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Journal articles on the topic 'Geometric Covering and Packing'

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Author: Grafiati

Published: 6 September 2023

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1

Chan, Timothy M., and Elyot Grant. "Exact algorithms and APX-hardness results for geometric packing and covering problems." Computational Geometry 47, no. 2 (February 2014): 112–24. http://dx.doi.org/10.1016/j.comgeo.2012.04.001.

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2

Ashok, Pradeesha, Aniket Basu Roy, and Sathish Govindarajan. "Local search strikes again: PTAS for variants of geometric covering and packing." Journal of Combinatorial Optimization 39, no. 2 (June 21, 2019): 618–35. http://dx.doi.org/10.1007/s10878-019-00432-y.

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3

Klevanskiy, Nikolay N., Sergey I. Tkachev, Ludmila A. Voloshchuk, Rouslan B. Nourgaziev, and Vladimir S. Mavzovin. "Regular Two-Dimensional Packing of Congruent Objects: Cognitive Analysis of Honeycomb Constructions." Applied Sciences 11, no. 11 (May 31, 2021): 5128. http://dx.doi.org/10.3390/app11115128.

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A new approach to investigate the two-dimensional, regular packing of arbitrary geometric objects (GOs), using cognitive visualization, is presented. GOs correspond to congruent non-convex polygons with their associated coordinate system. The origins of these coordinate systems are accepted by object poles. The approach considered is based on cognitive processes that are forms of heuristic judgments. According to the first heuristic judgment, regular packing of congruent GOs on the plane have a honeycomb structure, that is, each GO contacts six neighboring GO, the poles of which are vertices of the pole hexagon in the honeycomb construction of packing. Based on the visualization of the honeycomb constructions a second heuristic judgment is obtained, according to which inside the hexagon of the poles, there are fragments of three GOs. The consequence is a third heuristic judgment on the plane covering density with regular packings of congruent GOs. With the help of cognitive visualization, it is established that inside the hexagon of poles there are fragments of exactly three objects. The fourth heuristic judgment is related to the proposal of a triple lattice packing for regular packing of congruent GOs.

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4

Tyrin, Grigory, and Vladimir Frolovsky. "Research and application of the crow search algorithm for geometric covering optimization problems." Proceedings of the Russian higher school Academy of sciences, no. 1 (July 8, 2021): 54–61. http://dx.doi.org/10.17212/1727-2769-2021-1-54-61.

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The problem of geometric covering is a special case of the optimal design problem and belongs to the class of cutting and packing problems. The challenge is to position some geometric objects on the surface to be coated so that the entire surface is covered. The complexity of the problems under consideration is due to their belonging to the class of NP-hard problems, which excludes the possibility of solving them by exact methods and requires the development of approximate optimization methods and algorithms. This article discusses the problem of geometric covering of an area with circles from a given set of radii. To solve the problem of geometric covering, a hexagonal grid coverage method with optimization by a metaheuristic algorithm is used. The crow search algorithm is such an algorithm, which is a relatively new metaheuristic algorithm based on the intelligent behavior of crows in a flock. The crow search algorithm includes two control parameters: the awareness probability and the flight length. To study the solution method and check the efficiency, a problem was modeled on the basis of a real design of automatic irrigation systems, and the results of experiments with different values of control parameters were presented.

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5

Fejes Tóth, G., P. Gritzmann, and J. M. Wills. "Finite sphere packing and sphere covering." Discrete & Computational Geometry 4, no. 1 (January 1989): 19–40. http://dx.doi.org/10.1007/bf02187713.

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6

Swanepoel, Konrad J. "Simultaneous Packing and Covering in Sequence Spaces." Discrete & Computational Geometry 42, no. 2 (May 7, 2009): 335–40. http://dx.doi.org/10.1007/s00454-009-9189-8.

