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

Abstract

Permutation multiplication (or permutation composition) is perhaps the simplest of all algorithms in computer science. Yet for large permutations, the standard algorithm is not the fastest for disk or for flash, and surprisingly, it is not even the fastest algorithm for RAM on recent multi-core CPUs. On a recent commodity eight-core machine we demonstrate a novel algorithm that is 50% faster than the traditional algorithm. For larger permutations on flash or disk, the novel algorithm is orders of magnitude faster. A disk-parallel algorithm is demonstrated that can multiply two permutations with 12.8 billion points using 16 parallel local disks of a cluster in under one hour. Such large permutations are important in computational group theory, where they arise as the result of the well-known Todd-Coxeter coset enumeration algorithm. The novel algorithm emphasizes several passes of streaming access to the data instead of the traditional single pass using random access to the data. Similar novel algorithms are presented for permutation inverse and permutation multiplication by an inverse, thus providing a complete library of the underlying permutation operations needed for computations with permutation groups.