An Upper Bound on the Number of Circular Transpositions to Sort a Permutation

TL;DR

This paper establishes an improved upper bound of n^2/4 on the number of circular transpositions needed to sort any permutation, extending the classic adjacent transpositions bound.

Contribution

It introduces a new upper bound for sorting permutations with circular transpositions, answering an open question from prior research.

Findings

Upper bound of n^2/4 transpositions for sorting permutations

Inclusion of a circular transposition reduces the sorting steps

Addresses an open problem in permutation sorting theory

Abstract

We consider the problem of upper bounding the number of circular transpositions needed to sort a permutation. It is well known that any permutation can be sorted using at most $n(n-1)/2$ adjacent transpositions. We show that, if we allow all adjacent transpositions, as well as the transposition that interchanges the element in position 1 with the element in the last position, then the number of transpositions needed is at most $n^2/4$. This answers an open question posed by Feng, Chitturi and Sudborough (2010).