On a greedy algorithm to construct universal cycles for permutations

Gao, Alice L.L. and Kitaev, Sergey and Steiner, Wolfgang and Zhang, Philip B. (2019) On a greedy algorithm to construct universal cycles for permutations. International Journal of Foundations of Computer Science, 30 (1). pp. 61-72. ISSN 0129-0541

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    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 .