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)


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.


Convex Body Linear Time Convex Polygon Rigid Motion Maximum Area 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  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), CrossRefzbMATHMathSciNetGoogle 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), CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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), CrossRefzbMATHMathSciNetGoogle 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), CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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), zbMATHGoogle 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), 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), zbMATHMathSciNetGoogle 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), CrossRefzbMATHMathSciNetGoogle 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),
  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)CrossRefzbMATHMathSciNetGoogle 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), CrossRefGoogle Scholar
  19. 19.
    Sharir, M., Toledo, S.: Extremal polygon containment problems. Comput. Geom. Theory Appl. 4, 99–118 (1994)CrossRefzbMATHMathSciNetGoogle 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), 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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