Serafino Cicerone
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.
- Pankaj K. Agarwal and Micha Sharir. 2000. Arrangements and their applications. In Handbook of Computational Geometry. Elsevier, 49–119.Google Scholar
- 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.Google ScholarDigital Library
- 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.Google Scholar
- 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.Google ScholarDigital Library
- Xiuzhen Cheng, Ding-Zhu Du, Joon-Mo Kim, and Lu Ruan. 2005. Optimal rectangular partitions. In Handbook of Combinatorial Optimization. Springer, 313–327.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.2019.01.037Google ScholarCross Ref
- 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.Google ScholarDigital Library
- 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.Google Scholar
- Ding-Zhu Du and Yanjun Zhang. 1990. On heuristics for minimum length rectilinear partitions. Algorithmica 5, 1 (1990), 111–128.Google ScholarCross Ref
- 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.Google Scholar
- David Eppstein. 2010. Graph-theoretic solutions to computational geometry problems. In Graph-Theoretic Concepts in Computer Science. Springer, 1–16.Google Scholar
- 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.Google ScholarCross Ref
- 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/10.1137/0202019Google ScholarCross Ref
- Hiroshi Imai and Takao Asano. 1986. Efficient algorithms for geometric graph search problems. SIAM Journal on Computing 15, 2 (1986), 478–494.Google ScholarDigital Library
- 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.Google ScholarDigital Library
- J. Mark Keil. 2000. Polygon decomposition. In Handbook on Computational Geometry, J. R. Sack and J. Urrutia (Eds.). Elsevier Science, 491–518.Google ScholarCross Ref
- Y. Kurozumi and W. A. Davis. 1982. Polygonal approximation by the minimax method. Computer Graphics and Image Processing 19 (1982), 248–264.Google ScholarCross Ref
- 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.Google ScholarCross Ref
- 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.Google Scholar
- Witold Lipski. 1983. Finding a Manhattan path and related problems. Networks 13, 2 (1983), 399–409.Google ScholarCross Ref
- Witold Lipski. 1984. An Manhattan path algorithm. Information Processing Letters 19, 2 (1984), 99–102.Google ScholarDigital Library
- Maarten Löffler and Wouter Meulemans. 2017. Discretized approaches to schematization. In Proceedings of the 29th Canadian Conference on Computational Geometry (CCCG’17). 1–6.Google Scholar
- 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.2015.02.012Google ScholarCross Ref
- Catherine C. McGeoch. 2012. A Guide to Experimental Algorithmics. Cambridge University Press. Google Scholar
- 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.Google Scholar
- Wouter Meulemans. 2016. Discretized approaches to schematization. arXiv:1606.06488. http://arxiv.org/abs/1606.06488.Google Scholar
- 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.Google ScholarCross Ref
- Gabriele Neyer. 1999. Line simplification with restricted orientations. In Algorithms and Data Structures. Lecture Notes in Computer Science, Vol. 1663. Springer, 13–24.Google ScholarDigital Library
- 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. Google Scholar
- 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.Google ScholarDigital Library
- Tatsuo Ohtsuki. 1982. Minimum dissection of rectilinear regions. In Proceedings of the IEEE International Symposium on Circuits and Systems.Google Scholar
- T. Okoshi. 1985. Planar Circuits for Microwaves and Lightwaves. Springer-Verlag. https://doi.org/10.1007/978-3-642-70083-5Google Scholar
- Jeremy G. Siek, Lie-Quan Lee, and Andrew Lumsdaine. 2002. The Boost Graph Library: User Guide and Reference Manual. Pearson/Prentice Hall.Google Scholar
- Tomás Suk, Cyril Höschl IV, and Jan Flusser. 2012. Decomposition of binary images—A survey and comparison. Pattern Recognition 45, 12 (2012), 4279–4291.Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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.Google Scholar
- Dražen Tutić. 2009. Area preserving cartographic line generalization. Kartografija i Geoinformacije 8 (June 2009), 85–100.Google Scholar
- M. Visvalingam and J. D. Whyatt. 1993. Line generalisation by repeated elimination of points. Cartographic Journal 30, 1 (1993), 46–51. https://doi.org/10.1179/000870493786962263Google ScholarCross Ref
Index Terms
- Combining Polygon Schematization and Decomposition Approaches for Solving the Cavity Decomposition Problem
Recommendations
Lower bounds for approximate polygon decomposition and minimum gap
We consider the problem of decomposing polygons (with holes) into various types of simpler polygons. We focus on the problem of partitioning a rectilinear polygon, with holes, into rectangles, and show an Ω (n log n) lower bound on the time-complexity. ...
Polygon decomposition for efficient construction of Minkowski sums
Special issue on: Sixteenth European Workshop on Computational Geometry (EUROCG-2000)Several algorithms for computing the Minkowski sum of two polygons in the plane begin by decomposing each polygon into convex subpolygons. We examine different methods for decomposing polygons by their suitability for efficient construction of Minkowski ...
On partitioning rectilinear polygons into star-shaped polygons
AbstractWe present an algorithm for finding optimum partitions of simple monotone rectilinear polygons into star-shaped polygons. The algorithm may introduce Steiner points and its time complexity isO(n), wheren is the number of vertices in the polygon. ...
Comments