Approximating reversal distance for strings with bounded number of duplicates

• Authors: Petr Kolman
• Venue: MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
• Year: 2005

For a string A=a1... an, a reversalρ(i,j), 1≤ ij≤ 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(k2)-approximation algorithm running in time O(k · n). For instances with $3 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.