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Connecting Polygonizations via Stretches and Twangs

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Abstract

We show that the space of polygonizations of a fixed planar point set S of n points is connected by O(n 2) “moves” between simple polygons. Each move is composed of a sequence of atomic moves called “stretches” and “twangs,” which walk between weakly simple “polygonal wraps” of S. These moves show promise to serve as a basis for generating random polygons.

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References

  1. Auer, T., Held, M.: Heuristics for the generation of random polygons. In: Proc. 8th Canad. Conf. Comput. Geom., pp. 38–43 (1996)

  2. Bao, D.D.-W., Bao, D.: Sampler of Riemann-Finsler Geometry. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  3. Bose, P., Hurtado, F.: Flips in planar graphs. Comput. Geom. Theory. Appl. 42, 60–81 (2009)

    MATH  MathSciNet  Google Scholar 

  4. Czyzowicz, J., Hurtado, F., Urrutia, J., Zaguia, N.: On polygons enclosing point sets. Geombinatorics XI, 21–28 (2001)

    MathSciNet  Google Scholar 

  5. Damian, M., Flatland, R., O’Rourke, J., Ramaswami, S.: Connecting polygonizations via stretches and twangs. In: Proc. 25th Sympos. Theoretical Aspects Comput. Sci. (STACS), IBFI Schloss Dagstuhl, pp. 217–228 (2008)

  6. García, A., Noy, M., Tejel, J.: Lower bounds on the number of crossing-free subgraphs of K N . Comput. Geom. Theory Appl. 16, 211–221 (2000)

    MATH  Google Scholar 

  7. Hernando, C., Hurtado, F., Houle, M.: On local transformation of polygons with visibility properties. Theor. Comput. Sci. 289, 919–937 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kannan, R.: Markov chains and polynomial time algorithms. In: 35th IEEE Sympos. Foundations Comput. Sci (FOCS), pp. 656–671 (1994)

  9. Lovász, L.: Random walks on graphs: A survey. In: Combinatorics, Paul Erdös is Eighty. Bolyai Society Math. Studies, vol. 2, pp. 353–398 (1993)

  10. Molloy, M., Reed, B., Steiger, W.: On the mixing rate of the triangulation walk. In: DIMACS Series in Disc. Math. and Theoret. Comput. Sci., vol. 43, pp. 179–190 (1999)

  11. Randall, D.: Rapidly mixing Markov chains with applications in computer science and physics. Comput. Sci. Eng. 8(2), 30–41 (2006)

    Article  Google Scholar 

  12. Sinclair, A.: Algorithms for Random Generation and Counting: A Markov Chain Approach. Birkhäuser, Basel (1993)

    MATH  Google Scholar 

  13. van Leeuwen, J., Schoone, A.A.: Untangling a travelling salesman tour in the plane. In: Mühlbacher, J.R. (ed.) Proc. 7th Internat. Workshop Graph-Theoret. Concepts Comput. Sci., pp. 87–98. Hanser, München (1982)

    Google Scholar 

  14. Zhu, C., Sundaram, G., Snoeyink, J., Mitchell, J.S.B.: Generating random polygons with given vertices. Comput. Geom. Theory Appl. 6, 277–290 (1996)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dept. of Computer Science, Villanova Univ., Villanova, PA, 19085, USA

    Mirela Damian

  2. Dept. of Computer Science, Siena College, Loudonville, NY, 12211, USA

    Robin Flatland

  3. Dept. of Computer Science, Smith College, Northampton, MA, 01063, USA

    Joseph O’Rourke

  4. Dept. of Computer Science, Rutgers University, Camden, NJ, 08102, USA

    Suneeta Ramaswami

Authors
  1. Mirela Damian
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  2. Robin Flatland
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  3. Joseph O’Rourke
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  4. Suneeta Ramaswami
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Correspondence to Joseph O’Rourke.

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Damian, M., Flatland, R., O’Rourke, J. et al. Connecting Polygonizations via Stretches and Twangs. Theory Comput Syst 47, 674–695 (2010). https://doi.org/10.1007/s00224-009-9192-8

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  • DOI: https://doi.org/10.1007/s00224-009-9192-8

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