Sorting permutations with transpositions in O(n3) amortized time

Abstract

The problem of sorting a permutation π over alphabet ∑={1,2,3…n} with the minimum number of transpositions is known to be intractable. That is, the computation of transposition distance i.e. dt(π), is hard. Such sorting is equivalent to determining the distance between any α, β of the symmetric group Sn over ∑ under the transposition operation. It has applications in computer interconnection networks where a node is denoted by a permutation, and genetics where a genome is modeled by a permutation. The maximum distance between any pair of permutations in Sn is the diameter i.e. diam(Γ(Sn)), of the corresponding Cayley graph Γ(Sn). It corresponds to the maximum latency in the computer interconnection network modeled by Γ(Sn) and the maximum genetic dissimilarity between any pair α, β in Sn. We call transpositions, prefix transpositions, suffix transpositions and prefix/suffix transpositions as block-moves. We design an algorithm to compute dt(π) ∀π in Sn and compute diam(Γ(Sn)) with transpositions in O(n!n3) time and O(n!n2) space; at an amortized time of O(n3). The same can be computed for the remaining block moves in O(n2log⁡n) amortized time and O(n!n) space.