Combining Polygon Schematization and Decomposition Approaches for Solving the Cavity Decomposition Problem
Authors: 
Serafino Cicerone, Mattia D’emidio, Daniele Frigioni, Filippo Tirabassi PascucciAuthors Info & Claims
Serafino Cicerone, Mattia D’emidio, Daniele Frigioni, Filippo Tirabassi PascucciAuthors Info & ClaimsArticle No.: 22, Pages 1 - 37
Abstract
The cavity decomposition problem is a computational geometry problem, arising in the context of modern electronic CAD systems, that concerns detecting the generation and propagation of electromagnetic noise into multi-layer printed circuit boards. Algorithmically speaking, the problem can be formulated so as to contain, as sub-problems, the well-known polygon schematization and polygon decomposition problems. Given a polygon P and a finite set C of given directions, polygon schematization asks for computing a C-oriented polygon P′ with “low complexity” and “high resemblance” to P, whereas polygon decomposition asks for partitioning P into a set of basic polygonal elements (e.g., triangles) whose size is as small as possible.
In this article, we present three different solutions for the cavity decomposition problem, which are obtained by suitably combining existing algorithms for polygon schematization and decomposition, by considering different input parameters, and by addressing both methodological and implementation issues. Since it is difficult to compare the three solutions on a theoretical basis, we present an extensive experimental study, employing both real-world and random data, conducted to assess their performance. We rank the proposed solutions according to the results of the experimental evaluation, and provide insights on natural candidates to be adopted, in practice, as modules of modern printed circuit board design software tools, depending on the observed performance and on the different constraints on the desired output.
References
[1]
Pankaj K. Agarwal and Micha Sharir. 2000. Arrangements and their applications. In Handbook of Computational Geometry. Elsevier, 49–119.
[2]
Pierre Alliez and Andreas Fabri. 2016. CGAL: The computational geometry algorithms library. In Proceedings of the Special Interest Group on Computer Graphics and Interactive Techniques Conference (SIGGRAPH’16). ACM, New York, NY, Article 8, 8 pages.
[3]
Quirijn W. Bouts, Irina Kostitsyna, Marc J. van Kreveld, Wouter Meulemans, Willem Sonke, and Kevin Verbeek. 2016. Mapping polygons to the grid with small Hausdorff and Fréchet distance. In Proceedings of the 24th Annual European Symposium on Algorithms (ESA’16). Article 22, 16 pages.
[4]
Kevin Buchin, Wouter Meulemans, André van Renssen, and Bettina Speckmann. 2016. Area-preserving simplification and schematization of polygonal subdivisions. ACM Transactions on Spatial Algorithms and Systems 2, 1 (2016), Article 2, 36 pages.
[5]
Xiuzhen Cheng, Ding-Zhu Du, Joon-Mo Kim, and Lu Ruan. 2005. Optimal rectangular partitions. In Handbook of Combinatorial Optimization. Springer, 313–327.
[6]
Serafino Cicerone and Matteo Cermignani. 2012. Fast and simple approach for polygon schematization. In Computational Science and Its Applications. Lecture Notes in Computer Science, Vol. 7333. Springer, 267–279.
[7]
Serafino Cicerone and Gabriele Di Stefano. 2014. Decomposing octilinear polygons into triangles and rectangles. In Discrete and Computational Geometry and Graphs. Lecture Notes in Computer Science, Vol. 8845. Springer, 18–30.
[8]
Serafino Cicerone, Antonio Orlandi, Bruce Archambeault, Samuel Connor, Jun Fan, and James L. Drewniak. 2009. Cavities’ identification algorithm for power integrity analysis of complex boards. In Proceedings of the 20th International Zurich Symposium on Electromagnetic Compatibility (EMC-Zurich’09). IEEE, Los Alamitos, CA, 253–256.
[9]
Serafino Cicerone and Gabriele Di Stefano. 2019. Approximation algorithms for decomposing octilinear polygons. Theoretical Computer Science 779 (2019), 17–36. https://doi.org/10.1016/j.tcs.2019.01.037
[10]
Daniel Delling, Andreas Gemsa, Martin Nöllenburg, and Thomas Pajor. 2010. Path schematization for route sketches. In Algorithm Theory—SWAT 2010. Lecture Notes in Computer Science, Vol. 6139. Springer, 285–296.
[11]
David H. Douglas and Thomas K. Peucker. 1973. Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Canadian Geographer 10, 2 (1973), 112–122.
[12]
Ding-Zhu Du and Yanjun Zhang. 1990. On heuristics for minimum length rectilinear partitions. Algorithmica 5, 1 (1990), 111–128.
[13]
Stephane Durocher and Saeed Mehrabi. 2012. Computing partitions of rectilinear polygons with minimum stabbing number. In Computing and Combinatorics. Lecture Notes in Computer Science, Vol. 7434. Springer, 228–239.
[14]
David Eppstein. 2010. Graph-theoretic solutions to computational geometry problems. In Graph-Theoretic Concepts in Computer Science. Springer, 1–16.
[15]
Leonard A. Ferrari, Pathamadi V. Sankar, and Jack Sklansky. 1984. Minimal rectangular partitions of digitized blobs. Computer Vision, Graphics, and Image Processing 28, 1 (1984), 58–71.
