PUB - Publikationen an der Universität Bielefeld

Background Two genomes A and 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 A and B. Different dis- tances 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 ci and pj be respectively the numbers of cycles of length i and of paths of length j in the breakpoint graph of genomes A and B. Then, the breakpoint distance of A and B is equal to n∗ − (c2 + p0 2). Similarly, when the considered rearrangements are those modeled by the double-cut-and-join (DCJ) operation, the rearrangement distance of A and B is n∗ − (c + pe 2 ), where c is the total number of cycles and pe is the total number of paths of even length. Motivation The 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 σk distance, defined to be n∗ − ( c2 + c4 + . . . + c k + p0+p2+...+p k−2 2 ), and increasingly investigate the complexities of median and double distance for the σ4 distance, then the σ6 distance, and so on. Results While for the median much effort was done in our and in other research groups but no progress was obtained even for the σ4 distance, for solving the double distance under σ4 and σ6 distances we could devise linear time algorithms, which we present here