Upper Bounds for Sorting Permutations With a Transposition Tree

Abstract

An upper bound for sorting permutations with an operation estimates the diameter of the corresponding Cayley graph and an exact upper bound equals the diameter. Computing tight upper bounds for various operations is of theoretical and practical (e.g., interconnection networks, genetics) interest. Akers and Krishnamurthy gave a Ω(n! n2) time method that examines n! permutations to compute an upper bound, f(Γ), to sort any permutation with a given transposition tree T, where Γ is the Cayley graph corresponding to T. We compute two intuitive upper bounds γ and δ′ each in O(n2) time for the same, by working solely with the transposition tree. Recently, Ganesan computed β, an estimate of the exact upper bound for the same, in O(n2) time. Our upper bounds are tighter than f(Γ) and β, on average and in most of the cases. For a class of trees, we prove that the new upper bounds are tighter than β and f(Γ).

Sorting permutations with a transposition tree

“…The associated cost is idiam(T ) moves. It follows that x − i vertices from X are homed among themselves where the upper bound for the associated cost is (x − i)(diam(T ) − 1/2) due to Lemma 5 of [1]. When x − i is odd then cost reduces by 1/2.…”

“…Chitturi designed an algorithm Algorithm S that identifies the set of all vertices that have maximum eccentricity i.e. S in T in linear time [1]. The general idea of the algorithms in [1] is to delete a set of leaves say X and obtain an upper bound on the number of moves that suffice to home markers to all vertices in X.…”

“…Note that if all but one of the clusters is deleted then the diameter of the resultant tree decreases. Based on this idea, Algorithm D computes an upper bound δ that works by deleting all clusters in S except the largest cluster C * [1]. It employs a linear time algorithm F ast N C to identify all clusters [1].…”

“…Based on this idea, Algorithm D computes an upper bound δ that works by deleting all clusters in S except the largest cluster C * [1]. It employs a linear time algorithm F ast N C to identify all clusters [1]. Algorithm D has two versions, D v1 and D v2 which calculates δ v1 and δ v2 respectively.…”

“…Further, if there are dependencies such as the home of a is b, the home of b is c etc. then the dependency that yields the worst case is shown to be mutual swap of pairs of markers; that is, the home of a is b and the home of b is a (Lemma 5 of [1]). So, given a subset Y of S where the vertices of Y are to be homed within Y , the maximum number of pairs of Y are swapped employing their corresponding sequence of moves.…”