Proceedings 41st Annual Symposium on Foundations of Computer Science (2000)
Redondo Beach, California
Nov. 12, 2000 to Nov. 14, 2000
I. Pak , Dept. of Math., Yale Univ., New Haven, CT, USA
The product replacement algorithm is a heuristic designed to generate random group elements. The idea is to run a random walk on generating /spl kappa/-tuples of the group, and then output a random component. The algorithm was designed by C.R. Leedham-Green, and further investigated by F. Cellar et al. (1995). It was found to have an outstanding performance, much better than the previously known algorithms (P. Diaconis and L. Saloff-Coste, 1996). The algorithm is now included in two major group algebra packages: GAP (M. Scheonert et al., 1995) and MAGMA (W. Bosma et al., 1997). In spite of the many serious attempts and partial results, the analysis of the algorithm remains difficult at best. For small values of /spl kappa/, even graph connectivity becomes a serious obstacle. The most general results are due to Diaconis and Saloff-Coste, who used a state of the art analytic technique to obtain polynomial bounds in special cases, and (sub)-exponential bounds in the general case. The main result of the paper is a polynomial upper bound for the cost of the algorithm, provided /spl kappa/ is large enough.
random number generation; heuristic programming; group theory; polynomials; symbol manipulation; product replacement algorithm; heuristic; random group elements; random walk; generating /spl kappa/-tuples; random component; group algebra packages; GAP; MAGMA; graph connectivity; state of the art analytic technique; polynomial bounds; sub exponential bounds; polynomial upper bound
I. Pak, "The product replacement algorithm is polynomial," Proceedings 41st Annual Symposium on Foundations of Computer Science(FOCS), Redondo Beach, California, 2000, pp. 476.