For a string A = a1 ... an, a reversalρ (i, j), 1 ≤ i ≤ j ≤ n, transforms the string A into a string A′ = a1 ... ai - 1 aj aj - 1 ... ai aj + 1 ...an, that is, the reversal ρ (i, j) reverses the order of symbols in the substring ai ... aj of A. In the case of signed strings, where each symbol is given a sign + or -, the reversal operation also flips the sign of each symbol in the reversed substring. Given two strings, A and B, signed or unsigned, sorting by reversals (SBR) is the problem of finding the minimum number of reversals that transform the string A into the string B. Traditionally, the problem was studied for permutations, that is, for strings in which every symbol appears exactly once. We consider a generalization of the problem, k-SBR, and allow each symbol to appear at most k times in each string, for some k ≥ 1. The main result of the paper is an O (k2)-approximation algorithm running in time O (n). For instances with 3 < k ≤ O (sqrt(log n log* n)), this is the best known approximation algorithm for k-SBR and, moreover, it is faster than the previous best approximation algorithm. © 2006 Elsevier B.V. All rights reserved.
Kolman, P., & Waleń, T. (2007). Approximating reversal distance for strings with bounded number of duplicates. Discrete Applied Mathematics, 155(3), 327–336. https://doi.org/10.1016/j.dam.2006.05.011