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7

MATTILA, PERTTI, and R. DANIEL MAULDIN. "Measure and dimension functions: measurability and densities." Mathematical Proceedings of the Cambridge Philosophical Society 121, no. 1 (January 1997): 81–100. http://dx.doi.org/10.1017/s0305004196001089.

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During the past several years, new types of geometric measure and dimension have been introduced; the packing measure and dimension, see [Su], [Tr] and [TT1]. These notions are playing an increasingly prevalent role in various aspects of dynamics and measure theory. Packing measure is a sort of dual of Hausdorff measure in that it is defined in terms of packings rather than coverings. However, in contrast to Hausdorff measure, the usual definition of packing measure requires two limiting procedures, first the construction of a premeasure and then a second standard limiting process to obtain the measure. This makes packing measure somewhat delicate to deal with. The question arises as to whether there is some simpler method for defining packing measure and dimension. In this paper, we find a basic limitation on this possibility. We do this by determining the descriptive set-theoretic complexity of the packing functions. Whereas the Hausdorff dimension function on the space of compact sets is Borel measurable, the packing dimension function is not. On the other hand, we show that the packing dimension functions are measurable with respect to the σ-algebra generated by the analytic sets. Thus, the usual sorts of measurability properties used in connection with Hausdorff measure, for example measures of sections and projections, remain true for packing measure.

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8

Groemer, H. "Some basic properties of packing and covering constants." Discrete & Computational Geometry 1, no. 2 (June 1986): 183–93. http://dx.doi.org/10.1007/bf02187693.

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9

Böröczky, Károly. "Finite packing and covering by congruent convex domains." Discrete & Computational Geometry 30, no. 2 (July 10, 2003): 185–93. http://dx.doi.org/10.1007/s00454-003-0005-8.

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10

Schurmann, Achill, and Frank Vallentin. "Computational Approaches to Lattice Packing and Covering Problems." Discrete & Computational Geometry 35, no. 1 (October 12, 2005): 73–116. http://dx.doi.org/10.1007/s00454-005-1202-2.

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11

Chepoi, Victor, Bertrand Estellon, and Guyslain Naves. "Packing and Covering with Balls on Busemann Surfaces." Discrete & Computational Geometry 57, no. 4 (March 6, 2017): 985–1011. http://dx.doi.org/10.1007/s00454-017-9872-0.

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12

Januszewski, Janusz. "On-line packing and covering a disk with disks." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 52, no. 2 (May 1, 2011): 305–14. http://dx.doi.org/10.1007/s13366-011-0039-5.

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13

Smith, Edwin. "An Improvement of an Inequality Linking Packing and Covering Densities in 3-space." Geometriae Dedicata 117, no. 1 (March 29, 2006): 11–18. http://dx.doi.org/10.1007/s10711-005-9005-4.

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14

Ismailescu, D. "Inequalities between lattice packing and covering densities of centrally symmetric plane convex bodies." Discrete & Computational Geometry 25, no. 3 (April 2001): 365–88. http://dx.doi.org/10.1007/s004540010068.

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15

van Driel, W. D., G. Q. Zhang, J. H. J. Janssen, and L. J. Ernst. "Response Surface Modeling for Nonlinear Packaging Stresses." Journal of Electronic Packaging 125, no. 4 (December 1, 2003): 490–97. http://dx.doi.org/10.1115/1.1604149.

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The present study focuses on the development of reliable response surface models (RSM’s) for the major packaging processes of a typical electronic package. The major objective is to optimize the product/process designs against the possible failure mode of vertical die cracks. First, the finite element mode (FEM)-based physics of failure models are developed and the reliability of the predicted stress levels was verified by experiments. In the development of reliable thermo-mechanical simulation models, both the process (time and temperature) dependent material nonlinearity and geometric nonlinearity are taken into account. Afterwards, RSM’s covering the whole specified geometric design spaces are constructed. Finally, these RSM’s are used to predict, evaluate, optimize, and eventually qualify the thermo-mechanical behavior of this electronic package against the actual design requirements prior to major physical prototyping and manufacturing investments.

