Sorting Unsigned Permutations by Weighted Reversals,Transpositions, and Transreversals

Reversals, transpositions and transreversals are common events in genome rearrangement. The genome rearrangement sorting problem is to transform one genome into another using the minimum number of given rearrangement operations. An integer permutation is used to represent a genome in many cases. It can be divided into disjoint strips with each strip denoting a block of consecutive integers. A singleton is a strip of one integer. And the genome rearrangement problem turns into the problem of sorting a permutation into the identity permutation equivalently. Hannenhalli and Pevzner designed a polynomial time algorithm for the unsigned reversal sorting problem on those permutations with O(log n) singletons. In this paper, first we describe one case in which Hannenhalli and Pevzner’s algorithm may fail and propose a corrected approach. In addition, we propose a (1+ε)-approximation algorithm for sorting unsigned permutations with O(log n) singletons by reversals of weight 1 and transpositions/transreversals of weight 2.     

A 2-approximation algorithm for genome rearrangements by reversals and transpositions

Recently, a new approach to analyze genomes evolving which is based on comparision of gene orders versus traditional comparision of DNA sequences was proposed (Sankoff et al. 1992). The approach is based on the global rearrangements (e.g., inversions and transpositions of fragments). Analysis of genomes evolving by inversions and transpositions leads to a combinatorial problem of sorting by reversals and transpositions, i.e., sorting of a permutation using reversals and transpositions of arbitrary fragments. We study sorting of signed permutations by reversals and transpositions, a problem which adequately models genome rearrangements, as the genes in DNA are oriented. We establish a lower bound and give a 2-approximation algorithm for the problem.     

A new approximation algorithm for cut-and-paste sorting of unsigned circular permutations

A cut-and-paste operation can be a reversal, a transposition, or a transreversal on circular or linear permutations. There are several approximation algorithms for sorting signed permutations by combinations of these operations. For sorting unsigned permutations, we only know an algorithm with performance ratio 3 and its improved version with performance ratio $2.8386+\delta$ allowing reversals and transpositions. In this paper, we present a 2.25-approximation algorithm for sorting unsigned circular permutations by cut-and-paste operations. A structure called tie is proposed to represent the alternating path of length 5. We can classify the ties into 6 types and find ways to remove the breakpoints for each type of ties, so that every cut-and-paste operation can reduce at least $\frac{4}{3}$ breakpoints averagely. Our algorithm can be used to sort unsigned linear permutations and achieve the performance ratio 2.25 if another operation named revrev is allowed.     

Sorting by Short Block-Moves

Sorting permutations by operations such as reversals and block-moves has received much interest because of its applications in the study of genome rearrangements and in the design of interconnection networks. A short block-move is an operation on a permutation that moves an element at most two positions away from its original position. This paper investigates the problem of finding a minimum-length sorting sequence of short block-moves for a given permutation. A 4/3 -approximation algorithm for this problem is presented. Woven double-strip permutations are defined and a polynomial-time algorithm for this class of permutations is devised that employs graph matching techniques. A linear-time maximum matching algorithm for a special class of grid graphs improves the time complexity of the algorithm for woven double-strip permutations. Received June 1, 1997; revised July 25, 1998.     

短块移动排序算法研究

用短块移动操作对一个排列进行排序是一种染色体基因重排技术。怎样才能找出使用短块移动次数最少的排序算法是计算生物学等领域最热门的研究问题之一。给出了短块移动的最优解算法,对近似算法进行了修改。实验验证了最优解算法和近似算法在实际运行过程中都有较好的表现。     

A 14/11-approximation algorithm for sorting by short block-moves

Block-move is one of the popular operations for genome rearrangement. A short block-move is an operation on a permutation that moves an element at most two positions away from its original position. Heath and Vergara investigated the problem of finding a minimum-length sorting sequence of short block-moves for a given permutation and devised a 4/3-approximation algorithm. In this paper, we present a new 14/11-approximation algorithm for this problem. Firstly, we devise an exact polynomial time alg...     

