A parallel architecture for disk-based computing over the Baby Monster and other large finite simple groups

Abstract

We outline a distributed, disk-based technique for computing over very large matrix groups. This technique is used to compute a permutation representation for the Baby Monster, a sporadic simple group that acts on 13,571,955,000 points. Its group order is approximately 4 × 10 33 . This is a landmark because it is 100 times larger than any previous construction of a permutation representation. By using the computed on-disk data structures, computation over the Baby Monster is now feasible using the distributed disks of a cluster. Our work allows researchers to use either a matrix, a permutation, or a word representation for computing over the Baby Monster where previously only a matrix representation was available. The methodology is demonstrated by using as a signature the image of a vector that is stabilized by the maximal subgroup. The technique extends to finite simple groups and to other groups, through other signatures.

Fast multiplication of large permutations for disk, flash memory and RAM

“…For lower-level memory data, some of the main ideas of disk-based computing [14,16] have been used successfully in recent years to solve or make progress on important problems in computational group theory [9,10,14,15], where the size of the data is too large for one RAM subsystem or even the aggregate RAM of a cluster.…”

A comparative analysis of parallel disk-based Methods for enumerating implicit graphs

“…Robinson and Cooperman [18] introduced tiered duplicate detection as a method to speedup the enumeration of the Baby Monster sporadic simple group, and more recently applied it to the problem of the Fischer23 group [19].…”

“…The enumeration of the first subgroup [18] in the point stabilizer subgroup chain was performed using Tiered Duplicate Detection with a hash multiple of 2. It was done on a cluster of 32 nodes with just 8 terabytes of aggregate disk (250 gigabytes per node) and 16 gigabytes of aggregate RAM (500 megabytes per node).…”

“…This includes the constructions of permutation group representations from matrix group repre-sentations, condensation methods for representation theory, and variations of Sims's method of base and strong generating set for high degree permutations. These methods of computational group theory are often applied to the sporadic simple groups, a natural ladder of challenge problems [2,4,5,6,7,13,18,21,22]. (The finite simple groups form the building blocks from which all other finite groups can be constructed.…”

A disk-based parallel implementation for direct condensation of large permutation modules

“…A more recent research direction [6,22] looked at using distributed disk for large computations such as orbit enumeration. Here the disks of a cluster are accessed in a streaming manner for similar performance to a single memory module.…”