Efficient Algorithms for Some Variants of the Farthest String Problem

The farthest string problem (FARTHEST STRING) is one of the core problems in the field of consensus word analysis and several biological problems such as discovering potential drugs, universal primers, or unbiased consensus sequences. Given k strings of the same length L and a nonnegative integer d, FARTHEST STRING is to find a string s such that none of the given strings has a Hamming distance that is smaller than d from s. It has been shown to be NP-complete. In this paper, we study two variants of FARTHEST STRING. One is to search for a string s satisfying the condition that the hamming distances between s and all the given strings are greater than d. We give an O((|Σ|(L-d)) )-time algorithm, where | Σ| is the alphabet size. The other variant is to find a string s such that the sum of the hamming distances between s and all the given strings are maximized. We solve this problem in O( kL ) time.