短块移动排序的14/11近似算法

块移动是基因重组的一种重要形式.短块移动是将排列中的元素最多移动到偏离原来两个位置的块移动.Heath和Vergara最先给出短块移动排序近似度为4/3的多项式时间算法.本文设计了近似性能比为14/11的短块移动排序新算法.首先讨论了具有伞形结构排列图的子排列的排序方法,并将这种子排列称为‘伞’,设计了特殊子排列伞短块移动排序的多项式时间精确算法.然后给出关联伞子排列短块移动排序的贪心算法.讨论了5种特殊子排列的短块移动排序方法,证明了它们短块移动距离的新下界,从而证明此贪心算法的近似性能比为14/11,这是目前解答短块移动排序问题近似性能比最小的多项式近似算法.     

基因组Translocation排序问题的改进多项式算法

该文给出基因组Transhocation排序问题的一个改进多项式算法，原算法所有存储空间O（n)，时间复杂度为O(n^3)，文中改进算法仍采用O(n)存储空间，时间复杂度为O(n^2logn),具体地，将计算Translocation距离的时间复杂度由O(n^3)改进为O(n^2),将计算Translocation序列的时间复杂度由O(n^3)改进为O(n^2logn).     

Special Transpositions of Permutation Elements and Properties of Their Composition

We propose a strategy to solve the problem of optimization of a linear function on the set of cyclic permutations. The strategy is based on the properties of transpositions of special kind. The properties of this class of transpositions are investigated. Assertions about the impact of compositions of such transpositions on an arbitrary permutation are proved. For the approximate solutions obtained using the above strategy, estimation is substantiated.     

A more efficient algorithm for perfect sorting by reversals

We describe a new algorithm for the problem of perfect sorting a signed permutation by reversals. The worst-case time complexity of this algorithm is parameterized by the maximum prime degree d of the strong interval tree, i.e., f(d).nO(1). This improves the best known algorithm which complexity was based on a parameter always larger than or equal to d.     

计算椭圆曲线上多标量乘的快速算法

椭圆曲线密码体制最主要的运算就是椭圆曲线上的标量乘和多标量乘,在各种密码协议中起到了核心作用.文中设计了多个整数的一种新的联合带符号二进制表示的编码算法,它每次最多处理相邻的两列,因此在实现上是简单而快速的;在此基础上提出了计算椭圆曲线上多标量乘的一个新算法,并对这个算法进行了分析,最后将新算法和已有多标量乘算法进行了比较,指出新算法在一般情况下(m3时)效率可提高7%～15%.     

一些非一次相关置换和对合的产生方法

本文讨论非一次相关置换和对合的产生问题.对于置换，首先给出了由给定置换进行仿射变换产生一类相关次数相同置换的方法，然后给出了由低维非一次相关置换递归产生高维非一次相关置换的方法，并估计了这些方法产生的置换个数.对于对合，给出了一个从特定非一次相关对合的不动点上构作不相交p-组产生非一次相关对合的方法，并估计出一个对合个数的松下界.     

Exact and approximation algorithms for sorting by reversals,with application to genome rearrangement

Motivated by the problem in computational biology of reconstructing the series of chromosome inversions by which one organism evolved from another, we consider the problem of computing the shortest series of reversals that transform one permutation to another. The permutations describe the order of genes on corresponding chromosomes, and areversal takes an arbitrary substring of elements, and reverses their order.For this problem, we develop two algorithms: a greedy approximation algorithm, that finds a solution provably close to optimal inO(n ²) time and0(n) space forn-element permutations, and a branch- and-bound exact algorithm, that finds an optimal solution in0(mL(n, n)) time and0(n ²) space, wherem is the size of the branch- and-bound search tree, andL(n, n) is the time to solve a linear program ofn variables andn constraints. The greedy algorithm is the first to come within a constant factor of the optimum; it guarantees a solution that uses no more than twice the minimum number of reversals. The lower and upper bounds of the branch- and-bound algorithm are a novel application of maximum-weight matchings, shortest paths, and linear programming.In a series of experiments, we study the performance of an implementation on random permutations, and permutations generated by random reversals. For permutations differing byk random reversals, we find that the average upper bound on reversal distance estimatesk to within one reversal fork<1/2n andn<100. For the difficult case of random permutations, we find that the average difference between the upper and lower bounds is less than three reversals forn<50. Due to the tightness of these bounds, we can solve, to optimality, problems on 30 elements in a few minutes of computer time. This approaches the scale of mitochondrial genomes.This research was supported by a postdoctoral fellowship from the Program in Mathematics and Molecular Biology of the University of California at Berkeley under National Science Foundation Grant DMS-8720208, and by a fellowship from the Centre de recherches mathématiques of the Université de Montréal.This research was supported by grants from the Natural Sciences and Engineering Research Council of Canada, and the Fonds pour la formation de chercheurs et l'aide à la recherche (Québec). The author is a fellow of the Canadian Institute for Advanced Research.     

