# Technical Program

## Paper Detail

 Paper ID T.4.2 Paper Title Duplication with transposition distance to the root for q-ary strings Authors Nikita Polyanskii, Technical University of Munich, Germany; Ilya Vorobyev, Skolkovo Institute of Science and Technology, Russia Session T.4: Sequences Presentation Lecture Track Topics in Information Theory Manuscript Click here to download the manuscript Virtual Presentation Click here to watch in the Virtual Symposium Abstract We study the duplication with transposition distance between strings of length $n$ over a $q$-ary alphabet and their roots. In other words, we investigate the number of duplication operations of the form $x = (abcd) \to y = (abcbd)$, where $x$ and $y$ are strings and $a$, $b$, $c$ and $d$ are their substrings, needed to get a $q$-ary string of length $n$ starting from the set of strings without duplications. For exact duplication, we prove that the maximal distance between a string of length at most $n$ and its root has the asymptotic order $n/\log n$. For approximate duplication, where a $\beta$-fraction of symbols may be duplicated incorrectly, we show that the maximal distance has a sharp transition from the order $n/\log n$ to $\log n$ at $\beta=(q-1)/q$. The motivation for this problem comes from genomics, where such duplications represent a special kind of mutation and the distance between a given biological sequence and its root is the smallest number of transposition mutations required to generate the sequence.