ON PIERCING SETS OF AXIS-PARALLEL RECTANGLES AND RINGS

1999 ◽  
Vol 09 (03) ◽  
pp. 219-233 ◽  
Author(s):  
MICHAEL SEGAL
Keyword(s):  
Dimensional Space ◽  
Time Algorithm ◽  
Efficient Algorithms ◽  
Center Problem ◽  
Running Time ◽  
Higher Dimensional ◽  
Axis Parallel ◽  
The Given ◽  
Existing Result

We consider the p-piercing problem for axis-parallel rectangles. We are given a collection of axis-parallel rectangles in the plane and wish to determine whether there exists a set of p points whose union intersects all the given rectangles. We present efficient algorithms for finding a piercing set (i.e, a set of p points as above) for values of p=1,2,3,4,5. The result for 4 and 5-piercing improves an existing result of O(n  log 3 n) and O(n  log 4 n) to O(n  log  n) time. The result for 5-piercing can be applied find an O(n  log 2 n) time algorithm for planar rectilinear 5-center problem, in which we are given a set S of n points in the pane, and wish to find 5 axis-parallel congruent squares of smallest possible size whose union covers S. We improve the existing algorithm for general (but fixed) p to O(np-4 log  n) running time, and we also extend our algorithms to higher dimensional space. We also consider the problem of piercing a set of rectangular rings.

Line-Constrained k-Median, k-Means, and k-Center Problems in the Plane

2016 ◽  
Vol 26 (03n04) ◽  
pp. 185-210
Author(s):  
Haitao Wang ◽  
Jingru Zhang
Keyword(s):  
Time Algorithm ◽  
Efficient Algorithms ◽  
Additional Constraint ◽  
Distance Measures ◽  
Np Hard ◽  
Center Problem ◽  
Median Problem ◽  
One Dimensional ◽  
The Given

The (weighted) [Formula: see text]-median, [Formula: see text]-means, and [Formula: see text]-center problems in the plane are known to be NP-hard. In this paper, we study these problems with an additional constraint that requires the sought [Formula: see text] facilities to be on a given line. We present efficient algorithms for various distance measures such as [Formula: see text]. We assume that all [Formula: see text] weighted points are given sorted by their projections on the given line. For [Formula: see text]-median, our algorithms for [Formula: see text] and [Formula: see text] metrics run in [Formula: see text] time and [Formula: see text] time, respectively. For [Formula: see text]-means, which is defined only on the squared [Formula: see text] distance, we give an [Formula: see text] time algorithm. For [Formula: see text]-center, our algorithms run in [Formula: see text] time for all three metrics, and in [Formula: see text] time for the unweighted version under [Formula: see text] and [Formula: see text] metrics. While our results for the [Formula: see text]-center problem are optimal, the results for the [Formula: see text]-median problem almost match the best algorithms for the corresponding one-dimensional problems.

Approximating Convex Polyhedra with Axis-Parallel Boxes

1997 ◽  
Vol 07 (03) ◽  
pp. 253-267 ◽  
Author(s):  
Binhai Zhu
Keyword(s):  
Hausdorff Distance ◽  
Convex Polyhedron ◽  
Linear Time ◽  
Time Algorithm ◽  
Convex Polyhedra ◽  
Linear Time Algorithm ◽  
Brute Force ◽  
Axis Parallel ◽  
The Given ◽  
Fixed Center

In this paper, we present an O(n4 log 2n) time algorithm to compute an approximate discrete axis-parallel box of a given n-vertex convex polyhedron P such that the given polyhedron is minimized. Here, "discrete" means that each plane containing a face of the approximate box passes through a vertex of P (or, more generally, passes through a point of a set of given points). This algorithm is significantly faster than the brute force O(n7) time solution for computing the optimal approximate axis-parallel box A* of P such that the symmetric difference of the volume between P and A* is minimized. We present a linear time algorithm to compute a pseudo-optimal (with factor [Formula: see text] approximate axis-parallel box of a convex polyhedron under the Hausdorff distance criterion. We also present O(n) and O(n7 log n) time algorithms to compute the optimal approximate ball, with or without a fixed center, of a convex polyhedron under the Hausdorff distance criterion.

COMPUTING THE DISCRETE FRÉCHET DISTANCE WITH IMPRECISE INPUT

2012 ◽  
Vol 22 (01) ◽  
pp. 27-44 ◽  
Author(s):  
HEE-KAP AHN ◽  
CHRISTIAN KNAUER ◽  
MARC SCHERFENBERG ◽  
LENA SCHLIPF ◽  
ANTOINE VIGNERON
Keyword(s):  
Time Algorithm ◽  
Constant Factor ◽  
Fréchet Distance ◽  
Planar Case ◽  
Running Time ◽  
Frechet Distance ◽  
Axis Parallel ◽  
Polygonal Curves ◽  
Orthogonal Case ◽  
Improved Algorithm

