An Upper Bound For Sorting Permutations With A Transposition Tree

Abstract

Abstract Cayley graphs are constructed over n! vertices that denote n! permutations of the symmetric group Sn. The edge set of a particular Cayley graph Γ is determined by the corresponding operation that consists of a well defined generator set. One such operation is a transposition tree which is a spanning tree over n vertices. Cayley graphs have been extensively studied and they have applications in genomic studies and computer interconnection networks. In computer interconnection networks each vertex of the corresponding Γ denotes a computer and an existence of an edge between two vertices indicates that a direct two way communication exists between the corresponding computers. In 2013, upper bounds for Cayley graphs generated by transposition trees were computed in polynomial time. One such upper bound δ´ was known to be the tightest until recently. In 2019, δ* was shown to be better than δ’ in a cumulative sense. We improve upon δ’* to obtain δ’* i is better than δ’ in a cumulative sense, as indicated by the experimental results. Likewise, δ’* is comparable to δ*. For corresponding classes of trees, δ’* is shown to be tighter than δ’ and δ* respectively.