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：两个基因组若在同一组基因家族上各自恰好拥有每个家族的一个基因，则构成规范对。规范基因组的不同距离可通过称为断点图的结构推导得出，该结构将两个给定基因组的关系表示为偶数长度环和路径的集合。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的环的数量，p_0为长度为0的路径的数量。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为环的总数，p_e为偶数路径的总数。该距离公式是基因组进化与祖先重建中若干其他组合问题（如中位数或双距离）的基本单元。有趣的是，对于断点距离，中位数和双距离问题均可在多项式时间内求解，而对于重排距离，它们属于NP难问题。探索这两个极端之间复杂度空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距离的复杂度。尽管本课题组及其他研究组在中位数问题上投入了大量努力，但即使在σ_4距离下也未取得进展，而对于σ_4和σ_6距离下的双距离求解，我们设计了线性时间算法，现于本文中呈现。