We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time 2O(d2) m2n2 log 2(mn) the minimum Fréchet distance between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O(mn log 3(mn) + (m2+n2) log (mn)). In the d-dimensional orthogonal case, where points are modelled as axis-parallel boxes, and we use the L∞ distance, we give an O(dmn log (dmn))-time algorithm. We also give efficient O(dmn)-time algorithms to approximate the maximum Fréchet distance, as well as the minimum and maximum Fréchet distance under translation. These algorithms achieve constant factor approximation ratios in "realistic" settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).

scholarly journals An Efficient Algorithm of the Planar 3-Center Problem for a set of the convex position points

2018 ◽  
Vol 232 ◽  
pp. 03022
Author(s):  
Donglai Bian ◽  
Bo Jiang ◽  
Zhiying Cao
Keyword(s):  
Efficient Algorithm ◽  
Convex Polygon ◽  
Time Algorithm ◽  
Center Problem ◽  
Minimum Radius ◽  
Convex Position ◽  
The Given ◽  
Circular Disks

The planar 3-center problem for a set S of points given in the plane asks for three congruent circular disks with the minimum radius, whose union can cover all points of S completely. In this paper, we present an O(n2 log3n) time algorithm for a restricted planar 3-center problem in which the given points are in the convex positions , i.e. The given points are the vertices of a convex polygon exactly.

scholarly journals Computing the Discrete Compactness of Orthogonal Pseudo-Polytopes via Their D-EVM Representation

2010 ◽  
Vol 2010 ◽  
pp. 1-28 ◽  
Author(s):  
Ricardo Pérez-Aguila
Keyword(s):  
Dimensional Space ◽  
Efficient Algorithms ◽  
Video Sequences ◽  
Higher Dimensional ◽  
Discrete Compactness ◽  
Representation Scheme ◽  
2D And 3D ◽  
Orthogonal Polytopes

This work is devoted to present a methodology for the computation of Discrete Compactness in -dimensional orthogonal pseudo-polytopes. The proposed procedures take in account compactness' definitions originally presented for the 2D and 3D cases and extend them directly for considering the D case. There are introduced efficient algorithms for computing discrete compactness which are based on an orthogonal polytopes representation scheme known as the Extreme Vertices Model in the -Dimensional Space (D-EVM). It will be shown the potential of the application of Discrete Compactness in higher-dimensional contexts by applying it, through EVM-based algorithms, in the classification of video sequences, associated to the monitoring of a volcano's activity, which are expressed as 4D orthogonal polytopes in the space-color-time geometry.

THE ALIGNED K-CENTER PROBLEM

2011 ◽  
Vol 21 (02) ◽  
pp. 157-178 ◽  
Author(s):  
PETER BRASS ◽  
CHRISTIAN KNAUER ◽  
HYEON-SUK NA ◽  
CHAN-SU SHIN ◽  
ANTOINE VIGNERON
Keyword(s):  
Approximation Algorithms ◽  
Randomized Algorithms ◽  
Randomized Algorithm ◽  
Time Algorithm ◽  
Center Problem ◽  
Running Time ◽  
Maximum Radius ◽  
Arbitrary Line ◽  
Set Of Points

In this paper we study several instances of the alignedk-center problem where the goal is, given a set of points S in the plane and a parameter k ⩾ 1, to find k disks with centers on a line ℓ such that their union covers S and the maximum radius of the disks is minimized. This problem is a constrained version of the well-known k-center problem in which the centers are constrained to lie in a particular region such as a segment, a line, or a polygon. We first consider the simplest version of the problem where the line ℓ is given in advance; we can solve this problem in time O(n log 2 n). In the case where only the direction of ℓ is fixed, we give an O(n2 log 2 n)-time algorithm. When ℓ is an arbitrary line, we give a randomized algorithm with expected running time O(n4 log 2 n). Then we present (1+ε)-approximation algorithms for these three problems. When we denote T(k, ε) = (k/ε2+(k/ε) log k) log (1/ε), these algorithms run in O(n log k + T(k, ε)) time, O(n log k + T(k, ε)/ε) time, and O(n log k + T(k, ε)/ε2) time, respectively. For k = O(n1/3/ log n), we also give randomized algorithms with expected running times O(n + (k/ε2) log (1/ε)), O(n+(k/ε3) log (1/ε)), and O(n + (k/ε4) log (1/ε)), respectively.

Ultrashort pulses propagation through different approaches of the Split-Step Fourier method

2018 ◽  
Vol 1 (3) ◽  
pp. 2
Author(s):  
José Stênio De Negreiros Júnior ◽  
Daniel Do Nascimento e Sá Cavalcante ◽  
Jermana Lopes de Moraes ◽  
Lucas Rodrigues Marcelino ◽  
Francisco Tadeu De Carvalho Belchior Magalhães ◽  
...  
Keyword(s):  
Energy Conservation ◽  
Optical Pulse ◽  
Time Window ◽  
Fourier Method ◽  
Ultrashort Pulses ◽  
Nonlinear Effects ◽  
Single Mode ◽  
Time Algorithm ◽  
Optical Pulses ◽  
Running Time

Simulating the propagation of optical pulses in a single mode optical fiber is of fundamental importance for studying the several effects that may occur within such medium when it is under some linear and nonlinear effects. In this work, we simulate it by implementing the nonlinear Schrödinger equation using the Split-Step Fourier method in some of its approaches. Then, we compare their running time, algorithm complexity and accuracy regarding energy conservation of the optical pulse. We note that the method is simple to implement and presents good results of energy conservation, besides low temporal cost. We observe a greater precision for the symmetrized approach, although its running time can be up to 126% higher than the other approaches, depending on the parameters set. We conclude that the time window must be adjusted for each length of propagation in the fiber, so that the error regarding energy conservation during propagation can be reduced.

