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:通过基因家族集合上的两个基因组组成规范对,当它们各自从每个家族中恰好有一个基因时,它们会形成一个基因组对。规范基因组之间的不同距离可以从称为断点图的结构中导出,该图将给定的两个基因组之间的关系表示为偶数长度和路径的一组循环。然后,断点距离等于n-(c_2 + p_0/2),其中n为基因数,c_2表示长度为2的循环数,p_0表示长度为0的路径数。类似地,当所考虑的重排列是由双切和接(DCJ)操作建模的重排列时,重排列距离为n-(c + p_e/2),其中c为总循环数,p_e为总偶数路径数。距离公式是基因组演化和祖先重建的若干其他组合问题的基本单元,例如中位数或双距离。有趣的是,对于断点距离,中位数和双距离问题均可以在多项式时间内解决,而对于重排列距离,则是NP难的。探索这两个极端之间的复杂性空间的一种方法是考虑{\sigma}_k距离,定义为n-[c_2 + c_4 + … + c_k + (p_0 + p_2 + … +p_k)/2],并逐渐研究{\sigma}_4距离、{\sigma}_6距离等条件下中位数和双距离的复杂性。虽然对于中位数,我们和其他研究小组都投入了大量的精力,但甚至对于{\sigma}_4距离也没有取得进展,在解决{\sigma}_4和{\sigma}_6距离下的双距离问题时,我们可以设计出线性时间算法,我们在此提出。