On a greedy algorithm to construct universal cycles for permutations

Alice L.L. Gao, Sergey Kitaev, Wolfgang Steiner, Philip B. Zhang

Research output: Contribution to journalArticle

Abstract

A universal cycle for permutations of length n is a cyclic word or permutation, any factor of which is order-isomorphic to exactly one permutation of length n, and containing all permutations of length n as factors. It is well known that universal cycles for permutations of length n 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 n, which is based on applying a greedy algorithm to a permutation of length n - 1. We prove that this approach gives a unique universal cycle In for permutations, and we study properties of I n .

LanguageEnglish
Pages61-72
Number of pages12
JournalInternational Journal of Foundations of Computer Science
Volume30
Issue number1
DOIs
Publication statusPublished - 5 Mar 2019

Keywords

  • universal cycles
  • permutations
  • greedy algorithm
  • combinatorial generation

Cite this

Gao, Alice L.L. ; Kitaev, Sergey ; Steiner, Wolfgang ; Zhang, Philip B. / On a greedy algorithm to construct universal cycles for permutations. In: International Journal of Foundations of Computer Science. 2019 ; Vol. 30, No. 1. pp. 61-72.
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On a greedy algorithm to construct universal cycles for permutations. / Gao, Alice L.L.; Kitaev, Sergey; Steiner, Wolfgang; Zhang, Philip B.

In: International Journal of Foundations of Computer Science, Vol. 30, No. 1, 05.03.2019, p. 61-72.

Research output: Contribution to journalArticle

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T1 - On a greedy algorithm to construct universal cycles for permutations

AU - Gao, Alice L.L.

AU - Kitaev, Sergey

AU - Steiner, Wolfgang

AU - Zhang, Philip B.

PY - 2019/3/5

Y1 - 2019/3/5

N2 - A universal cycle for permutations of length n is a cyclic word or permutation, any factor of which is order-isomorphic to exactly one permutation of length n, and containing all permutations of length n as factors. It is well known that universal cycles for permutations of length n 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 n, which is based on applying a greedy algorithm to a permutation of length n - 1. We prove that this approach gives a unique universal cycle In for permutations, and we study properties of I n .

AB - A universal cycle for permutations of length n is a cyclic word or permutation, any factor of which is order-isomorphic to exactly one permutation of length n, and containing all permutations of length n as factors. It is well known that universal cycles for permutations of length n 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 n, which is based on applying a greedy algorithm to a permutation of length n - 1. We prove that this approach gives a unique universal cycle In for permutations, and we study properties of I n .

KW - universal cycles

KW - permutations

KW - greedy algorithm

KW - combinatorial generation

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