探究双距离问题的复杂度
Investigating the complexity of the double distance problems
AI总结：
本文研究了基因组双距离问题的复杂度，针对介于断点距离和重排距离之间的 σ_4 和 σ_6 距离，提出了线性时间求解算法。
摘要：
当两个基因组在相同的基因家族集合上各自恰好包含每个家族的一个基因时，它们构成一个规范对。规范基因组的不同距离可从一种称为断点图的结构中导出，该结构将两个给定基因组之间的关系表示为偶数长度环和路径的集合。于是，断点距离等于 n - (c_2 + p_0/2)，其中 n 是基因数量，c_2 是长度为 2 的环的数量，p_0 是长度为 0 的路径的数量。类似地，当所考虑的重排是由双切接合（DCJ）操作建模时，重排距离为 n - (c + p_e/2)，其中 c 是环的总数，p_e 是偶数路径的总数。距离公式是若干与基因组进化和祖先重建相关的其他组合问题的基本单元，例如中位数或双距离问题。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，它们是 NP 困难的。探索这两个极端之间复杂度空间的一种方式是考虑 σ_k 距离，其定义为 n - [c_2 + c_4 + ... + c_k + (p_0 + p_2 + ... + p_k)/2]，并逐步研究中位数和双距离在 σ_4 距离、然后是 σ_6 距离等情况下的复杂度。尽管我们及其他研究团队在中位数问题上投入了大量精力，但即使对于 σ_4 距离也未取得任何进展，而对于在 σ_4 和 σ_6 距离下求解双距离问题，我们成功设计了线性时间算法，并在此予以展示。
英文摘要：
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 σ_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 σ_4 distance, then the σ_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 σ_4 distance, for solving the double distance under σ_4 and σ_6 distances we could devise linear time algorithms, which we present here.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
