# Approximating the Maximum Overlap of Polygons under Translation

Article

First Online:

- 199 Downloads
- 3 Citations

## 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 a \((1-\varepsilon )\)-approximation algorithm, for finding the translation of \(Q\), which maximizes its area of overlap with \(P\). Our algorithm runs in \(O\left( {c n}\right) \) time, where *c* is a constant that depends only on *k* and \(\varepsilon \). This suggests that for polygons that are “close” to being convex, the problem can be solved (approximately), in near linear time.

## Keywords

Approximation algorithms Computational geometry Polygon overlap## Notes

### Acknowledgments

The authors would like to thank the anonymous referees for their insightful comments. In particular, the improved construction of Lemma 12 was suggested by an anonymous referee.

## References

- 1.Ahn, H.K., Cheng, S.W., Kweon, H.J., Yon, J.: Overlap of convex polytopes under rigid motion. Computat. Geom. Theory Appl.
**47**(1), 15–24 (2014). doi: 10.1016/j.comgeo.2013.08.001 MathSciNetCrossRefzbMATHGoogle Scholar - 2.Ahn, H.K., Cheng, S.W., Reinbacher, I.: Maximum overlap of convex polytopes under translation. Computat. Geom. Theory Appl.
**46**(5), 552–565 (2013). doi: 10.1016/j.comgeo.2011.11.003 MathSciNetCrossRefzbMATHGoogle Scholar - 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. Computat. Geom. Theory Appl.
**37**(1), 3–15 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 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 - 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, Amsterdam (2000)CrossRefGoogle Scholar
- 6.Avis, D., Bose, P., Toussaint, G.T., Shermer, T.C., Zhu, B., Snoeyink, J.: On the sectional area of convex polytopes. In: Proceedings of the 12th Annual Symposium on Computational Geometry SoCG, pp. 411–412 (1996)Google Scholar
- 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 - 8.Barequet, G., Har-Peled, S.: Polygon containment and translational min-hausdorff-distance between segment sets are 3sum-hard. Int. J. Comput. Geom. Appl.
**11**(4), 465–474 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Brodal, G.S., Jacob, R.: Dynamic planar convex hull. In: Proceedings of the 43th Annual IEEE Symposium on Foundations of Computer Science FOCS, pp. 617–626 (2002)Google Scholar
- 10.Chazelle, B., Liu, D., Magen, A.: Sublinear geometric algorithms. SIAM J. Comput.
**35**(3), 627–646 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Cheng, S.W., Lam, C.K.: Shape matching under rigid motion. Comput. Geom. Theory Appl.
**46**(6), 591–603 (2013). doi: 10.1016/j.comgeo.2013.01.002 MathSciNetCrossRefzbMATHGoogle Scholar - 12.Cheong, O., Efrat, A., Har-Peled, S.: On finding a guard that sees most and a shop that sells most. Discrete Comput. Geom.
**37**(4), 545–563 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 13.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: Scandinavian Workshop on Algorithms. LNCS, vol. 3111, pp. 138–149 (2004)Google Scholar
- 14.de Berg, M., Cheong, O., Devillers, O., van Kreveld, M., Teillaud, M.: Computing the maximum overlap of two convex polygons under translations. Theor. Comput. Sci.
**31**, 613–628 (1998)Google Scholar - 15.Gritzmann, P., Klee, V.: Inner and outer \(j\)-radii of convex bodies in finite-dimensional normed spaces. Discrete Comput. Geom.
**7**, 255–280 (1992)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Har-Peled, S.: Taking a walk in a planar arrangement. SIAM J. Comput.
**30**(4), 1341–1367 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Har-Peled, S.: Geometric approximation algorithms. In: Mathematical Surveys and Monographs, vol. 173. American Mathematical Society, Providence, RI (2011)Google Scholar
- 18.Har-Peled, S., Roy, S.: Approximating the maximum overlap of polygons under translation. In: Proceedings of the 22nd Annual European Symposium on Algorithms (ESA), pp. 542–553 (2014)Google Scholar
- 19.Har-Peled, S., Roy, S.: Approximating the Maximum Overlap of Polygons Under Translation. CoRR abs/1406.5778. http://arxiv.org/abs/1406.5778 (2014)
- 20.Keil, J.M., Snoeyink, J.: On the time bound for convex decomposition of simple polygons. Int. J. Comput. Geom. Appl.
**12**(3), 181–192 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Mount, D.M., Silverman, R., Wu, A.Y.: On the area of overlap of translated polygons. Comput. Vis. Image Underst. (CVIU)
**64**(1), 53–61 (1996). http://www.cs.umd.edu/~mount/Papers/overlap.ps - 22.Sharir, M., Toledo, S.: Extremal polygon containment problems. Comput. Geom. Theory Appl.
**4**, 99–118 (1994)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Toussaint, G.T.: Solving geometric problems with the rotating calipers. In: Proceedings of the IEEE MELECON ’83, pp. A10.02/1–A10.02/4Google Scholar
- 24.Vigneron, A.: Geometric optimization and sums of algebraic functions. ACM Trans. Algorithms
**10**(1), 4 (2014). doi: 10.1145/2532647 MathSciNetCrossRefzbMATHGoogle Scholar

## Copyright information

© Springer Science+Business Media New York 2016