Finding optimal alignment and consensus of circular strings

Abstract

We consider the problem of finding the optimal alignment and consensus (string) of circular strings. Circular strings are different from linear strings in that the first (leftmost) symbol of a circular string is wrapped around next to the last (rightmost) symbol. In nature, for example, bacterial and mitochondrial DNAs typically form circular strings. The consensus string problem is finding a representative string (consensus) of a given set of strings, and it has been studied on linear strings extensively. However, only a few efforts have been made for the consensus problem for circular strings, even though circular strings are biologically important. In this paper, we introduce the consensus problem for circular strings and present novel algorithms to find the optimal alignment and consensus of circular strings under the Hamming distance metric. They are O(n2 log n)-time algorithms for three circular strings and an O(n3 log n)-time algorithm for four circular strings. Our algorithms are O(n/log n) times faster than the n?ive algorithm directly using the solutions for the linear consensus problems, which takes O(n3) time for three circular strings and O(n4) time for four circular strings. We achieved this speedup by adopting a convolution and a system of linear equations into our algorithms to reflect the characteristics of circular strings that we found.