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16

Smith, E. H. "A Bound on the Ratio between the Packing and Covering Densities of a Convex Body." Discrete & Computational Geometry 23, no. 3 (March 2000): 325–31. http://dx.doi.org/10.1007/pl00009503.

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17

Hladky, Jozef, Michael Stengel, Nicholas Vining, Bernhard Kerbl, Hans-Peter Seidel, and Markus Steinberger. "QuadStream." ACM Transactions on Graphics 41, no. 6 (November 30, 2022): 1–13. http://dx.doi.org/10.1145/3550454.3555524.

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Streaming rendered 3D content over a network to a thin client device, such as a phone or a VR/AR headset, brings high-fidelity graphics to platforms where it would not normally possible due to thermal, power, or cost constraints. Streamed 3D content must be transmitted with a representation that is both robust to latency and potential network dropouts. Transmitting a video stream and reprojecting to correct for changing viewpoints fails in the presence of disocclusion events; streaming scene geometry and performing high-quality rendering on the client is not possible on limited-power mobile GPUs. To balance the competing goals of disocclusion robustness and minimal client workload, we introduce QuadStream , a new streaming content representation that reduces motion-to-photon latency by allowing clients to efficiently render novel views without artifacts caused by disocclusion events. Motivated by traditional macroblock approaches to video codec design, we decompose the scene seen from positions in a view cell into a series of quad proxies , or view-aligned quads from multiple views. By operating on a rasterized G-Buffer, our approach is independent of the representation used for the scene itself; the resulting QuadStream is an approximate geometric representation of the scene that can be reconstructed by a thin client to render both the current view and nearby adjacent views. Our technical contributions are an efficient parallel quad generation, merging, and packing strategy for proxy views covering potential client movement in a scene; a packing and encoding strategy that allows masked quads with depth information to be transmitted as a frame-coherent stream; and an efficient rendering approach for rendering our QuadStream representation into entirely novel views on thin clients. We show that our approach achieves superior quality compared both to video data streaming methods, and to geometry-based streaming.

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18

Marx, Dániel, and Michał Pilipczuk. "Optimal Parameterized Algorithms for Planar Facility Location Problems Using Voronoi Diagrams." ACM Transactions on Algorithms 18, no. 2 (April 30, 2022): 1–64. http://dx.doi.org/10.1145/3483425.

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We study a general family of facility location problems defined on planar graphs and on the two-dimensional plane. In these problems, a subset of k objects has to be selected, satisfying certain packing (disjointness) and covering constraints. Our main result is showing that, for each of these problems, the n O (√ k ) time brute force algorithm of selecting k objects can be improved to n O (√ k ) time. The algorithm is based on an idea that was introduced recently in the design of geometric QPTASs, but was not yet used for exact algorithms and for planar graphs. We focus on the Voronoi diagram of a hypothetical solution of k objects, guess a balanced separator cycle of this Voronoi diagram to obtain a set that separates the solution in a balanced way, and then recurse on the resulting subproblems. The following list is an exemplary selection of concrete consequences of our main result. We can solve each of the following problems in time n O (√ k ), where n is the total size of the input: d -Scattered Set : find k vertices in an edge-weighted planar graph that pairwise are at distance at least d from each other ( d is part of the input). d -Dominating Set (or ( k,d )-Center): find k vertices in an edge-weighted planar graph such that every vertex of the graph is at distance at most d from at least one selected vertex ( d is part of the input). Given a set D of connected vertex sets in a planar graph G , find k disjoint vertex sets in D . Given a set D of disks in the plane (of possibly different radii), find k disjoint disks in D . Given a set D of simple polygons in the plane, find k disjoint polygons in D . Given a set D of disks in the plane (of possibly different radii) and a set P of points, find k disks in D that together cover the maximum number of points in P . Given a set D of axis-parallel squares in the plane (of possibly different sizes) and a set P of points, find k squares in D that together cover the maximum number of points in P . It is known from previous work that, assuming the Exponential Time Hypothesis (ETH), there is no f ( k ) n o (√ k ) time algorithm for any computable function f for any of these problems. Furthermore, we give evidence that packing problems have n O (√ k ) time algorithms for a much more general class of objects than covering problems have. For example, we show that, assuming ETH, the problem where a set D of axis-parallel rectangles and a set P of points are given, and the task is to select k rectangles that together cover the entire point set, does not admit an f ( k ) n o ( k ) time algorithm for any computable function f .

