利用前缀转置的基因组重排问题的上界
Upper Bounds to Genome Rearrangement Problem using Prefix Transpositions
摘要：
基因组重排问题研究生物体中一组DNA的大规模突变。在过去的四十年里，计算生物学家和计算机科学家广泛研究了各种重排，如反转、换位、易位、裂变、融合和组合以及不同的变异。从数学的角度来看，基因组由排列表示。基因组重排问题被解释为一个问题，将一个排列转化为另一个在一定的约束条件下，取决于所选择的重排在最小数量的移动。找到最小移动次数等价于用给定的重排对置换进行排序。转置是对置换的操作，其将置换的子列表移动到同一置换中的不同位置。\emph{Prefix Transposition}，顾名思义，是移动作为置换前缀的子列表的置换。在这篇论文中，我们研究了置换上的前缀转置，并给出了一个更好的前缀转置置换排序的上界。称为\emph{generalised sequence length algorithm}的贪心算法被定义为序列长度算法的扩展，其中还考虑了合适的交替移动。该算法用于将上界依次改进为$n-\log_{3.3} n$和$n-\log_3 n$。在论文的后半部分，我们定义了一个\emph{block}的概念。我们将其与广义序列长度算法的贪婪移动一起使用，以获得$n-\log_2 n$的上界，从而通过前缀转置对排列进行排序。
英文摘要：
A Genome rearrangement problem studies large-scale mutations on a set of DNAs in living organisms. Various rearrangements like reversals, transpositions, translocations, fissions, fusions, and combinations and different variations have been studied extensively by computational biologists and computer scientists over the past four decades. From a mathematical point of view, a genome is represented by a permutation. The genome rearrangement problem is interpreted as a problem that transforms one permutation into another in a minimum number of moves under certain constraints depending on the chosen rearrangements. Finding the minimum number of moves is equivalent to sorting the permutation with the given rearrangement. A transposition is an operation on a permutation that moves a sublist of a permutation to a different position in the same permutation. A \emph{Prefix Transposition}, as the name suggests, is a transposition that moves a sublist which is a prefix of the permutation. In this thesis, we study prefix transpositions on permutations and present a better upper bound for sorting permutations with prefix transpositions. A greedy algorithm called the \emph{generalised sequence length algorithm} is defined as an extension of the sequence length algorithm where suitable alternate moves are also considered. This algorithm is used to sequentially improve the upper bound to $n-\log_{3.3} n$ and $n-\log_3 n$. In the latter part of the thesis, we defined the concept of a \emph{block}. We used it along with the greedy moves of the generalised sequence length algorithm to get an upper bound of $n-\log_2 n$ to sort permutations by prefix transpositions.
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