### 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} .

Language | English |
---|---|

Pages | 61-72 |

Number of pages | 12 |

Journal | International Journal of Foundations of Computer Science |

Volume | 30 |

Issue number | 1 |

DOIs | |

Publication status | Published - 5 Mar 2019 |

### Keywords

- universal cycles
- permutations
- greedy algorithm
- combinatorial generation

### Cite this

*International Journal of Foundations of Computer Science*,

*30*(1), 61-72. https://doi.org/10.1142/S0129054119400033

}

*International Journal of Foundations of Computer Science*, vol. 30, no. 1, pp. 61-72. https://doi.org/10.1142/S0129054119400033

**On a greedy algorithm to construct universal cycles for permutations.** / Gao, Alice L.L.; Kitaev, Sergey; Steiner, Wolfgang; Zhang, Philip B.

Research output: Contribution to journal › Article

TY - JOUR

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

UR - https://arxiv.org/abs/1711.10820

UR - https://www.worldscientific.com/worldscinet/ijfcs

U2 - 10.1142/S0129054119400033

DO - 10.1142/S0129054119400033

M3 - Article

VL - 30

SP - 61

EP - 72

JO - International Journal of Foundations of Computer Science

T2 - International Journal of Foundations of Computer Science

JF - International Journal of Foundations of Computer Science

SN - 0129-0541

IS - 1

ER -