  Two genomes over the same set of gene families form a canonical pair when
each of them has exactly one gene from each family. 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. Then, the breakpoint distance is equal to n -
(c_2 + p_0/2), where n is the number of genes, c_2 is the number of cycles of
length 2 and p_0 is the number of paths of length 0. Similarly, when the
considered rearrangements are those modeled by the double-cut-and-join (DCJ)
operation, the rearrangement distance is n - (c + p_e/2), where c is the total
number of cycles and p_e is the total number of even paths.
  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 the {\sigma}_k
distance, defined to be n - [c_2 + c_4 + ... + c_k + (p_0 + p_2 + ... +p_k)/2],
and increasingly investigate the complexities of median and double distance for
the {\sigma}_4 distance, then the {\sigma}_6 distance, and so on. While 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.
