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Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm

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Abstract

We show that every orthogonal polyhedron homeomorphic to a sphere can be unfolded without overlap while using only polynomially many (orthogonal) cuts. By contrast, the best previous such result used exponentially many cuts. More precisely, given an orthogonal polyhedron with n vertices, the algorithm cuts the polyhedron only where it is met by the grid of coordinate planes passing through the vertices, together with Θ(n 2) additional coordinate planes between every two such grid planes.

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References

  1. Abel, Z., Demaine, E.D.: Edge-unfolding orthogonal polyhedra is strongly np-complete. In: Proceedings of the 23rd Canadian Conference on Computational Geometry (2011)

  2. Aloupis, G., Bose, P.K., Collette, S., Demaine, E.D., Demaine, M.L., Douieb, K., Dujmović, V., Iacono, J., Langerman, S., Morin, P.: Common unfoldings of polyominoes and polycube. In: Revised Papers from the China-Japan Joint Conference on Computational Geometry, Graphs and Applications. Lecture Notes in Computer Science, vol 7033, pp 44–54 (2010)

  3. Bern M., Demaine E.D., Eppstein D., Kuo E., Mantler A., Snoeyink J.: Ununfoldable polyhedra with convex faces. Comput. Geom. Theory Appl. 24(2), 51–62 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Biedl,~T., Demaine,~E., Demaine, M., Lubiw,~A., Overmars,~M., O’Rourke,~J., Robbins,~S., Whitesides,~S.: Unfolding some classes of orthogonal polyhedra. In: Proceedings of the 10th Canadian Conference on Computational Geometry (1998)

  5. Damian, M., Flatland, R., Meijer, H., O’Rourke, J.: Unfolding well-separated orthotrees. In: Abstracts from the 15th Annual Fall Workshop on Computational Geometry (2005)

  6. Damian M., Flatland R., O’Rourke J.: Epsilon-unfolding orthogonal polyhedra. Graphs Comb. 23(1), 179–194 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Damian M., Flatland R., O’Rourke J.: Unfolding Manhattan towers. Comput. Geom. Theory Appl. 40, 102–114 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Damian, M., Meijer, H.: Edge-unfolding orthostacks with orthogonally convex slabs. In: Abstracts from the 14th Annual Fall Workshop on Computational Geometry, pp. 20–21 (2004). http://cgw2004.csail.mit.edu/talks/34.ps

  9. Demaine, E.D.., O’Rourke, J.: A survey of folding and unfolding in computational geometry. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Discrete and Computational Geometry, Mathematical Sciences Research Institute Publications, pp. 167–211. Cambridge University Press, Cambridge (2005)

  10. Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, Cambridge (2007)

  11. O’Rourke, J.: wblock Unfolding orthogonal polyhedra. In: Goodman, J.E., Pach, J., Pollack, R. (eds.) Surveys on Discrete and Computational Geometry: Twenty Years Later, pp. 231–255. American Mathematical Society, New York (2008)

  12. Sleator D.D., Tarjan R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 24(3), 362–391 (1983)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dept. of Computing Sciences, Villanova University, 800 Lancaster Avenue, Villanova, PA, 19085, USA

    Mirela Damian

  2. Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, 32 Vassar St, Cambridge, MA, 02139, USA

    Erik D. Demaine

  3. Department of Computer Science, Siena College, 515 Loudon Road, Loudonville, NY, 12211, USA

    Robin Flatland

Authors
  1. Mirela Damian
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  2. Erik D. Demaine
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  3. Robin Flatland
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Corresponding author

Correspondence to Robin Flatland.

Additional information

E. D. Demaine was partially supported by NSF CAREER award CCF-0347776.

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Damian, M., Demaine, E.D. & Flatland, R. Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm. Graphs and Combinatorics 30, 125–140 (2014). https://doi.org/10.1007/s00373-012-1257-9

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  • DOI: https://doi.org/10.1007/s00373-012-1257-9

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