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

Abstract

Through the use of a new disk-based method for enumerating very large orbits, condensation for orbits with tens of billions of elements can be performed. The algorithm is novel in that it offers efficient access to data using distributed diskbased data structures. This provides fast access to hundreds of gigabytes of data, which allows for computing without worrying about memory limitations.The new algorithm is demonstrated on one of the longstanding open problems in the Modular Atlas Project [11]: the Brauer tree of the principal 17-block the sporadic simple Fischer group F i23. The tree is completed by computing three orbit counting matrices for the F i23-orbit of size 11, 739, 046, 176 acting on vectors of dimension 728 over GF (2). The construction of these matrices requires 3-1/2 days on a cluster of 56 computers, and uses 8 GB of disk storage and 800 MB of memory per machine.

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].…”

“…While this technique required some additional computation time, it allowed the entire search space to be stored on disk. This allowed for the enumeration of the first subgroup [19] to be performed using Tiered Duplicate Detection as well. The branching factor was again around 1.75.…”