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19

Shragai, Nadav, and Gershon Elber. "Geometric covering." Computer-Aided Design 45, no. 2 (February 2013): 243–51. http://dx.doi.org/10.1016/j.cad.2012.10.007.

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20

Ghosh, S. K., and P. E. Haxell. "Packing and covering tetrahedra." Discrete Applied Mathematics 161, no. 9 (June 2013): 1209–15. http://dx.doi.org/10.1016/j.dam.2010.05.027.

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21

Strauss, Rouven, Florin Isvoranu, and Gershon Elber. "Geometric multi-covering." Computers & Graphics 38 (February 2014): 222–29. http://dx.doi.org/10.1016/j.cag.2013.10.018.

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22

Ene, Alina, Sariel Har-Peled, and Benjamin Raichel. "Geometric Packing under Nonuniform Constraints." SIAM Journal on Computing 46, no. 6 (January 2017): 1745–84. http://dx.doi.org/10.1137/120898413.

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23

Kwon, O.-joung, and Jean-Florent Raymond. "Packing and Covering Induced Subdivisions." SIAM Journal on Discrete Mathematics 35, no. 2 (January 2021): 597–636. http://dx.doi.org/10.1137/18m1226166.

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24

Cohen, G., I. Honkala, S. Litsyn, and P. Sole. "Long packing and covering codes." IEEE Transactions on Information Theory 43, no. 5 (1997): 1617–19. http://dx.doi.org/10.1109/18.623161.

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25

McDonald, Jessica, Gregory J. Puleo, and Craig Tennenhouse. "Packing and Covering Directed Triangles." Graphs and Combinatorics 36, no. 4 (April 11, 2020): 1059–63. http://dx.doi.org/10.1007/s00373-020-02167-8.

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26

Lonc, Zbigniew. "Majorization, packing, covering and matroids." Discrete Mathematics 121, no. 1-3 (October 1993): 151–57. http://dx.doi.org/10.1016/0012-365x(93)90548-8.

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27

Chang, Gerard J., and George L. Nemhauser. "Covering, Packing and Generalized Perfection." SIAM Journal on Algebraic Discrete Methods 6, no. 1 (January 1985): 109–32. http://dx.doi.org/10.1137/0606012.

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28

Chee, Yeow Meng, Charles J. Colbourn, Alan C. H. Ling, and Richard M. Wilson. "Covering and packing for pairs." Journal of Combinatorial Theory, Series A 120, no. 7 (September 2013): 1440–49. http://dx.doi.org/10.1016/j.jcta.2013.04.005.

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29

Alon, Noga, Yair Caro, and Raphael Yuster. "Packing and covering dense graphs." Journal of Combinatorial Designs 6, no. 6 (1998): 451–72. http://dx.doi.org/10.1002/(sici)1520-6610(1998)6:6<451::aid-jcd6>3.0.co;2-e.

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30

Hojny, Christopher. "Packing, partitioning, and covering symresacks." Discrete Applied Mathematics 283 (September 2020): 689–717. http://dx.doi.org/10.1016/j.dam.2020.03.002.

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31

Cranston, Daniel W., Jiaxi Nie, Jacques Verstraëte, and Alexandra Wesolek. "On asymptotic packing of geometric graphs." Discrete Applied Mathematics 322 (December 2022): 142–52. http://dx.doi.org/10.1016/j.dam.2022.07.030.

