Bounds for sorting by prefix reversal

Abstract

For a permutation σ of the integers from 1 to n, let ƒ(σ) be the smallest number of prefix reversals that will transform σ to the identity permutation, and let ƒ(n) be the largest such ƒ(σ) for all σ in (the symmetric group) S_n. We show that $\text{ƒ(n)⩽}\text{(5n+5)}\text{3}$, and that $\text{ƒ(n)⩾}\text{17n}\text{16}$ for n a multiple of 16. If, furthermore, each integer is required to participate in an even number of reversed prefixes, the corresponding function g(n) is shown to obey $\text{3n}\text{2−1}\text{⩽g(n)⩽2n+3}$.