scholarly journals Approximation Algorithms for Sorting λ-Permutations by λ-Operations

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 175
Author(s):  
Guilherme Henrique Santos Miranda ◽  
Alexsandro Oliveira Alexandrino ◽  
Carla Negri Lintzmayer ◽  
Zanoni Dias
Keyword(s):  
Comparative Genomics ◽  
Approximation Algorithms ◽  
Evolutionary Distance ◽  
Biological Relevance ◽  
A Value ◽  
Identity Permutation ◽  
Minimum Number

Understanding how different two organisms are is one question addressed by the comparative genomics field. A well-accepted way to estimate the evolutionary distance between genomes of two organisms is finding the rearrangement distance, which is the smallest number of rearrangements needed to transform one genome into another. By representing genomes as permutations, one of them can be represented as the identity permutation, and, so, we reduce the problem of transforming one permutation into another to the problem of sorting a permutation using the minimum number of rearrangements. This work investigates the problems of sorting permutations using reversals and/or transpositions, with some additional restrictions of biological relevance. Given a value λ, the problem now is how to sort a λ-permutation, which is a permutation whose elements are less than λ positions away from their correct places (regarding the identity), by applying the minimum number of rearrangements. Each λ-rearrangement must have size, at most, λ, and, when applied to a λ-permutation, the result should also be a λ-permutation. We present algorithms with approximation factors of O(λ2), O(λ), and O(1) for the problems of Sorting λ-Permutations by λ-Reversals, by λ-Transpositions, and by both operations.

Sorting permutations by prefix and suffix rearrangements

2017 ◽  
Vol 15 (01) ◽  
pp. 1750002 ◽  
Author(s):  
Carla Negri Lintzmayer ◽  
Guillaume Fertin ◽  
Zanoni Dias
Keyword(s):  
Approximation Algorithms ◽  
Genome Rearrangements ◽  
Combinatorial Problems ◽  
Signed Permutations ◽  
Identity Permutation ◽  
A Genome ◽  
Minimum Number

Some interesting combinatorial problems have been motivated by genome rearrangements, which are mutations that affect large portions of a genome. When we represent genomes as permutations, the goal is to transform a given permutation into the identity permutation with the minimum number of rearrangements. When they affect segments from the beginning (respectively end) of the permutation, they are called prefix (respectively suffix) rearrangements. This paper presents results for rearrangement problems that involve prefix and suffix versions of reversals and transpositions considering unsigned and signed permutations. We give 2-approximation and ([Formula: see text])-approximation algorithms for these problems, where [Formula: see text] is a constant divided by the number of breakpoints (pairs of consecutive elements that should not be consecutive in the identity permutation) in the input permutation. We also give bounds for the diameters concerning these problems and provide ways of improving the practical results of our algorithms.

Sorting permutations by fragmentation-weighted operations

2020 ◽  
Vol 18 (02) ◽  
pp. 2050006 ◽  
Author(s):  
Alexsandro Oliveira Alexandrino ◽  
Carla Negri Lintzmayer ◽  
Zanoni Dias
Keyword(s):  
Approximation Algorithms ◽  
Computational Biology ◽  
Cost Function ◽  
Traditional Approach ◽  
Upper Bounds ◽  
Evolutionary Distance ◽  
Lower And Upper Bounds ◽  
Approximation Factor ◽  
New Type ◽  
The Cost

One of the main problems in Computational Biology is to find the evolutionary distance among species. In most approaches, such distance only involves rearrangements, which are mutations that alter large pieces of the species’ genome. When we represent genomes as permutations, the problem of transforming one genome into another is equivalent to the problem of Sorting Permutations by Rearrangement Operations. The traditional approach is to consider that any rearrangement has the same probability to happen, and so, the goal is to find a minimum sequence of operations which sorts the permutation. However, studies have shown that some rearrangements are more likely to happen than others, and so a weighted approach is more realistic. In a weighted approach, the goal is to find a sequence which sorts the permutations, such that the cost of that sequence is minimum. This work introduces a new type of cost function, which is related to the amount of fragmentation caused by a rearrangement. We present some results about the lower and upper bounds for the fragmentation-weighted problems and the relation between the unweighted and the fragmentation-weighted approach. Our main results are 2-approximation algorithms for five versions of this problem involving reversals and transpositions. We also give bounds for the diameters concerning these problems and provide an improved approximation factor for simple permutations considering transpositions.

APPROXIMATION ALGORITHMS FOR A VARIANT OF DISCRETE PIERCING SET PROBLEM FOR UNIT DISKS

2013 ◽  
Vol 23 (06) ◽  
pp. 461-477 ◽  
Author(s):  
MINATI DE ◽  
GAUTAM K. DAS ◽  
PAZ CARMI ◽  
SUBHAS C. NANDY
Keyword(s):  
Approximation Algorithms ◽  
Simple Algorithm ◽  
Constant Factor ◽  
Performance Ratio ◽  
Approximation Result ◽  
Worst Case ◽  
Approximation Factor ◽  
Minimum Number ◽  
Unit Disks ◽  
Set Of Points

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio [Formula: see text] in nO(k) time.

