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41st Annual Symposium on Foundations of Computer Science
The product replacement algorithm is polynomial
Redondo Beach, California
November 12November 14
ISBN: 0769508502
ASCII Text  x  
I. Pak, "The product replacement algorithm is polynomial," 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pp. 476, 41st Annual Symposium on Foundations of Computer Science, 2000.  
BibTex  x  
@article{ 10.1109/SFCS.2000.892135, author = {I. Pak}, title = {The product replacement algorithm is polynomial}, journal ={2013 IEEE 54th Annual Symposium on Foundations of Computer Science}, volume = {0}, year = {2000}, issn = {02725428}, pages = {476}, doi = {http://doi.ieeecomputersociety.org/10.1109/SFCS.2000.892135}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  CONF JO  2013 IEEE 54th Annual Symposium on Foundations of Computer Science TI  The product replacement algorithm is polynomial SN  02725428 SP EP A1  I. Pak, PY  2000 KW  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 VL  0 JA  2013 IEEE 54th Annual Symposium on Foundations of Computer Science ER   
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. LeedhamGreen, 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. SaloffCoste, 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 SaloffCoste, 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.
Index Terms:
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
Citation:
I. Pak, "The product replacement algorithm is polynomial," focs, pp.476, 41st Annual Symposium on Foundations of Computer Science, 2000
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