On a Greedy Algorithm to Construct Universal Cycles for Permutations

Author:

Gao Alice L. L.1,Kitaev Sergey2,Steiner Wolfgang3,Zhang Philip B.4

Affiliation:

1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, P. R. China

2. Department of Computer and Information Sciences, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK

3. IRIF, CNRS UMR 8243, Université Paris Diderot – Paris 7, 75205 Paris Cedex 13, Paris, France

4. College of Mathematical Science, Tianjin Normal University, Tianjin 300387, P. R. China

Abstract

A universal cycle for permutations of length [Formula: see text] is a cyclic word or permutation, any factor of which is order-isomorphic to exactly one permutation of length [Formula: see text], and containing all permutations of length [Formula: see text] as factors. It is well known that universal cycles for permutations of length [Formula: see text] exist. However, all known ways to construct such cycles are rather complicated. For example, in the original paper establishing the existence of the universal cycles, constructing such a cycle involves finding an Eulerian cycle in a certain graph and then dealing with partially ordered sets. In this paper, we offer a simple way to generate a universal cycle for permutations of length [Formula: see text], which is based on applying a greedy algorithm to a permutation of length [Formula: see text]. We prove that this approach gives a unique universal cycle [Formula: see text] for permutations, and we study properties of [Formula: see text].

Publisher

World Scientific Pub Co Pte Lt

Subject

Computer Science (miscellaneous)

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