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A Linear Time Algorithm for the Maximum Overlap of Two Convex Polygons Under Translation

Authors Timothy M. Chan , Isaac M. Hair

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Convex polygons
  • shape matching
  • prune-and-search
  • parametric search

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Abstract

Given two convex polygons P and Q with n and m edges, the maximum overlap problem is to find a translation of P that maximizes the area of its intersection with Q. We give the first randomized algorithm for this problem with linear running time. Our result improves the previous two-and-a-half-decades-old algorithm by de Berg, Cheong, Devillers, van Kreveld, and Teillaud (1998), which ran in O((n+m)log(n+m)) time, as well as multiple recent algorithms given for special cases of the problem.

Cite As Get BibTex

Timothy M. Chan and Isaac M. Hair. A Linear Time Algorithm for the Maximum Overlap of Two Convex Polygons Under Translation. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.31

Author Details

Timothy M. Chan
  • Siebel School of Computing and Data Science, University of Illinois at Urbana-Champaign, IL, USA
Isaac M. Hair
  • Department of Computer Science, University of California, Santa Barbara, CA, USA

Funding

  • Chan, Timothy M.: Work supported by NSF Grant CCF-2224271.

References

  1. Pankaj K. Agarwal, Torben Hagerup, Rahul Ray, Micha Sharir, Michiel H. M. Smid, and Emo Welzl. Translating a planar object to maximize point containment. In Proc. 10th Annual European Symposium on Algorithms (ESA), volume 2461 of Lecture Notes in Computer Science, pages 42-53. Springer, 2002. URL: https://doi.org/10.1007/3-540-45749-6_8.
  2. Pankaj K. Agarwal and Micha Sharir. Efficient algorithms for geometric optimization. ACM Comput. Surv., 30(4):412-458, 1998. URL: https://doi.org/10.1145/299917.299918.
  3. Hee-Kap Ahn, Peter Brass, and Chan-Su Shin. Maximum overlap and minimum convex hull of two convex polyhedra under translations. Comput. Geom., 40(2):171-177, 2008. URL: https://doi.org/10.1016/J.COMGEO.2007.08.001.
  4. Hee-Kap Ahn, Siu-Wing Cheng, Hyuk Jun Kweon, and Juyoung Yon. Overlap of convex polytopes under rigid motion. Comput. Geom., 47(1):15-24, 2014. URL: https://doi.org/10.1016/J.COMGEO.2013.08.001.
  5. Hee-Kap Ahn, Siu-Wing Cheng, and Iris Reinbacher. Maximum overlap of convex polytopes under translation. Comput. Geom., 46(5):552-565, 2013. URL: https://doi.org/10.1016/J.COMGEO.2011.11.003.
  6. Hee-Kap Ahn, Otfried Cheong, Chong-Dae Park, Chan-Su Shin, and Antoine Vigneron. Maximizing the overlap of two planar convex sets under rigid motions. Comput. Geom., 37(1):3-15, 2007. URL: https://doi.org/10.1016/J.COMGEO.2006.01.005.
  7. Helmut Alt and Leonidas J. Guibas. Discrete geometric shapes: Matching, interpolation, and approximation. In Jörg-Rüdiger Sack and Jorge Urrutia, editors, Handbook of Computational Geometry, pages 121-153. North Holland / Elsevier, 2000. URL: https://doi.org/10.1016/B978-044482537-7/50004-8.
  8. Nancy M. Amato and Franco P. Preparata. A time-optimal parallel algorithm for three-dimensional convex hulls. Algorithmica, 14(2):169-182, 1995. URL: https://doi.org/10.1007/BF01293667.
  9. Gill Barequet, Matthew T. Dickerson, and Petru Pau. Translating a convex polygon to contain a maximum number of points. Comput. Geom., 8:167-179, 1997. URL: https://doi.org/10.1016/S0925-7721(96)00011-9.
