Investigating the complexity of the double distance problems

Abstract

BackgroundTwo genomes mathbb {A} and mathbb {B} over the same set of gene families form a canonical pair when each of them has exactly one gene from each family. Denote by n_* the number of common families of mathbb {A} and mathbb {B}. Different distances of canonical genomes can be derived from a structure called breakpoint graph, which represents the relation between the two given genomes as a collection of cycles of even length and paths. Let c_i and p_j be respectively the numbers of cycles of length i and of paths of length j in the breakpoint graph of genomes mathbb {A} and mathbb {B}. Then, the breakpoint distance of mathbb {A} and mathbb {B} is equal to n_*-left( c_2+frac{p_0}{2}right). Similarly, when the considered rearrangements are those modeled by the double-cut-and-join (DCJ) operation, the rearrangement distance of mathbb {A} and mathbb {B} is n_*-left( c+frac{p_e }{2}right), where c is the total number of cycles and p_e is the total number of paths of even length.MotivationThe distance formulation is a basic unit for several other combinatorial problems related to genome evolution and ancestral reconstruction, such as median or double distance. Interestingly, both median and double distance problems can be solved in polynomial time for the breakpoint distance, while they are NP-hard for the rearrangement distance. One way of exploring the complexity space between these two extremes is to consider a sigma _k distance, defined to be n_*-left( c_2+c_4+ldots +c_k+frac{p_0+p_2+ldots +p_{k-2}}{2}right), and increasingly investigate the complexities of median and double distance for the sigma _4 distance, then the sigma _6 distance, and so on.ResultsWhile for the median much effort was done in our and in other research groups but no progress was obtained even for the sigma _4 distance, for solving the double distance under sigma _4 and sigma _6 distances we could devise linear time algorithms, which we present here.