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

Font Size:  Small  Medium  Large

The Cycles of the Multiway Perfect Shuffle Permutation

John Ellis, Hongbing Fan, Jeffrey Shallit


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  &bmod; (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.

Full Text: GZIP Compressed PostScript PostScript PDF original HTML abstract page