Search Numbers in Networks with Special Topologies

2019 ◽  
Vol 19 (01) ◽  
pp. 1940004
Author(s):  
BOTING YANG ◽  
RUNTAO ZHANG ◽  
YI CAO ◽  
FARONG ZHONG
Keyword(s):  
Approximation Algorithms ◽  
Search Strategy ◽  
Linear Time ◽  
Time Algorithm ◽  
Time Transformation ◽  
Linear Time Algorithm ◽  
Optimal Layout ◽  
Explicit Formulas ◽  
Minimum Number ◽  
Search Numbers

In this paper, we consider the problem of finding the minimum number of searchers to sweep networks/graphs with special topological structures. Such a number is called the search number. We first study graphs, which contain only one cycle, and present a linear time algorithm to compute the vertex separation and the optimal layout of such graphs; by a linear-time transformation, we can find the search number of this kind of graphs in linear time. We also investigate graphs, in which every vertex lies on at most one cycle and each cycle contains at most three vertices of degree more than two, and we propose a linear time algorithm to compute their search number and optimal search strategy. We prove explicit formulas for the search number of the graphs obtained from complete k-ary trees by replacing vertices by cycles. We also present some results on approximation algorithms.

Entrained ion transport systems generate the membrane component of chaotic agonist-induced vasomotion

1997 ◽  
Vol 273 (2) ◽  
pp. H909-H920 ◽  
Author(s):  
D. H. Edwards ◽  
T. M. Griffith
Keyword(s):  
Fractal Dimension ◽  
Ion Transport ◽  
Transport Systems ◽  
Oscillatory Activity ◽  
Resistance Arteries ◽  
Dynamic Complexity ◽  
K Channels ◽  
A Value ◽  
Minimum Number ◽  
Membrane Oscillator

We have analyzed the contribution of membrane ion transport systems to chaotic vasomotion induced by histamine in isolated rabbit ear resistance arteries. Dynamic complexity was monitored as a fractal correlation dimension that provides an estimate of the minimum number of control variables contributing to an irregular time series and generally took a value between 2 and 4. A distinct subcomponent of the overall oscillatory activity (frequency approximately 0.06 Hz) was selectively suppressed by blockade of Ca(2+)-activated K+ channels (KCa) with tetraethylammonium, Ca(2+)-activated Cl- channels with low extracellular Cl- concentration and niflumic acid, the Na(+)-K+ adenosine-triphosphatase (ATPase) with ouabain, and Na+/Ca2+ exchange with low-Na+ buffer. Each of these interventions caused a fall in average fractal dimension to a value < 2, whereas inhibition of voltage-dependent K+ channels with 4-aminopyridine or the Ca(2+)-ATPase extrusion pump with vanadate were without effect on the form and complexity of the vasomotion. There was no systematic correlation between the changes in fractal dimension induced by the various interventions and their effects on perfusion pressure. Our findings suggest that nonlinearity in the kinetics of multiple coupled ion transport systems leads to entrainment and the emergence of a composite membrane oscillator, thus accounting for the low fractal dimension of the vasomotion observed in these arteries.

scholarly journals Respiratory Stress of Sub-Lethal Concentration of Chlorine on Oreochromis niloticus

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Asna Salam ◽  
A.U. Arun ◽  
Shalu Soman
Keyword(s):  
Body Weight ◽  
Oreochromis Niloticus ◽  
Movement Control ◽  
Aquatic Organisms ◽  
Dilution Method ◽  
A Value ◽  
Significant Difference ◽  
Minimum Number ◽  
Serial Dilution Method ◽  
And Control

In aquatic environment, one of the most significant manifestations of the toxic stress on aquatic organisms, especially fishes are the over stimulation or depression of respiratory activities. These variations in respiratory activities have been used as in indicator of environmental stress. This study was aimed to assess the sub-lethal toxicity of chlorine through respiratory stress on Oreochromis niloticus. It was noticed that in all dosage experiments as the concentration increased rate of gill movement decreased. The rate of gill movement during the first minute after the dosage in control was 128 per minute. Serial dilution method employed in this study helped to assess the LC50 of chlorine and the value noted was 2 ppm. The minimum number of gill movement noted was in 20ppm dosage and the value was 98 per minute whereas the maximum noted was in 5ppm dosage and the value was 101 per minute. In 60th minutes after dosage also the control showed a value of 126 per minute and 20 ppm dosage showed a very low value such as 32 per minute whereas 5ppm dosage of chlorine showed a reasonably good value of 61 per minute. It was noted that in the case of average gill movement, control showed a very high value such as 127 per minute, whereas 5ppm dosage of chlorine showed a reasonably good value such as 84 per minute, and 20 ppm dosage showed a low value 72 per minute. When a comparison was made between control and differently dosed fishes, it was observed that in all dosed cases the average gill movement was very low when compared with control. A significant difference exists between gill movement in the dosed group and control (P<0.05). Consumption of oxygen increased with an increase in dosage and decreased with increased period of exposure. In control the average oxygen consumption was 0.012 mg/ml/gm body weight, in 5ppm it was 0.014 mg/ml/gm body weight, in 10 ppm it was 0.0155 mg/ml/gm body weight, in 15 ppm it was 0.021 mg/ml/gm body weight and in 20ppm it was0.022 mg/ml/gm body weight.

