Advertisement

ISAAC 2010: Algorithms and Computation pp 97-108

# Maximum Overlap of Convex Polytopes under Translation

• Hee-Kap Ahn
• Siu-Wing Cheng
• Iris Reinbacher
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6507)

## Abstract

We study the problem of maximizing the overlap of two convex polytopes under translation in $${\mathbb R}^d$$ for some constant d ≥ 3. Let n be the number of bounding hyperplanes of the polytopes. We present an algorithm that, for any ε> 0, finds an overlap at least the optimum minus ε and reports a translation realizing it. The running time is $$O(n^{{\lfloor d/2 \rfloor}+1} \log^d n)$$ with probability at least 1 − n − O(1), which can be improved to O(nlog3.5 n) in $${\mathbb R}^3$$. The time complexity analysis depends on a bounded incidence condition that we enforce with probability one by randomly perturbing the input polytopes. This causes an additive error ε, which can be made arbitrarily small by decreasing the perturbation magnitude. Our algorithm in fact computes the maximum overlap of the perturbed polytopes. All bounds and their big-O constants are independent of ε.

## Keywords

Convex Polygon Convex Polytopes Steep Ascent Face Pair Time Complexity Analysis
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.

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Ahn, H.-K., Brass, P., Shin, C.-S.: Maximum overlap and minimum convex hull of two convex polyhedra under translation. Comput. Geom. Theory and Appl. 40, 171–177 (2008)
2. 2.
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 and Appl. 37, 3–15 (2007)
3. 3.
de Berg, M., Cheong, O., Devillers, O., van Kreveld, M., Teillaud, M.: Computing the Maximum Overlap of Two Convex Polygons under Translations. Theory of Comput. Syst. 31, 613–628 (1998)
4. 4.
Chazelle, B.: An optimal algorithm for intersecting three-dimensional convex polyhedra. SIAM J. Computing 21, 671–696 (1992)
5. 5.
Chazelle, B.: An optimal convex hull algorithm in any fixed dimension. Discr. Comput. Geom. 9, 377–409 (1993)
6. 6.
Chazelle, B.: Cutting Hyperplanes for Divide-and-Conquer. Discr. Comput. Geom. 9, 145–159 (1993)
7. 7.
Dobkin, D.P., Kirkpatrick, D.G.: Determining the separation of preprocessed polyhedra – a unified approach. In: Proc. 17th Internat. Colloq. Automata Lang. Program., pp. 400–413 (1990)Google Scholar
8. 8.
Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Heidelberg (1987)
9. 9.
Fukuda, K., Uno, T.: Polynomial time algorithms for maximizing the intersection volume of polytopes. Pacific J. Optimization 3, 37–52 (2007)
10. 10.
Haussler, D., Welzl, E.: Epsilon-nets and simplex range queries. Discr. Comput. Geom. 2, 127–151 (1987)
11. 11.
Heijmans, H.J.A.M., Tuzikov, A.V.: Similarity and symmetry measures for convex shapes using Minkowski addition. IEEE Trans. PAMI 20, 980–993 (1998)
12. 12.
Megiddo, N.: Linear programming in linear time when the dimension is fixed. J. ACM 31, 114–127 (1984)
13. 13.
Meyer, F., Bouthemy, P.: Region-based tracking in an image sequence. In: Sandini, G. (ed.) ECCV 1992. LNCS, vol. 588, pp. 476–484. Springer, Heidelberg (1992)
14. 14.
Mount, D.M., Silverman, R., Wu, A.Y.: On the area of overlap of translated polygons. Computer Vision and Image Understanding 64, 53–61 (1996)
15. 15.
Sangwine-Yager, J.R.: Mixed Volumes. In: Gruber, P.M., Wills, J.M. (eds.) Handbook on Convex Geometry, vol. A, pp. 43–71. Elsevier, Amsterdam (1993)
16. 16.
Vigneron, A.: Geometric optimization and sums of algebraic functions. In: Proc. ACM–SIAM Sympos. Alg., pp. 906–917 (2010)Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2010

## Authors and Affiliations

• Hee-Kap Ahn
• 1
• Siu-Wing Cheng
• 2
• Iris Reinbacher
• 1
1. 1.Department of Computer Science and EngineeringPOSTECHKorea
2. 2.Department of Computer Science and EngineeringHKUSTHong Kong