Minimum Rectilinear Polygons for Given Angle Sequences

  • William S. Evans
  • Krzysztof Fleszar
  • Philipp Kindermann
  • Noushin Saeedi
  • Chan-Su Shin
  • Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9943)


A rectilinear polygon is a polygon whose edges are axis-aligned. Walking counterclockwise on the boundary of such a polygon yields a sequence of left turns and right turns. The number of left turns always equals the number of right turns plus 4. It is known that any such sequence can be realized by a rectilinear polygon. In this paper, we consider the problem of finding realizations that minimize the perimeter or the area of the polygon or the area of the bounding box of the polygon. We show that all three problems are NP-hard in general. Then we consider the special cases of x-monotone and xy-monotone rectilinear polygons. For these, we can optimize the three objectives efficiently.


Vertical Edge Full Version Simple Polygon Horizontal Edge Minimum Height 
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  1. 1.
    Bae, S.W., Okamoto, Y., Shin, C.-S.: Area bounds of rectilinear polygons realized by angle sequences. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 629–638. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-35261-4_65 CrossRefGoogle Scholar
  2. 2.
    Biedl, T.C., Durocher, S., Snoeyink, J.: Reconstructing polygons from scanner data. Theor. Comput. Sci. 412(32), 4161–4172 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, D.Z., Wang, H.: An improved algorithm for reconstructing a simple polygon from its visibility angles. Comput. Geom. 45(5–6), 254–257 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Culberson, J.C., Rawlins, G.J.E.: Turtlegons: generating simple polygons from sequences of angles. In: Proceedings of 1st Annual ACM Symposium on Computational Geometry (SoCG 1985), pp. 305–310 (1985)Google Scholar
  5. 5.
    Disser, Y., Mihalák, M., Widmayer, P.: A polygon is determined by its angles. Comput. Geom. 44(8), 418–426 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Evans, W.S., Fleszar, K., Kindermann, P., Saeedi, N., Shin, C.S., Wolff, A.: Minimum rectilinear polygons for given angle sequences. Arxiv report (2016).
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman & Co., New York (1979)zbMATHGoogle Scholar
  8. 8.
    Hartley, R.I.: Drawing polygons given angle sequences. Inform. Process. Lett. 31(1), 31–33 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Patrignani, M.: On the complexity of orthogonal compaction. Comput. Geom. 19(1), 47–67 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Sack, J.R.: Rectilinear computational geometry. Ph.D. thesis, School of Computer Science, McGill University (1984).
  11. 11.
    Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Vijayan, G., Wigderson, A.: Rectilinear graphs and their embeddings. SIAM J. Comput. 14(2), 355–372 (1985)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • William S. Evans
    • 1
  • Krzysztof Fleszar
    • 2
  • Philipp Kindermann
    • 2
    • 3
  • Noushin Saeedi
    • 1
  • Chan-Su Shin
    • 4
  • Alexander Wolff
    • 2
  1. 1.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada
  2. 2.Lehrstuhl für Informatik IUniversität WürzburgWürzburgGermany
  3. 3.LG Theoretische InformatikFernUniversität in HagenHagenGermany
  4. 4.Division of Computer and Electronic SystemsHankuk University of Foreign StudiesYonginSouth Korea

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