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32

Cooper, Jacob W., Andrzej Grzesik, Adam Kabela, and Daniel Král’. "Packing and covering directed triangles asymptotically." European Journal of Combinatorics 101 (March 2022): 103462. http://dx.doi.org/10.1016/j.ejc.2021.103462.

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33

Stein, Sherman. "Tiling, packing, and covering by clusters." Rocky Mountain Journal of Mathematics 16, no. 2 (June 1986): 277–322. http://dx.doi.org/10.1216/rmj-1986-16-2-277.

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34

Conway, J. H., and S. Torquato. "Packing, tiling, and covering with tetrahedra." Proceedings of the National Academy of Sciences 103, no. 28 (July 3, 2006): 10612–17. http://dx.doi.org/10.1073/pnas.0601389103.

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35

Gai, Ling, and Guochuan Zhang. "Hardness of lazy packing and covering." Operations Research Letters 37, no. 2 (March 2009): 89–92. http://dx.doi.org/10.1016/j.orl.2008.12.007.

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36

Rödl, Vojtěch. "On a Packing and Covering Problem." European Journal of Combinatorics 6, no. 1 (March 1985): 69–78. http://dx.doi.org/10.1016/s0195-6698(85)80023-8.

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37

Yin, Jianxing. "Cyclic Difference Packing and Covering Arrays." Designs, Codes and Cryptography 37, no. 2 (November 2005): 281–92. http://dx.doi.org/10.1007/s10623-004-3991-3.

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38

Graham, Niall, and Frank Harary. "Covering and packing in graphs—V." Computers & Mathematics with Applications 15, no. 4 (1988): 267–70. http://dx.doi.org/10.1016/0898-1221(88)90211-8.

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39

Chen, Jianer, Henning Fernau, Peter Shaw, Jianxin Wang, and Zhibiao Yang. "Kernels for packing and covering problems." Theoretical Computer Science 790 (October 2019): 152–66. http://dx.doi.org/10.1016/j.tcs.2019.04.018.

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40

Jana, Satyabrata, and Supantha Pandit. "Covering and packing of rectilinear subdivision." Theoretical Computer Science 840 (November 2020): 166–76. http://dx.doi.org/10.1016/j.tcs.2020.07.038.

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41

Haxell, P. E. "Packing and covering triangles in graphs." Discrete Mathematics 195, no. 1-3 (January 1999): 251–54. http://dx.doi.org/10.1016/s0012-365x(98)00183-6.

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42

Björklund, Andreas, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. "Covering and packing in linear space." Information Processing Letters 111, no. 21-22 (November 2011): 1033–36. http://dx.doi.org/10.1016/j.ipl.2011.08.002.

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43

Fan, Genghua, Hongbi Jiang, Ping Li, Douglas B. West, Daqing Yang, and Xuding Zhu. "Extensions of matroid covering and packing." European Journal of Combinatorics 76 (February 2019): 117–22. http://dx.doi.org/10.1016/j.ejc.2018.09.010.

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44

Jungnickel, D., and L. Storme. "Packing and Covering Groups with Subgroups." Journal of Algebra 239, no. 1 (May 2001): 191–214. http://dx.doi.org/10.1006/jabr.2000.8640.

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45

Bollauf, Maiara F., Ram Zamir, and Sueli I. R. Costa. "Multilevel Constructions: Coding, Packing and Geometric Uniformity." IEEE Transactions on Information Theory 65, no. 12 (December 2019): 7669–81. http://dx.doi.org/10.1109/tit.2019.2933219.

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46

Paiva, Aureliano Sancho S., Rafael S. Oliveira, and Roberto F. S. Andrade. "Two-phase fluid flow in geometric packing." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, no. 2056 (December 13, 2015): 20150111. http://dx.doi.org/10.1098/rsta.2015.0111.

