Is Your Permutation Algorithm Unbiased for n ≠ 2m?
Many papers on parallel random permutation algorithms assume the input size n to be a power of two and imply that these algorithms can be easily generalized to arbitrary n. We show that this simplifying assumption is not necessarily correct since it may result in a bias. Many of these algorithms are, however, consistent, i.e., iterating them ultimately converges against an unbiased permutation. We prove this convergence along with proving exponential convergence speed. Furthermore, we present an analysis of iterating applied to a butterfly permutation network, which works in-place and is well-suited for implementation on many-core systems such as GPUs. We also show a method that improves the convergence speed even further and yields a practical implementation of the permutation network on current GPUs.
Keywordsparallel random permutation butterfly network bias consistency GPU
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- 1.Anderson, R.: Parallel algorithms for generating random permutations on a shared memory machine. In: Proc. SPAA 1990, pp. 95–102. ACM (1990)Google Scholar
- 2.Blelloch, G.E.: Prefix sums and their applications. Tech. Rep. CMU-CS-90-190, School of Computer Science, Carnegie Mellon University (November 1990)Google Scholar
- 3.Cong, G., Bader, D.A.: An empirical analysis of parallel random permutation algorithms on SMPs. In: Oudshoorn, M.J., Rajasekaran, S. (eds.) ISCA PDCS, pp. 27–34 (2005)Google Scholar
- 4.CUDPP – CUDA data parallel primitives library, http://code.google.com/p/cudpp/
- 8.Knuth, D.E.: The art of computer programming, 3rd edn., vol. 2 (1997)Google Scholar
- 9.Knuth, D.E.: The art of computer programming, volume 3 (2nd ed.) (1998)Google Scholar
- 10.Leighton, F.: Introduction to parallel algorithms and architectures: arrays, trees, hypercubes, vol. (1). M. Kaufmann Publishers (1992)Google Scholar
- 11.Meyer, C.: Matrix Analysis and Applied Linear Algebra. SIAM (2000)Google Scholar
- 12.NVIDIA: NVIDIA CUDA C programming guide, version 3.2 (2011)Google Scholar
- 17.Zoubir, A.M.: Model selection: A bootstrap approach. In: Proc. ICASSP (1999)Google Scholar