Subwords in Reverse-Complement Order

Abstract

We examine finite words over an alphabet $$ \Gamma = \{ a,\overline a ;b,\overline b \} $$ of pairs of letters, where each word w1w2 ... w t is identified with its reverse complement $$ \overline{w} _{t} \cdots \overline{w} _{2} \overline{w} _{1} $$ where ( $$ \overline{\overline a} = a,\overline{\overline b} = b $$ ). We seek the smallest k such that every word of length n, composed from Γ, is uniquely determined by the set of its subwords of length up to k. Our almost sharp result (k~ 2n = 3) is an analogue of a classical result for “normal” words. This problem has its roots in bioinformatics.