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We investigate how a plug of obstacles inside a two-dimensional channel affects the drainage of high viscous fluid (oil) when the channel is invaded by a less viscous fluid (water). The plug consists of an Apollonian packing with, at most, 17 circles of different sizes, which is intended to model an inhomogeneous porous region. The work aims to quantify the amount of retained oil in the region where the flow is influenced by the packing. The investigation, carried out with the help of the computational fluid dynamics package ANSYS-FLUENT , is based on the integration of the complete set of equations of motion. The study considers the effect of both the injection speed and the number and size of obstacles, which directly affects the porosity of the system. The results indicate a complex dependence in the fraction of retained oil on the velocity and geometric parameters. The regions where the oil remains trapped is very sensitive to the number of circles and their size, which influence in different ways the porosity of the system. Nevertheless, at low values of Reynolds and capillary numbers Re <4 and n c ≃10 −5 , the overall expected result that the volume fraction of oil retained decreases with increasing porosity is recovered. A direct relationship between the injection speed and the fraction of oil is also obtained.

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47

Huang, Aiping, and William Zhu. "Geometric Lattice Structure of Covering-Based Rough Sets through Matroids." Journal of Applied Mathematics 2012 (2012): 1–25. http://dx.doi.org/10.1155/2012/236307.

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Covering-based rough set theory is a useful tool to deal with inexact, uncertain, or vague knowledge in information systems. Geometric lattice has been widely used in diverse fields, especially search algorithm design, which plays an important role in covering reductions. In this paper, we construct three geometric lattice structures of covering-based rough sets through matroids and study the relationship among them. First, a geometric lattice structure of covering-based rough sets is established through the transversal matroid induced by a covering. Then its characteristics, such as atoms, modular elements, and modular pairs, are studied. We also construct a one-to-one correspondence between this type of geometric lattices and transversal matroids in the context of covering-based rough sets. Second, we present three sufficient and necessary conditions for two types of covering upper approximation operators to be closure operators of matroids. We also represent two types of matroids through closure axioms and then obtain two geometric lattice structures of covering-based rough sets. Third, we study the relationship among these three geometric lattice structures. Some core concepts such as reducible elements in covering-based rough sets are investigated with geometric lattices. In a word, this work points out an interesting view, namely, geometric lattice, to study covering-based rough sets.

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48

Xu, Jie, Tao Wu, Jianwei Zhang, Hao Chen, Wei Sun, and Chuang Peng. "Microstructure Measurement and Microgeometric Packing Characterization of Rigid Polyurethane Foam Defects." Cellular Polymers 36, no. 4 (July 2017): 183–204. http://dx.doi.org/10.1177/026248931703600402.

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Streak and blister cell defects pose extensive surface problems for rigid polyurethane foams. In this study, these morphological anomalies were visually inspected using 2D optical techniques, and the cell microstructural coefficients including degree of anisotropy cell circumdiameter, and the volumetric isoperimetric quotient were calculated from the observations. A geometric regular polyhedron approximation method was developed based on relative density equations, in order to characterize the packing structures of both normal and anomalous cells. The reversely calculated cell volume constant, Cc, from polyhedron geometric voxels was compared with the empirical polyhedron cell volume value, Ch. The geometric relationship between actual cells and approximated polyhedrons was characterized by the defined volumetric isoperimetric quotient. Binary packing structures were derived from deviation comparisons between the two cell volume constants, and the assumed partial relative density ratios of the two individual packing polyhedrons. The modelling results show that normal cells have a similar packing to the Weaire-Phelan model, while anomalous cells have a dodecahedron/icosidodecahedron binary packing.

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49

Godbole, Anant, Thomas Grubb, Kyutae Han, and Bill Kay. "Threshold progressions in covering and packing contexts." Journal of Combinatorics 13, no. 3 (2022): 303–31. http://dx.doi.org/10.4310/joc.2022.v13.n3.a1.

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50

Sole, P. "Packing radius, covering radius, and dual distance." IEEE Transactions on Information Theory 41, no. 1 (1995): 268–72. http://dx.doi.org/10.1109/18.370102.

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