完全逆序排序的平均-事例分析

Average-case analysis of perfect sorting by reversals (Journal Version)

Perfect sorting by reversals, a problem originating in computational genomics, is the process of sorting a signed permutation to either the identity or to the reversed identity permutation, by a sequence of reversals that do not break any common interval. Bérard et al. (2007) make use of strong interval trees to describe an algorithm for sorting signed permutations by reversals. Combinatorial properties of this family of trees are essential to the algorithm analysis. Here, we use the expected value of certain tree parameters to prove that the average run-time of the algorithm is at worst, poly…

Sorting by Reversals and the Theory of 4-Regular Graphs

We show that the theory of sorting by reversals fits into the well-established theory of circuit partitions of 4-regular multigraphs (which also involves the combinatorial structures of circle graphs and delta-matroids). In this way, we expose strong connections between the two theories that have not been fully appreciated before. We also discuss a generalization of sorting by reversals involving the double-cut-and-join (DCJ) operation. Finally, we also show that the theory of sorting by reversals is closely related to that of gene assembly in ciliates.

Sorting Permutations with Fixed Pinnacle Set

We give a positive answer to a question raised by Davis et al. ({\em Discrete Mathematics} 341, 2018), concerning permutations with the same pinnacle set. Given $π\in S_n$, a {\em pinnacle} of $π$ is an element $π_i$ ($i\neq 1,n$) such that $π_{i-1}<π_i>π_{i+1}$. The question is: given $π,π'\in S_n$ with the same pinnacle set $S$, is there a sequence of operations that transforms $π$ into $π'$ such that all the intermediate permutations have pinnacle set $S$? We introduce {\em balanced reversals}, defined as reversals that do not modify the pinnacle set of the permutation to which they are app…

二进制和三进制字符串的前缀反转

Prefix reversals on binary and ternary strings

给定一个置换pi，前缀反转f^（i）对pi的应用颠倒了pi的前i个元素的顺序。由Gates和Papadimitriou（英语：Papadimitriou）提出的前缀反转排序（也称为煎饼翻转）问题（Bounds for sorting by prefix Reversals），离散数学27，pp. 47-57），要求对给定排列的元素进行排序所需的前缀反转的最小数量。在本文中，我们研究了这个问题的一个变种，前缀反转不作用于排列，但字符串在一个固定大小的字母表。我们确定了二进制和三进制字符串排序所需的最小前缀反转数，并给出了这些排序问题的多项式时间算法;证明了计算两个二进制字符串之间的最小前缀反转距离是NP-难的;给出了二进制字符串前缀反转直径的精确表达式，并给出了三进制字符串前缀反转直径的界.我们还考虑了一种较弱的排序形式称为分组（相同的符号），并给出了多项式时间算法的最佳分组二进制和三进制字符串。一些有趣的开放问题也进行了讨论。

On a lower bound for sorting signed permutations by reversals

Computing the reversal distances of signed permutations is an important topic in Bioinformatics. Recently, a new lower bound for the reversal distance was obtained via the plane permutation framework. This lower bound appears different from the existing lower bound obtained by Bafna and Pevzner through breakpoint graphs. In this paper, we prove that the two lower bounds are equal. Moreover, we confirm a related conjecture on skew-symmetric plane permutations, which can be restated as follows: let $p=(0,-1,-2,\ldots -n,n,n-1,\ldots 1)$ and let $$ \tilde{s}=(0,a_1,a_2,\ldots a_n,-a_n,-a_{n-1},\l…

Polynomial-time sortable stacks of burnt pancakes

Pancake flipping, a famous open problem in computer science, can be formalised as the problem of sorting a permutation of positive integers using as few prefix reversals as possible. In that context, a prefix reversal of length k reverses the order of the first k elements of the permutation. The burnt variant of pancake flipping involves permutations of signed integers, and reversals in that case not only reverse the order of elements but also invert their signs. Although three decades have now passed since the first works on these problems, neither their computational complexity nor the maxim…