Approximating Reversal Distance for Strings with Bounded Number of Duplicates

For a string A = a1 . . . an, a reversal ρ(i, j), 1 ≤ i < j ≤ n, transforms the string A into a string A′ = a1 . . . ai−1ajaj−1 . . . aiaj+1 . . . an, that is, the reversal ρ(i, j) reverses the order of symbols in the substring ai . . . aj of A. In a 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 a simple O(k)-approximation algorithm running in time O(k · n). For instances with 3 < k ≤ O( √ 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. In particular, for k = O(1) which is of interest for DNA comparisons, we have a linear time O(1)-approximation algorithm.