On the Toric Graph as a Tool to Handle the Problem of Sorting by Transpositions

• Authors: Rodrigo A. Hausen;Luerbio Faria;Celina M. Figueiredo;Luis Antonio Kowada
• Venue: BSB '08 Proceedings of the 3rd Brazilian symposium on Bioinformatics: Advances in Bioinformatics and Computational Biology
• Year: 2008

To this date, neither a polynomial algorithm to sort a permutation by transpositions has been found, nor a proof that it is an NP-hard problem has been given. Therefore, determining the exact transposition distance dt(茂戮驴) of a generic permutation 茂戮驴, relative to the identity, is generally done by an exhaustive search on the space Snof all permutations of nelements. In a 2001 paper, Eriksson et al.[1] made a breakthrough by proposing a structure named by them as toric graph, which allowed the reduction of the search space, speeding-up the process, such that greater instances could be solved. Surprisingly, Eriksson et al.were able to exhibit a counterexample to a conjecture by Meidanis et al.[2] that the transposition diameter would be equal to the distance of the reverse permutation $\lfloor{n/2}\rfloor+1$. The goal of the present paper is to further study the toric graph, focusing on the case when n+ 1 is prime. We observe that the transposition diameter problem for n= 16 is still open. We determine that there are exactly $\frac{n!-n}{n+1} + n$ vertices in the toric graph and find a lower bound $d_t(\pi) \ge \lfloor{n/2}\rfloor$ on the transposition distance for every permutation 茂戮驴in a unitary toric class that is not the identity permutation. We provide experimental data on the exact distance of those permutations to back our conjecture that $d_t(\pi) \le \lfloor{n/2}\rfloor + 1$, where 茂戮驴belongs to a unitary toric class, and that $\lfloor{n/2}\rfloor + 1$ is equal to the transposition diameter when n+ 1 is prime.