Duplication Distance to the Root for Binary Sequences

We study the tandem duplication distance between binary sequences and their roots. In other words, the quantity of interest is the number of tandem duplication operations of the form <inline-formula> <tex-math notation="LaTeX">$ {x}_{} = {a}_{} {b}_{} {c}_{} \to {y}_{} = {a}_{} {b}_{} {b}_{} {c}_{}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$ {x}_{}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$ {y}_{}$ </tex-math></inline-formula> are sequences and <inline-formula> <tex-math notation="LaTeX">$ {a}_{}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$ {b}_{}$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$ {c}_{}$ </tex-math></inline-formula> are their substrings, needed to generate a binary sequence of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> starting from a square-free sequence from the set {0, 1, 01, 10, 010, 101}. This problem is a restricted case of finding the duplication/deduplication distance between two sequences, defined as the minimum number of duplication and deduplication operations required to transform one sequence to the other. We consider both exact and approximate tandem duplications. For exact duplication, denoting the maximum distance to the root of a sequence of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> by <inline-formula> <tex-math notation="LaTeX">$f(n)$ </tex-math></inline-formula>, we prove that <inline-formula> <tex-math notation="LaTeX">$f(n)=\Theta (n)$ </tex-math></inline-formula>. For the case of approximate duplication, where a <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula>-fraction of symbols may be duplicated incorrectly, we show that the maximum distance has a sharp transition from linear in <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> to logarithmic at <inline-formula> <tex-math notation="LaTeX">$\beta =1/2$ </tex-math></inline-formula>. We also study the duplication distance to the root for the set of sequences arising from a given root and for special classes of sequences, namely, the De Bruijn sequences, the Thue–Morse sequence, and the Fibonacci words. The problem is motivated by genomic tandem duplication mutations and the smallest number of tandem duplication events required to generate a given biological sequence.