Approximating Reversal Distance for Strings with Bounded Number of Duplicates

Abstract

For a string A=a ₁... a _n , a reversalρ(i,j), 1≤ i<j≤ n, transforms the string A into a string A′=a ₁... a _{i − − 1} a _j a _{j − − 1} ... a _i a _j + 1 ... a _n , that is, the reversal ρ(i,j) reverses the order of symbols in the substring a _i ... a _j 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 ²)-approximation algorithm running in time O(k · n). For instances with $3 < k \leq O(\sqrt{log n log^* n})$, this is the best known approximation algorithm for k-SBR and, moreover, it is faster than the previous best approximation algorithm. In particular, for k=O(1) which is of interest for DNA comparisons, we have a linear time O(1)-approximation algorithm.