss.1998.1614, available online at http:##www.idealibrary.com on 260 0022-0000#99 #30.00 Copyright # 1999 by Academic Press All rights of reproduction in any form reserved. * Research conducted while visiting IAS and DIMACS and supported in part by an NSF Mathematical Sciences

We present a fully automatic framework for identifying symmetries in structural descriptions of digital circuits and CTL* formulas and using them in a model checker. We show how the set of sub-formulas of a formula can be partitioned into equivalence classes so that truth values for only one sub-formula in any class need be evaluated for model checking. We unify and extend the theories developed by Clarke et al [CEFJ96] and Emerson and Sistla [ES96] for symmetries in Kripke structures. We formalize the notion of structural symmetries in net-list descriptions of digital circuits and CTL* formulas. We show how they relate to symmetries in the corresponding Kripke structures. We also show how such symmetries can automatically be extracted by constructing a suitable directed labeled graph and computing its automorphism group. We present a novel fast algorithm for solving the graph automorphism problem for directed labeled graphs.

The goal of this work is to overcome the learning barriers faced when first using parallelism. Currently, in order to parallelize a system such as GAP, one must embed a message passing library such as MPI, with many routines and many parameters. GAP/MPI provides a simple, task-oriented interface sitting above the MPI library. The system presents the end-user with a single SPMD (single program, multiple data) environment in GAP: an existing, familiar interactive language. In GAP/MPI one describes the end application in terms of high level tasks, which are invoked by a single procedure call in GAP/MPI. This eliminates the complexities of a message passing library, such as encoding a message in a suitable data structure, message synchronization, communication topologies and deadlock avoidance.

Abstract: This note presents an elementary version of Sims’s algorithm for computing strong generators of a given perm group, together with a proof of correctness and some notes about appropriate low-level data structures. Upper and lower bounds on the running time are also obtained. (Following a suggestion of Vaughan Pratt, we adopt the convention that perm = permutation, perhaps thereby saving millions of syllables in future research.) 1. A data structure for perm groups. A “perm, ” for the purposes of this paper, is a one-toone mapping of a set onto itself. If α and β are perms such that α takes i ↦ → j and β takes j ↦ → k, the product αβ takes i ↦ → k. We write α − for the inverse of the perm α; hence αβ = γ iff α = γβ −. Let Π(k) be the set of all perms of the positive integers that fix all points> k. Consider the following data structure: For 1 ≤ j ≤ k, either σkj = ∅ or σkj is a perm of Π(k) that takes k ↦ → j. Let Σ(k) be the set of all non- ∅ perms σkj. We assume that σkk is the identity perm; hence Σ(k) is always nonempty. We write Γ(k) for the set of all perms that can be written as products of the form σ1... σk where each σi is in Σ(i). There is an easy way to test if a given perm π ∈ Π(k) is a member of Γ(k): Let π take k ↦ → j. Then if σkj = ∅ we have π � ∈ Γ(k); otherwise if k = 1 we have π ∈ Γ(k);

) 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 k-tuple of G if the elements generate G (we write hg1 ; : : : ; gk i = G). Let Nk (G) be the set of all generating k-tuples of G, and let Nk (G) = jNk (G)j. We consider two related problems on generating k-tuples. 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...