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.

翻译：两个共享相同基因家族集合的基因组，当每个基因组中各基因家族恰好拥有一个基因时，构成规范对。规范基因组的不同距离可通过称为断点图的结构导出，该结构将两个给定基因组的关系表示为偶数长度环与路径的集合。此时，断点距离等于 n - (c_2 + p_0/2)，其中 n 为基因数量，c_2 为长度为2的环数，p_0 为长度为0的路径数。类似地，当考虑由双切割与连接(DCJ)操作建模的重排时，重排距离为 n - (c + p_e/2)，其中 c 为环总数，p_e 为偶数路径总数。该距离公式是基因组进化与祖先重建中多个组合问题（如中位数问题或双重距离问题）的基本单元。有趣的是，对于断点距离，中位数与双重距离问题均可在多项式时间内求解；而对于重排距离，两者均为NP难问题。探索这两类极端复杂度区间的一种方法是考虑σ_k距离，定义为 n - [c_2 + c_4 + ... + c_k + (p_0 + p_2 + ... +p_k)/2]，并逐步研究σ_4距离、σ_6距离等情形下中位数与双重距离的复杂度。尽管我们与其他研究团队在中位数问题上投入了大量努力，但即使在σ_4距离下仍未取得进展；而对于σ_4与σ_6距离下的双重距离求解，我们设计了线性时间算法，本文对此进行阐述。