On the average number of reversals needed to sort signed permutations

Abstract

It is well-known that signed permutations can be sorted in polynomial time, while sorting unsigned ones is an NP-hard problem.Although sorting signed permutations by reversals is an “easy” computational problem, the study of the average reversal distance is justified because it gives rise to interesting combinatorial questions and also because in applications related with genome similarity analysis, exact solutions for this problem have been used to design approximate algorithms for its unsigned NP-hard version. Thus, the average number of reversals needed to sort signed permutations represents a good control mechanism for the quality of approximate solutions for the unsigned case. This paper analyzes the average number of reversals needed to sort signed permutations by exploring combinatorial properties of structures related with signed permutations, such as breakpoint graphs, and provides a recurrence formula for this average number.The recurrence formula is built based on the analysis of the probability that two nodes of the breakpoint graph belong to the same alternating cycle among all the breakpoint graphs related with permutations of length n. Through this analysis it is possible to compute the average reversal distance for signed variations of the identity permutation. Also, lower and upper bounds of the average reversal distance for signed permutations are provided. Additionally, based on computational data, it is shown how these bounds can be used in order to propose concrete upper and lower bounds.