[16]
John E. Hopcroft and Richard M. Karp. 1973. An algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing 2, 4 (1973), 225–231. https://doi.org/10.1137/0202019
[17]
Hiroshi Imai and Takao Asano. 1986. Efficient algorithms for geometric graph search problems. SIAM Journal on Computing 15, 2 (1986), 478–494.
[18]
Hiroshi Imai and Masao Iri. 1986. Computational-geometric methods for polygonal approximations of a curve. Computer Vision, Graphics, and Image Processing 36, 1 (1986), 31–41.
[19]
J. Mark Keil. 2000. Polygon decomposition. In Handbook on Computational Geometry, J. R. Sack and J. Urrutia (Eds.). Elsevier Science, 491–518.
[20]
Y. Kurozumi and W. A. Davis. 1982. Polygonal approximation by the minimax method. Computer Graphics and Image Processing 19 (1982), 248–264.
[21]
Guang-Tsai Lei, Robert W. Techentin, and Barry K. Gilbert. 1999. High-frequency characterization of power/ground plane structures. IEEE Transactions on Microwave Theory and Techniques 47 (1999), 562–569.
[22]
W. T. Liou, J. J. Tan, and R. C. Lee. 1989. Minimum partitioning simple rectilinear polygons in -time. In Proceedings of the 5th Annual Symposium on Computational Geometry (SCG’89). ACM, New York, NY, 344–353.
[23]
Witold Lipski. 1983. Finding a Manhattan path and related problems. Networks 13, 2 (1983), 399–409.
[24]
Witold Lipski. 1984. An Manhattan path algorithm. Information Processing Letters 19, 2 (1984), 99–102.
[25]
Maarten Löffler and Wouter Meulemans. 2017. Discretized approaches to schematization. In Proceedings of the 29th Canadian Conference on Computational Geometry (CCCG’17). 1–6.
[26]
Sudipta Maity and Bhaskar Gupta. 2015. Cavity model analysis of 30-60-90 triangular microstrip antenna. AEU: International Journal of Electronics and Communications 69, 6 (2015), 923–932. https://doi.org/10.1016/j.aeue.2015.02.012
[27]
Catherine C. McGeoch. 2012. A Guide to Experimental Algorithmics. Cambridge University Press.
[28]
Damian Merrick and Joachim Gudmundsson. 2006. Path simplification for metro map layout. In Graph Drawing. Lecture Notes in Computer Science, Vol. 4372. Springer, 258–269.
[29]
Wouter Meulemans. 2016. Discretized approaches to schematization. arXiv:1606.06488. http://arxiv.org/abs/1606.06488.
[30]
Nanju Na, Jinseong Choi, Sungjun Chun, Madhavan Swaminathan, and Jegannathan Srinivasan. 2000. Modeling and transient simulation of planes in electronic packages. IEEE Transactions on Advanced Packaging 23, 3 (2000), 340–352.
[31]
Gabriele Neyer. 1999. Line simplification with restricted orientations. In Algorithms and Data Structures. Lecture Notes in Computer Science, Vol. 1663. Springer, 13–24.
[32]
Soeren Nickel and Martin Nöllenburg. 2020. Towards data-driven multilinear metro maps. In Diagrammatic Representation and Inference, Ahti-Veikko Pietarinen, Peter Chapman, Leonie Bosveld-de Smet, Valeria Giardino, James Corter, and Sven Linker (Eds.). Springer International, Cham, Switzerland, 153–161.
[33]
M. Nollenburg and A. Wolff. 2011. Drawing and labeling high-quality metro maps by mixed-integer programming. IEEE Transactions on Visualization and Computer Graphics 17, 5 (2011), 626–641.
[34]
Tatsuo Ohtsuki. 1982. Minimum dissection of rectilinear regions. In Proceedings of the IEEE International Symposium on Circuits and Systems.
[35]
T. Okoshi. 1985. Planar Circuits for Microwaves and Lightwaves. Springer-Verlag. https://doi.org/10.1007/978-3-642-70083-5
[36]
Jeremy G. Siek, Lie-Quan Lee, and Andrew Lumsdaine. 2002. The Boost Graph Library: User Guide and Reference Manual. Pearson/Prentice Hall.
[37]
Tomás Suk, Cyril Höschl IV, and Jan Flusser. 2012. Decomposition of binary images—A survey and comparison. Pattern Recognition 45, 12 (2012), 4279–4291.
[38]
Madhavan Swaminathan, Kim Joungho, Istvan Novak, and James P. Libous. 2004. Power distribution networks for system-on-package: Status and challenges. IEEE Transactions on Advanced Packaging 27, 2 (2004), 286–300.
[39]
Jerry Swan, Suchith Anand, J. Mark Ware, and Mike Jackson. 2007. Automated schematization for web service applications. In Web and Wireless Geographical Information Systems. Lecture Notes in Computer Science, Vol. 4857. Springer, 216–226.
[40]
Dražen Tutić. 2009. Area preserving cartographic line generalization. Kartografija i Geoinformacije 8 (June 2009), 85–100.
[41]
M. Visvalingam and J. D. Whyatt. 1993. Line generalisation by repeated elimination of points. Cartographic Journal 30, 1 (1993), 46–51. https://doi.org/10.1179/000870493786962263
Index Terms
- Combining Polygon Schematization and Decomposition Approaches for Solving the Cavity Decomposition Problem
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Publication History
Published: 16 August 2021
Accepted: 01 April 2021
Revised: 01 March 2021
Received: 01 September 2019
Published in TSAS Volume 7, Issue 4
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