Bases of Motifs for Generating Repeated Patterns with Don't Cares

We investigate the problem of determining the basis of repeated motifs with don't cares in an input string. We give new upper and lower bounds on the problem, introducing a new notion of basis that is provably smaller than (and contained in) previously defined ones. Our basis can be computed in less time and space, and is still able to generate the same set of motifs. We also prove that the number of motifs in all these bases grows exponentially with the quorum, the minimal number of times a motif must appear, which was unnoticed in previous work. We show that a polynomial-time algorithm exists only for fixed quorum.