scholarly journals On the Complexity of Some Variations of Sorting by Transpositions

2020 ◽  
Vol 26 (9) ◽  
pp. 1076-1094
Author(s):  
Alexsandro Alexandrino ◽  
Andre Oliveira ◽  
Ulisses Dias ◽  
Zanoni Dias
Keyword(s):  
Comparative Genomics ◽  
Computational Biology ◽  
Cost Function ◽  
Large Scale ◽  
Minimum Cost ◽  
Evolutionary Distance ◽  
The Other ◽  
Np Hard ◽  
The Past

One of the main challenges in Computational Biology is to find the evolutionary distance between two organisms. In the field of comparative genomics, one way to estimate such distance is to find a minimum cost sequence of rearrangements (large scale mutations) needed to transform one genome into another, which is called the rearrangement distance. In the past decades, these problems were studied considering many types of rearrangements (such as reversals, transpositions, transreversals, and revrevs) and considering the same weight for all rearrangements, or different weights depending on the types of rearrangements. The complexity of the problems involving reversals, transpositions, and both rearrangements is known, even though the hardness proof for the problem combining reversals and transpositions was recently given. In this paper, we enhance the knowledge for these problems by proving that models involving transpositions alongside reversals, transreversals, and revrevs are NP-hard, considering weights w1 for reversals and w2 for the other rearrangements such that w2/w1 ≤ 1.5. In addition, we address a cost function related to the number of fragmentations caused by a rearrangement, proving that the problem of finding a minimum cost sorting sequence, considering the fragmentation cost function with some restrictions, is NP-hard for transpositions and the combination of reversals and transpositions.

Enumeration and Classification of Anomaly/Peak Bases in Two-Dimensional Intensity Histograms. Application of Graph Theory to Crystallographic Data Imaging

1996 ◽  
Vol 29 (3) ◽  
pp. 241-245
Author(s):  
J. H. Reibenspies
Keyword(s):  
Shape Parameter ◽  
Crystallographic Data ◽  
Two Dimensional ◽  
Area Detector ◽  
A Value ◽  
Direct Indication ◽  
Minimum Number ◽  
Shaped Graph ◽  
Insight Into

Bragg-event peaks, spikes, intensity streaks and other anomalies generate abnormal regions in two-dimensional intensity histograms from area-detector images. Examination of the shapes of these regions can contribute to the identification of the types of phenomena that generated them. The points that define the anomaly bases, when connected with imaginary lines, form unique graphs. Individual graphs, in turn, can be enumerated by employing graph-theoretical notation and the graph shapes classified. The number of lines in any given graph can also be determined by summing the degrees of the graph points and dividing by two. The ratio of the number of lines per point is a direct indication of the shape of the anomaly base. Long linear and curved shapes, like those associated with intensity streaks and powder rings, will have small lines-per-point ratios, while compact round, square or oval shapes, similar to those belonging to Bragg-event peaks, will have larger lines-per-point ratios. For any given number of points (Np ), for any given graph, the minimum number of lines (q) will equal Np − 1, while the maximum number of lines (q max, Np ) is determined from a round-shaped graph. A graph-shape parameter (GS) can thus be defined as (q − Np − 1)/(q max. Np − Np − 1), where a value near one indicates a round graph shape and a value near zero indicates a linear graph shape. The application of graph-theoretical techniques to anomaly bases can thus provide insight into the nature of the intensities distributed throughout the two-dimensional crystallographic data image.

APPROXIMATE BLOCK SORTING

2006 ◽  
Vol 17 (02) ◽  
pp. 337-355 ◽  
Author(s):  
MEENA MAHAJAN ◽  
RAGHAVAN RAMA ◽  
VENKATESH RAMAN ◽  
S. VIJAYKUMAR
Keyword(s):  
Approximation Algorithm ◽  
Optimization Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Fixed Parameter Tractable ◽  
Identity Permutation ◽  
Fixed Parameter ◽  
Minimum Number ◽  
Increasing Sequences ◽  
Better Than

We consider the problem BLOCK-SORTING: Given a permutation, sort it by using a minimum number of block moves, where a block is a maximal substring of the permutation which is also a substring of the identity permutation, and a block move repositions the chosen block so that it merges with another block. Although this problem has recently been shown to be NP-hard [3], nothing better than a trivial 3-approximation was known. We present here the first non-trivial approximation algorithm to this problem. For this purpose, we introduce the following optimization problem: Given a set of increasing sequences of distinct elements, merge them into one increasing sequence with a minimum number of block moves. We show that the merging problem has a polynomial time algorithm. Using this, we obtain an O(n3) time 2-approximation algorithm for BLOCK-SORTING. We also observe that BLOCK-SORTING, as well as sorting by transpositions, are fixed-parameter-tractable in the framework of [6].

Sign in / Sign up

Export Citation Format

Share Document