Celestial Tapestry

2020 ◽  
Author(s):  
Nicholas Mee
Keyword(s):  
Mathematics Curriculum ◽  
Dimensional Space ◽  
Golden Section ◽  
Single Point ◽  
Historical Context ◽  
Computer Art ◽  
Lightning Strike ◽  
Secret Message ◽  
Axiomatic Method ◽  
Higher Dimensional

Celestial Tapestry places mathematics within a vibrant cultural and historical context, highlighting links to the visual arts and design, and broader areas of artistic creativity. Threads are woven together telling of surprising influences that have passed between the arts and mathematics. The story involves many intriguing characters: Gaston Julia, who laid the foundations for fractals and computer art while recovering in hospital after suffering serious injury in the First World War; Charles Howard, Hinton who was imprisoned for bigamy but whose books had a huge influence on twentieth-century art; Michael Scott, the Scottish necromancer who was the dedicatee of Fibonacci’s Book of Calculation, the most important medieval book of mathematics; Richard of Wallingford, the pioneer clockmaker who suffered from leprosy and who never recovered from a lightning strike on his bedchamber; Alicia Stott Boole, the Victorian housewife who amazed mathematicians with her intuition for higher-dimensional space. The book includes more than 200 colour illustrations, puzzles to engage the reader, and many remarkable tales: the secret message in Hans Holbein’s The Ambassadors; the link between Viking runes, a Milanese banking dynasty, and modern sculpture; the connection between astrology, religion, and the Apocalypse; binary numbers and the I Ching. It also explains topics on the school mathematics curriculum: algorithms; arithmetic progressions; combinations and permutations; number sequences; the axiomatic method; geometrical proof; tessellations and polyhedra, as well as many essential topics for arts and humanities students: single-point perspective; fractals; computer art; the golden section; the higher-dimensional inspiration behind modern art.

scholarly journals Aperiodic Crystals

2014 ◽  
Vol 70 (a1) ◽  
pp. C1-C1 ◽  
Author(s):  
Ted Janssen ◽  
Aloysio Janner
Keyword(s):  
Physical Properties ◽  
Dimensional Space ◽  
X Rays ◽  
New Techniques ◽  
New Family ◽  
Modulated Phases ◽  
Open Questions ◽  
Higher Dimensional ◽  
Lattice Periodicity ◽  
Aperiodic Crystals

2014 is the International Year of Crystallography. During at least fifty years after the discovery of diffraction of X-rays by crystals, it was believed that crystals have lattice periodicity, and crystals were defined by this property. Now it has become clear that there is a large class of compounds with interesting properties that should be called crystals as well, but are not lattice periodic. A method has been developed to describe and analyze these aperiodic crystals, using a higher-dimensional space. In this lecture the discovery of aperiodic crystals and the development of the formalism of the so-called superspace will be described. There are several classes of such materials. After the incommensurate modulated phases, incommensurate magnetic crystals, incommensurate composites and quasicrystals were discovered. They could all be studied using the same technique. Their main properties of these classes and the ways to characterize them will be discussed. The new family of aperiodic crystals has led also to new physical properties, to new techniques in crystallography and to interesting mathematical questions. Much has been done in the last fifty years by hundreds of crystallographers, crystal growers, physicists, chemists, mineralogists and mathematicians. Many new insights have been obtained. But there are still many questions, also of fundamental nature, to be answered. We end with a discussion of these open questions.

Implementation of Selection Sort Algorithm in Various Programming Languages

2021 ◽  
Vol 10 (4) ◽  
pp. 2249-2255
Keyword(s):  
Programming Languages ◽  
Data Science ◽  
Efficient Algorithms ◽  
Huge Number ◽  
Running Time ◽  
Fast Running ◽  
Python Language ◽  
Sort Algorithm

Sorting algorithmdeals with the arrangement of alphanumeric data in static order.It plays an important roleinthe field of data science. Selection sort is one ofthe simplest and efficient algorithms which can be applied for the huge number of elements it works likeby giving list of unsorted information, the calculation which breaksintotwo partitions. One section has all the sorted information and another sectionhas all thestaying unsorted information. The calculation rehashes itself, by finding the smallestcomponentinside the rundown of unsorted information and swappingitwith the furthest left component, in the end setting everything straight information.This researchpresents the implementationof selection sort usingC/C++, Python, and Rust and measuredthetime complexity. After experiment,we have collectedtheresults in terms of running time, andanalyzed the outcomes.It was observed that python language hasvery smallamount of line of code, and it also consumesless storage and fast running time then other two languages.

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