Larry Finkelstein

New methods for using Cayley graphs in interconnection networks

Abstract 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.

Constructing Permutation Representations for Matrix Groups

New techniques, both theoretical and practical, are presented for constructing permutation representations for computing with matrix groups defined over finite fields. The permutation representation is constructed on a conjugacy class of subgroups of prime order. We construct a base for the permutation representation, which in turn simplifies the computation of a strong generating set. In addition, we present an elementary test for checking the simplicity of the permutation image. The theory has been successfully tested on a representation of the sporadic simple groupLy, discovered by Lyons(1972). With noa prioriassumptions, we find a permutation representation of degree 9606125 on a conjugacy class of subgroups of order 3, find the order of the resulting permutation group, and verify simplicity. A Monte Carlo variation of the algorithm was used to achieve better space and time efficiency. The construction of the permutation representation required four CPU days on a SPARCserver 670MP with 64 MB. The permutation representation was used implicitly in the sense that the group element was stored as a matrix, and its permutation action on a “point” was determined using a pre-computed data structure. Thus, additional computations required little additional space. The algorithm has also been implemented using the MasPar MP-1 SIMD parallel computer and 8 SPARC-2s running under MPI. The results of those parallel experiments are briefly reviewed.

Nearly linear time algorithms for permutation groups with a small base

A base of a permutation group G is a subset B of the permutation domain such that only the identity of G fixes B pointwise. The permutation representations of important classes of groups, including all finite simple groups other than the alternating groups, admit O(log n) size bases, where n is the size of the permutation domain. Groups with very small bases dominate the work on permutation groups in much of computational group theory. A series of new combinatorial results allows us to present Monte Carlo algorithms achieving O(n log’ n) (c a constant) time and space performance for such groups with respect to the fundamental operations of finding order and testing membership. (The input is a list of generators of the group.) Previous methods have achieved similar space performance only at the expense of increased time performance. Adaptations of a ‘(cube-doubling” technique [BSZ] and a local expansion property of groups [Ba3] (cf. [Ba4]) are the key to theoretically reducing the time complexity to O(rI log’ n.). The shared principal novelty of the new ideas is in their abilitv to build and manitmlate certain chains of subsets of a grou~, which are not themselves subgroups, in order to build the point stabilizer subgroup chain. Further combinatorial ideas are used to lower the constant c. Comparative timing estimates, based on asymptotic worst-case analysis, lead us to expect a new implementation to be faster than previous implementations for groups of high degree.

Applications of Cayley Graphs

This paper demonstrates the power of the Cayley graph approach to solve specific applications, such as rearrangement problems and the design of interconnection networks for parallel CPUs. Recent results of the authors for efficient use of Cayley graphs are used here in exploratory analysis to extend recent results of Babai et al. on a family of trivalent Cayley graphs associated with PSL2(p). This family and its subgroups are important as a model for interconnection networks of parallel CPUs. The methods have also been used to solve for the first time problems which were previously too large, such as the diameter of Rubiks 2 × 2 × 2 cube. New results on how to generalize the methods to rearrangement problems without a natural group structure are also presented.

A new base change algorithm for permutation groups

The computation of a strong generating set for a permutation group acting on a set

Constructing permutation representations for large matrix groups

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A random base change algorithm for permutation groups

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

A strong generating test and short presentations for permutation groups

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Reduction of group constructions to point stabilizers

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

Applying the Classification Theorem for Finite Simple Groups to Minimize Pin Count in Uniform Permutation Architectures

O(n^5 )