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Approximation Algorithms for Finding a Minimum Perimeter Polygon Intersecting a Set of Line Segments

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Algorithms and Data Structures (WADS 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5664))

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Abstract

Let S denote a set of line segments in the plane. We say that a polygon P intersects S if every segment in S has a non-empty intersection with the interior or boundary of P. Currently, the best known algorithm finding a minimum perimeter polygon intersecting a set of line segments has a worst case exponential running time. It is also still unknown whether this problem is NP-hard. In this note we explore several approximation algorithms. We present efficient approximation algorithms that yield good empirical results, but can perform very poorly on pathological examples. We also present an O(n logn) algorithm with a guaranteed worst case performance bound that is at most π/2 times that of the optimum.

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References

  1. Aurenhammer, F., Drysdale, R.L.S., Krasser, H.: Farthest line segment Voronoi diagrams. Information Processing Letters 100(6), 220–225 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bhattacharya, B., Toussaint, G.: Computing shortest transversals. In: Proc. 18th Internat. Colloq. Automat. Lang. Program., pp. 649–660. Springer, Heidelberg (1991)

    Google Scholar 

  3. Burkard, R.E., Rote, G., Yao, E.Y., Yu, Z.L.: Shortest polygonal paths in space. Computing 45, 5–68 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chan, T.M.: Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete and Computational Geometry 16, 36–368 (1996)

    MathSciNet  Google Scholar 

  5. Czyzowicz, J., Egyed, P., Everett, H., Rappaport, D., Shermer, T.C., Souvaine, D.L., Toussaint, G.T., Urrutia, J.: The aquarium keeper’s problem. In: Symp. on Discrete Algorithms (SODA), pp. 459–464 (1999)

    Google Scholar 

  6. Goodrich, M., Snoeyink, J.: Stabbing parallel segments with a convex polygon. Comput. Vis. Graph. Image Process. 49, 152–170 (1990)

    Article  MATH  Google Scholar 

  7. Graham, R.L.: An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters 1(4), 132–133 (1972)

    Article  MATH  Google Scholar 

  8. Hearn, D.W., Vijay, J.: Efficient algorithms for the (weighted) minimum circle problem. Operations Research 30(4), 777–795 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jarvis, R.A.: On the identification of the convex hull of a finite set of points in the plane. Information Processing Letters 2, 18–21 (1973)

    Article  MATH  Google Scholar 

  10. Löffler, M., van Kreveld, M.: Largest and smallest convex hulls for imprecise points. Algorithmica (in press)

    Google Scholar 

  11. Löffler, M., van Kreveld, M.: Approximating largest convex hulls for imprecise points. In: Kaklamanis, C., Skutella, M. (eds.) WAOA 2007. LNCS, vol. 4927, pp. 89–102. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  12. Löffler, M., van Kreveld, M.: Largest bounding box, smallest diameter, and related problems on imprecise points. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 446–457. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  13. Meijer, H., Rappaport, D.: Minimum polygon covers of parallel line segments. In: Proc. 2nd Canadian Conference on Comp. Geom., pp. 324–327 (1990)

    Google Scholar 

  14. Mukhopadhyay, A., Kumar, C., Greene, E., Bhattacharya, B.: On intersecting a set of parallel line segments with a convex polygon of minimum area. Information Processing Letters 105, 58–64 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rappaport, D.: Minimum polygon transversals of line segments. Int. J. Comput. Geometry Appl. 5(3), 243–256 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tamir, A.: Improved complexity bounds for center location problems on networks by using dynamic data structures. SIAM J. Discrete Math. 1(3), 377–396 (1988)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Queen’s University, Kingston, Ontario, Canada

    Farzad Hassanzadeh & David Rappaport

Authors
  1. Farzad Hassanzadeh
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  2. David Rappaport
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Editor information

Editors and Affiliations

  1. Carleton University, Ottawa, Canada

    Frank Dehne

  2. Department of Computer Science, University of Calgary, 2500 University Drive N.W., T2N 1N4, Calgary, AB, Canada

    Marina Gavrilova

  3. School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, P.O. Box, K1S 5B6, Ontario, Canada

    Jörg-Rüdiger Sack

  4. University of Calgary, Calgary, AB, Canada

    Csaba D. Tóth

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© 2009 Springer-Verlag Berlin Heidelberg

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Hassanzadeh, F., Rappaport, D. (2009). Approximation Algorithms for Finding a Minimum Perimeter Polygon Intersecting a Set of Line Segments. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_32

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  • DOI: https://doi.org/10.1007/978-3-642-03367-4_32

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03366-7

  • Online ISBN: 978-3-642-03367-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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