基于PAR 的排序算法自动生成研究

排序是计算机学科中的一类特殊问题,其算法设计策略的灵活性使得求解算法更具多样性.基于形式化方法PAR(partition-and-recur),研究了排序算法的自动生成问题.刻画了排序问题的代数性质,形式化构建了排序算法领域的泛型类型构件和算法构件,建立了排序领域特定语言和算法生成形式化模型,以参数替换的方式自动生成了一组排序算法,包括快速排序、堆排序、Shell排序等典型的已知算法以及增量选择排序等若干未见于现有文献的算法,并在程序生成系统中予以了实现.通过上层框架研究和底层构件支持,显著提高了特定领域算法的开发效率和可靠性.     

有效编码算法的LDPC编码器的VerilogHDL设计

低密度奇偶校验码(Low-Density-Parity-Checkcodes,简称LDPC码)是第四代通信系统强有力的竞争者,它是一种逼近香农限的线性分组码,译码的复杂度较低;但它的直接编码运算量较大,通常具有码长的二次方复杂度。本文创新点在于如何构造有效的编码,以降低LDPC码的编码复杂度;并研究和设计了用大规模集成电路去实现一个LDPC码的编码。文章中以(12,3,6)码为例,采用基于下三角矩阵的有效编码算法,通过重排列的顺序得到一个新的校验矩阵,以控制编码运算量为线性复杂度,并在QuartusII5.0软件平台上采用基于CPLD的VerilogHDL语言编程仿真实现了有效编码的过程,给出了编码的结构图和仿真波形,为LDPC码的硬件实现和实际应用提供了依据。     

枚举有符号基因组的可行交互移位算法

交互移位排序问题(SRT)是寻找一个使一个基因组转变为另一个基因组的最短交互移位序列。现在已有多个多项式时间的SRT算法,但大多数问题实例都有许多个最短交互移位序列,因此寻找所有最短交互移位序列问题是SRT一个自然的推广。这个问题可以归约为寻找一个基因组相对于另一个基因组的全部可行交互移位,即所有移位ρ满足:在一个基因组上执行ρ之后,所得基因组相对于另一个基因组的移位距离会减少。本文提出一个用来寻找全部可行交互移位的有效算法,尽管新算法的时间复杂度比穷举法改进不大,但实验结果表明,其在实际运行中表现更好。     

基因组一般移位排序问题的多项式时间算法

基因组移位排序在基因组重组排序计算研究中占有重要位置.交互型移位和非交互型移位均为移位的特殊形式.目前见到的多种移位排序算法均是针对交互型移位而得到的,未见基因组一般移位排序计算的研究结果.文中讨论包括交互型移位和非交互型移位的一般移位排序问题的求解方法,给出该问题的一个多项式时间算法.算法的关键在于将一般移位排序问题在线性时间内归约为交互型移位排序问题,利用交互型移位排序的算法来求解一般移位排序.作者的算法证实了Ozery-Flato等关于一般移位排序问题可以多项式时间解决的猜测.     

Improved bounds on sorting by length-weighted reversals

We study the problem of sorting binary sequences and permutations by length-weighted reversals. We consider a wide class of cost functions, namely f(ℓ)=ℓ^α for all α0, where ℓ is the length of the reversed subsequence. We present tight or nearly tight upper and lower bounds on the worst-case cost of sorting by reversals. Then we develop algorithms to approximate the optimal cost to sort a given input. Furthermore, we give polynomial-time algorithms to determine the optimal reversal sequence for a restricted but interesting class of sequences and cost functions. Our results have direct application in computational biology to the field of comparative genomics.     

Remark on one problem in extremal combinatorics

We reduce the problem of determining the maximum number of permutations of a finite set such that any pair of permutations has at least t common transpositions to the problem of determining the maximum number of permutations of finite set such that any pair has at least t common fixed points. The latter problem was solved by the author in [1].