Discrete Mathematics & Theoretical Computer Science, Vol 5 (2002)

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DMTCS vol 5 no 1 (2002), pp. 169-180

Discrete Mathematics & Theoretical Computer Science


Volume 5 n° 1 (2002), pp. 169-180

author:John Ellis and Hongbing Fan and Jeffrey Shallit
title:The Cycles of the Multiway Perfect Shuffle Permutation
keywords:perfect shuffle permutation, cycle decomposition
abstract:The (k,n)-perfect shuffle, a generalisation of the 2-way perfect shuffle, cuts a deck of kn cards into k equal size decks and interleaves them perfectly with the first card of the last deck at the top, the first card of the second-to-last deck as the second card, and so on. It is formally defined to be the permutation ρk,n: i → ki (&mod;kn+1), for 1 ≤ i ≤ kn. We uncover the cycle structure of the (k,n)-perfect shuffle permutation by a group-theoretic analysis and show how to compute representative elements from its cycles by an algorithm using O(kn) time and O((&log; kn)2) space. Consequently it is possible to realise the (k,n)-perfect shuffle via an in-place, linear-time algorithm. Algorithms that accomplish this for the 2-way shuffle have already been demonstrated.

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reference: John Ellis and Hongbing Fan and Jeffrey Shallit (2002), The Cycles of the Multiway Perfect Shuffle Permutation , Discrete Mathematics and Theoretical Computer Science 5, pp. 169-180
bibtex:For a corresponding BibTeX entry, please consider our BibTeX-file.
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