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Decomposing Octilinear Polygons into Triangles and Rectangles

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Discrete and Computational Geometry and Graphs (JCDCGG 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8845))

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Abstract

In this paper we study the minimal decomposition of octilinear polygons with holes into octilinear triangles and rectangles. This new problem is relevant in the context of modern electronic CAD systems, where it arises when the generation and propagation of electromagnetic noise into multi-layer PCBs has to be detected. It can be seen as a generalization of a problem deeply investigated in the last decades: the minimal decomposition of rectilinear polygons into rectangles, which is solvable in polynomial time. We show that the new problem is NP-hard. We also show the NP-hardness of a related problem, that is the decomposition of an octilinear polygon with holes into octilinear convex polygons. For both problems, we propose efficient approximation algorithms.

Work supported by the Research Grant 2012C4E3KT “PRIN 2012” Amanda (Algorithmics for MAssive and Networked DAta) from the Italian Ministry of University and Research.

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Notes

  1. 1.

    The PCB consists of 16 layers and its \({\approx }\)13,000 polygons (i.e., cavities) have been extracted from a Cadence® Allegro® PCB designer project file. The polygons have been approximated into octilinear polygons by using the schematization algorithm proposed in [3]. Disregarding the polygons having the area below a given threshold, we get the final dataset of \({\approx }\)1,000 octilinear polygons with \({\approx }\)100,000 total vertices.

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Authors and Affiliations

  1. Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, L’Aquila, Italy

    Serafino Cicerone & Gabriele Di Stefano

Authors
  1. Serafino Cicerone
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  2. Gabriele Di Stefano
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Correspondence to Serafino Cicerone .

Editor information

Editors and Affiliations

  1. Tokyo University of Science, Tokyo, Japan

    Jin Akiyama

  2. The University of Electro-Communications, Tokyo, Japan

    Hiro Ito

  3. Tokai University, Tokyo, Japan

    Toshinori Sakai

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Cicerone, S., Di Stefano, G. (2014). Decomposing Octilinear Polygons into Triangles and Rectangles. In: Akiyama, J., Ito, H., Sakai, T. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2013. Lecture Notes in Computer Science(), vol 8845. Springer, Cham. https://doi.org/10.1007/978-3-319-13287-7_3

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  • DOI: https://doi.org/10.1007/978-3-319-13287-7_3

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  • Online ISBN: 978-3-319-13287-7

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