Approximating the Maximum Overlap of Polygons under Translation

  • Sariel Har-Peled
  • Subhro Roy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8737)

Abstract

Let P and Q be two simple polygons in the plane of total complexity n, each of which can be decomposed into at most k convex parts. We present an (1 − ε)-approximation algorithm, for finding the translation of Q, which maximizes its area of overlap with P. Our algorithm runs in Ocn time, where c is a constant that depends only on k and ε.

This suggest that for polygons that are “close” to being convex, the problem can be solved (approximately), in near linear time.

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References

  1. 1.
    Ahn, H.K., Cheng, S.W., Kweon, H.J., Yon, J.: Overlap of convex polytopes under rigid motion. Comput. Geom. Theory Appl. 47(1), 15–24 (2014), http://dx.doi.org/10.1016/j.comgeo.2013.08.001 CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Ahn, H.K., Cheng, S.W., Reinbacher, I.: Maximum overlap of convex polytopes under translation. Comput. Geom. Theory Appl. 46(5), 552–565 (2013), http://dx.doi.org/10.1016/j.comgeo.2011.11.003 CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Ahn, H.K., Cheong, O., Park, C.D., Shin, C.S., Vigneron, A.: Maximizing the overlap of two planar convex sets under rigid motions. Comput. Geom. Theory Appl. 37(1), 3–15 (2007)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Alt, H., Fuchs, U., Rote, G., Weber, G.: Matching convex shapes with respect to the symmetric difference. Algorithmica 21, 89–103 (1998), http://citeseer.nj.nec.com/267158.html CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Alt, H., Guibas, L.J.: Discrete geometric shapes: Matching, interpolation, and approximation. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 121–153. Elsevier (2000)Google Scholar
  6. 6.
    Avis, D., Bose, P., Toussaint, G.T., Shermer, T.C., Zhu, B., Snoeyink, J.: On the sectional area of convex polytopes. In: Proc. 12th Annu. Sympos. Comput. Geom., pp. 411–412 (1996)Google Scholar
  7. 7.
    Barequet, G., Har-Peled, S.: Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. J. Algorithms 38, 91–109 (2001), http://cs.uiuc.edu/~sariel/research/papers/98/bbox.html CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Barequet, G., Har-Peled, S.: Polygon containment and translational min-hausdorff-distance between segment sets are 3sum-hard. Internat. J. Comput. Geom. Appl. 11(4), 465–474 (2001)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    de Berg, M., Cabello, S., Giannopoulos, P., Knauer, C., van Oostrum, R., Veltkamp, R.C.: Maximizing the area of overlap of two unions of disks under rigid motion. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 138–149. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    de Berg, M., Cheong, O., Devillers, O., van Kreveld, M., Teillaud, M.: Computing the maximum overlap of two convex polygons under translations. Theo. Comp. Sci. 31, 613–628 (1998), http://link.springer-ny.com/link/service/journals/00224/bibs/31n5p613.html MATHGoogle Scholar
  11. 11.
    Chazelle, B., Liu, D., Magen, A.: Sublinear geometric algorithms. SIAM J. Comput. 35(3), 627–646 (2005)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Vigneron, A.: Geometric optimization and sums of algebraic functions. ACM Trans. Algo. 4, 1–4 (2014), http://doi.acm.org/10.1145/2532647 MathSciNetGoogle Scholar
  13. 13.
    Cheong, O., Efrat, A., Har-Peled, S.: On finding a guard that sees most and a shop that sells most. Discrete Comput. 37(4), 545–563 (2007), http://link.springer-ny.com/link/service/journals/00454/ MATHMathSciNetGoogle Scholar
  14. 14.
    Gritzmann, P., Klee, V.: Inner and outer j-radii of convex bodies in finite-dimensional normed spaces. Discrete Comput. Geom. 7, 255–280 (1992), http://link.springer-ny.com/link/service/journals/00454/ CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Har-Peled, S.: Geometric Approximation Algorithms. Mathematical Surveys and Monographs, vol. 173. Amer. Math. Soc. (2011)Google Scholar
  16. 16.
    Har-Peled, S., Roy, S.: Approximating the maximum overlap of polygons under translation. CoRR abs/1406.5778 (2014), http://arxiv.org/abs/1406.5778
  17. 17.
    Keil, J.M., Snoeyink, J.: On the time bound for convex decomposition of simple polygons. Internat. J. Comput. Geom. Appl. 12(3), 181–192 (2002)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Mount, D.M., Silverman, R., Wu, A.Y.: On the area of overlap of translated polygons. Computer Vision and Image Understanding: CVIU 64(1), 53–61 (1996), http://www.cs.umd.edu/~mount/Papers/overlap.ps CrossRefGoogle Scholar
  19. 19.
    Sharir, M., Toledo, S.: Extremal polygon containment problems. Comput. Geom. Theory Appl. 4, 99–118 (1994)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Toussaint, G.T.: Solving geometric problems with the rotating calipers. In: Proc. IEEE MELECON 1983. pp. A10.02/1–4 (1983)Google Scholar
  21. 21.
    Vigneron, A.: Geometric optimization and sums of algebraic functions. ACM Trans. Algo. 4, 1–4 (2014), http://doi.acm.org/10.1145/2532647 MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Sariel Har-Peled
    • 1
  • Subhro Roy
    • 1
  1. 1.University of Illinois, Urbana-ChampaignUSA

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