Finding consensus and optimal alignment of circular strings

Circular strings are different from linear strings in that the last symbol is considered to precede the first symbol. Even though circular strings are biologically important, only a few efforts have been made to solve computational problems on circular strings. In this paper, we introduce consensus problems for circular strings of length n and present the first non-trivial algorithms to find a consensus and an optimal alignment for circular strings by the Hamming distance. They are O(n2logn)-time algorithms for three circular strings and an O(n3logn)-time algorithm for four circular strings. Our algorithms are O(n/logn) times faster than the naïve algorithms directly using the solutions for the linear consensus problems, which take O(n3) time for three circular strings and O(n4) time for four circular strings. This speedup was achieved by reducing the problems into correlations and by formulating and solving systems of linear equations. Moreover, our algorithms use only O(n) space.