Transposition diameter on circular binary strings

TL;DR

This paper studies transposition distance $d(S,T)$ on circular binary strings, where shifts are considered equivalent. It introduces a partition-based representation $S=(0^{s_{1}}1\dots0^{s_{k}}1)$ and defines the statistics $f_1(S,r)$ and $f_2(S,r)$ to derive lower and upper bounds on $d(S,T)$ via partitions; it also characterizes the transposition diameter with a sharp condition $d(S,T)=k-1$ when $f_1(S,1) > f_1(T,1) \ge f_2(T,1) > f_2(S,k-1)$. The results yield polynomial-time determination of distances when the pair attains the diameter and connect the problem to NP-hardness for the general decision version, offering avenues to extend the approach to arbitrary strings through combinatorial analysis.

Abstract

On the string of finite length, a (genomic) transposition is defined as the operation of exchanging two consecutive substrings. The minimum number of transpositions needed to transform one into the other is the transposition distance, that has been researched in recent years. In this paper, we study transposition distances on circular binary strings. A circular binary string is the string that consists of symbols $0$ and $1$ and regards its circular shifts as equivalent. The property of transpositions which partition strings is observed. A lower bound on the transposition distance is represented in terms of partitions. An upper bound on the transposition distance follows covering of the set of partitions. The transposition diameter is given with a necessary and sufficient condition.