Algorithms for Closest and Farthest String Problems via Rank Distance
A new distance between strings, termed rank distance, was introduced by Dinu (Fundamenta Informaticae, 2003). Since then, the properties of rank distance were studied in several papers. In this article, we continue the study of rank distance. More precisely we tackle three problems that concern the distance between strings.
The first problem that we study is String with Fixed Rank Distance (SFRD): given a set of strings S and an integer d decide if there exists a string that is at distance d from every string in S. For this problem we provide a polynomial time exact algorithm.
The second problem that we study is named is the Closest String Problem under Rank Distance (CSRD). The input consists of a set of strings S, asks to find the minimum integer d and a string that is at distance at most d from all strings in S. Since this problem is NP-hard (Dinu and Popa, CPM 2012) it is likely that no polynomial time algorithm exists. Thus, we propose three different approaches: a heuristic approach and two integer linear programming formulations, one of them using geometric interpretation of the problem.
Finally, we approach the Farthest String Problem via Rank Distance (FSRD) that asks to find two strings with the same frequency of characters (i.e. the same Parikh vector) that have the largest possible rank distance. We provide a polynomial time exact algorithm for this problem.
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