Sorting Circular Permutations by Bounded Transpositions

Abstract

A k-bounded (k ≥ 2) transposition is an operation that switches two elements that have at most k - 2 elements in between. We study the problem of sorting a circular permutation π of length n for k = 2, i.e., adjacent swaps and k = 3, i.e., short swaps. These transpositions mimic microrearrangements of gene order in viruses and bacteria. We prove a (1/4)n (2) lower bound for sorting by adjacent swaps. We show upper bounds of (5/32)n (2) + O(n log n) and (7/8)n + O(log n) for sequential and parallel sorting, respectively, by short swaps.

An Upper Bound for Sorting $$R_n$$ with LE

“…In LRE operation, both left and right rotate cyclically shifts the entire permutation. In contrast, [12] an extended bubblesort is considered, where an additional swap is allowed between elements in positions 1 and n. We call an operation say Ψ symmetric if for any generator of Ψ its inverse is also in Ψ. Exchange operation is inverse of itself whereas left and right rotate are inverses of one another, thus, LRE is symmetric.…”

A tight upper bound on the number of cyclically adjacent transpositions to sort a permutation

“…We first show how, given a displacement vector d that satisfies (1) and (2), we can find a sequence of cyclically adjacent transpositions that sort π and have net displacement vector d. We then give the expression given by Jerrum for c(i, j), the net numer of times swap (i, j) occurs in this sequence, as a function of π and d. Finally, we give Jerrum's main result which characterizes the displacement vector d that corresponds to the minimum length sequence of transpositions that sort π. Lemma 2. Given a displacement vector d that satisfies (1) and (2) with respect to some permutation π, a sequence of cyclically adjacent swaps that sort π and has net displacements given by d is found by repeatedly swapping cyclically adjacent elements (i, j) such that d(i) > d(j), and decreasing d(i) by 1 and increasing d(j) by 1.…”

Task swapping networks in distributed systems

“…end end end graph of ring topology having n agents with their task assignment represented by permutation p ∈ S n as Γ R(n) p . We call the corresponding permutation p ∈ S n for Γ R(n) p as a circular permutation [14] in which position i of a circular permutation p ∈ S n is referred to as verex (agent) i for 1 ≤ i ≤ n in Γ R(n) p . For example, a task swapping graph of Figure 6 is obtained by finding a minimum-length permutation factorization of π −1 1 π 2 using the generating set S 6 = {(i i+1) : 1 ≤ i < n}∪{(1 n)}.…”

“…If d s − d t > n for each pair of indices s and t, then we renew d s as d s − n and d t as d t + n, respectively. This process is called strictly contracting transformation [14]. If a displacment vector d admits no strictly contracting transformation, we say that a displacement vector d is stable, denotedd.…”

“…This sorting procedure is continued by using Algorithm 6 until arriving at the identity permutation (see below): is subject to the diameter of the modified bubble-sort graph M BS n discussed in Section 3. To the best of our knowledge, the formula of the diameter of M BS n is not known [14,30,40]. Nevertheless, a minimum-length sequence of sorting an aribtrary circular permutation p ∈ S n into the identity permutation I using the generating set S 6 = {(i i + 1) : 1 ≤ i < n} ∪ {(1 n)} in Algorithm 6 is obtained in polynomial time [26].…”