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Computing the Fréchet Distance Between Polygons with Holes

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Automata, Languages, and Programming (ICALP 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9134))

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Abstract

We study the problem of computing the Fréchet distance between subsets of Euclidean space. Even though the problem has been studied extensively for 1-dimensional curves, very little is known for \(d\)-dimensional spaces, for any \(d\ge 2\). For general polygons in \(\mathbb {R}^2\), it has been shown to be NP-hard, and the best known polynomial-time algorithm works only for polygons with at most a single puncture [Buchin et al., 2010]. Generalizing [Buchin et al., 2008] we give a polynomial-time algorithm for the case of arbitrary polygons with a constant number of punctures. Moreover, we show that approximating the Fréchet distance between polyhedral domains in \(\mathbb {R}^3\) to within a factor of \(n^{1/\log \log n}\) is NP-hard.

A. Nayyeri—Part of this work was done while the author was a postdoctoral fellow at CMU. Research supported in part by the NSF grants CCF 1065106 and CCF 09-15519.

A. Sidiropoulos—Research supported in part by the NSF grants CCF 1423230 and CAREER 1453472.

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References

  1. Agarwal, P.K., Avraham, R.B, Kaplan, H., Sharir, M.: Computing the discrete Fréchet distance in subquadratic time. In: SODA 2013, pp. 156–167. SIAM (2013)

    Google Scholar 

  2. Alt, H., Buchin, M.: Semi-computability of the Fréchet distance between surfaces. In: EWCG 2005, Eindhoven, Netherlands, pp. 45–48

    Google Scholar 

  3. Alt, H., Godau, M.: Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geometry Appl. 5, 75–91 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Aronov, B., Har-Peled, S., Knauer, C., Wang, Y., Wenk, C.: Fréchet distance for curves, revisited. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 52–63. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  5. Buchin, K., Buchin, M., Schulz, A.: Fréchet distance of surfaces: some simple hard cases. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part II. LNCS, vol. 6347, pp. 63–74. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  6. Buchin, K., Buchin, M., Wenk, C.: Computing the Fréchet distance between simple polygons. Comp. Geom. Theo. Appl. 41(1–2), 2–20 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chambers, E.W., de Verdière, E.C., Erickson, J., Lazard, S., Lazarus, F., Thite, S.: Homotopic Fréchet distance between curves or, walking your dog in the woods in polynomial time. Comput. Geom. Theory Appl. 43(3), 295–311 (2010)

    Article  MATH  Google Scholar 

  8. Chazal, F., Lieutier, A., Rossignac, J., Whited, B.: Ball-map: Homeomorphism between compatible surfaces. Int. J. Comput. Geometry Appl. 20(3), 285–306 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, D., Driemel, A., Guibas, L.J., Nguyen, A., Wenk, C.: Approximate map matching with respect to the Fréchet distance. In: ALENEX 2011, pp. 75–83 (2011)

    Google Scholar 

  10. Cook IV, A.F., Driemel, A., Har-Peled, S., Sherette, J., Wenk, C.: Computing the Fréchet distance between folded polygons. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 267–278. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  11. Éric Colin de Verdière and Jeff Erickson: Tightening non-simple paths and cycles on surfaces. SIAM J. Comput. 39(8), 3784–3813 (2010)

    Article  MathSciNet  Google Scholar 

  12. Dey, T.K., Ranjan, P., Wang, Y.: Convergence, stability, and discrete approximation of laplace spectra. In: SODA 2010, pp. 650–663 (2010)

    Google Scholar 

  13. Dinur, I.: Approximating svp\(_{\text{ infinity }}\) to within almost-polynomial factors is np-hard. Theor. Comput. Sci. 285(1), 55–71 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Driemel, A., Har-Peled, S.: Jaywalking your dog: computing the Fréchet distance with shortcuts. In: SODA 2012, pp. 318–337. SIAM (2012)

    Google Scholar 

  15. Driemel, A., Har-Peled, S., Wenk, C.: Approximating the Fréchet distance for realistic curves in near linear time. Discrete & Computational Geometry 48(1), 94–127 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Erickson, J., Nayyeri, A.: Shortest non-crossing walks in the plane. In: SODA 2011, pp. 297–308. SIAM (2011)

    Google Scholar 

  17. Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling, Mathematics and Visualization, pp. 157–186. Springer, Heidelberg (2005)

    Google Scholar 

  18. Godau, M.: On the Complexity of Measuring the Similarity Between Geometric Objects in Higher Dimensions. Ph.D thesis, Freie Universität Berlin (1998)

    Google Scholar 

  19. Har-Peled, S., Nayyeri, A., Salavatipour, M., Sidiropoulos, A.: How to walk your dog in the mountains with no magic leash. In: SoCG 2012, pp. 121–130. ACM, New York (2012)

    Google Scholar 

  20. Hass, J., Scott, P.: Intersections of curves on surfaces. Israel Journal of Mathematics 51(1–2), 90–120 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hershberger, J., Snoeyink, J.: Computing minimum length paths of a given homotopy class. Comput. Geom. Theory Appl. 4(2), 63–97 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  22. Schaefer, M., Sedgwick, E., Štefankovič, D.: Spiraling and folding: The word view. Algorithmica (2009) (in press)

    Google Scholar 

  23. Schaefer, M., Štefankovič, D.: Decidability of string graphs. J. Comput. Syst. Sci. 68(2), 319–334 (2004)

    Article  MATH  Google Scholar 

  24. van Kaick, O., Zhang, H., Hamarneh, G., Cohen-Or, D.: A survey on shape correspondence. Computer Graphics Forum 30(6), 1681–1707 (2011)

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, USA

    Amir Nayyeri

  2. Department of Computer Science & Engineering and Department of Mathematics, The Ohio State University, Columbus, USA

    Anastasios Sidiropoulos

Authors
  1. Amir Nayyeri
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  2. Anastasios Sidiropoulos
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Correspondence to Anastasios Sidiropoulos .

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Editors and Affiliations

  1. Reykjavik University, Reykjavik, Iceland

    Magnús M. Halldórsson

  2. Kyoto University, Kyoto, Japan

    Kazuo Iwama

  3. The University of Tokyo, Tokyo, Japan

    Naoki Kobayashi

  4. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Bettina Speckmann

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Nayyeri, A., Sidiropoulos, A. (2015). Computing the Fréchet Distance Between Polygons with Holes. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_81

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  • DOI: https://doi.org/10.1007/978-3-662-47672-7_81

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