  10. Sergio Cabello, Mark de Berg, Panos Giannopoulos, Christian Knauer, René van Oostrum, and Remco C. Veltkamp. Maximizing the area of overlap of two unions of disks under rigid motion. Int. J. Comput. Geom. Appl., 19(6):533-556, 2009. URL: https://doi.org/10.1142/S0218195909003118.
  11. Timothy M. Chan. A simpler linear-time algorithm for intersecting two convex polyhedra in three dimensions. Discret. Comput. Geom., 56(4):860-865, 2016. URL: https://doi.org/10.1007/S00454-016-9785-3.
  12. Timothy M. Chan. (Near-)linear-time randomized algorithms for row minima in Monge partial matrices and related problems. In Proc. of the 32nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1465-1482, 2021. URL: https://doi.org/10.1137/1.9781611976465.88.
  13. Timothy M. Chan and Isaac M. Hair. Convex polygon containment: Improving quadratic to near linear time. In Proc. 40th International Symposium on Computational Geometry (SoCG), volume 293 of LIPIcs, pages 34:1-34:15, 2024. URL: https://doi.org/10.4230/LIPICS.SOCG.2024.34.
  14. Bernard Chazelle. The polygon containment problem. Advances in Computing Research, 1(1):1-33, 1983. URL: https://www.cs.princeton.edu/~chazelle/pubs/PolygContainmentProb.pdf.
  15. Bernard Chazelle. Triangulating a simple polygon in linear time. Discret. Comput. Geom., 6:485-524, 1991. URL: https://doi.org/10.1007/BF02574703.
  16. Bernard Chazelle. Cuttings. In Dinesh P. Mehta and Sartaj Sahni, editors, Handbook of Data Structures and Applications. Chapman and Hall/CRC, 2004. URL: https://doi.org/10.1201/9781420035179.CH25.
  17. Bernard Chazelle and Leonidas J. Guibas. Fractional cascading: I. A data structuring technique. Algorithmica, 1(2):133-162, 1986. URL: https://doi.org/10.1007/BF01840440.
  18. Siu-Wing Cheng and Chi-Kit Lam. Shape matching under rigid motion. Comput. Geom., 46(6):591-603, 2013. URL: https://doi.org/10.1016/J.COMGEO.2013.01.002.
  19. Kenneth L. Clarkson. New applications of random sampling in computational geometry. Discret. Comput. Geom., 2:195-222, 1987. URL: https://doi.org/10.1007/BF02187879.
  20. Kenneth L. Clarkson and Peter W. Shor. Application of random sampling in computational geometry, II. Discret. Comput. Geom., 4:387-421, 1989. URL: https://doi.org/10.1007/BF02187740.
  21. Mark de Berg, Otfried Cheong, Olivier Devillers, Marc J. van Kreveld, and Monique Teillaud. Computing the maximum overlap of two convex polygons under translations. Theory Comput. Syst., 31(5):613-628, 1998. URL: https://doi.org/10.1007/PL00005845.
  22. Martin E. Dyer. Linear time algorithms for two- and three-variable linear programs. SIAM J. Comput., 13(1):31-45, 1984. URL: https://doi.org/10.1137/0213003.
  23. Martin E. Dyer. On a multidimensional search technique and its application to the euclidean one-centre problem. SIAM J. Comput., 15(3):725-738, 1986. URL: https://doi.org/10.1137/0215052.
  24. Greg N. Frederickson and Donald B. Johnson. Generalized selection and ranking: Sorted matrices. SIAM J. Comput., 13(1):14-30, 1984. URL: https://doi.org/10.1137/0213002.
  25. Sariel Har-Peled and Subhro Roy. Approximating the maximum overlap of polygons under translation. Algorithmica, 78(1):147-165, 2017. URL: https://doi.org/10.1007/S00453-016-0152-9.
  26. Shreesh Jadhav and Asish Mukhopadhyay. Computing a centerpoint of a finite planar set of points in linear time. Discret. Comput. Geom., 12:291-312, 1994. URL: https://doi.org/10.1007/BF02574382.
  27. Mook Kwon Jung, Seokyun Kang, and Hee-Kap Ahn. Minimum convex hull and maximum overlap of two convex polytopes. In Proc. 36th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2025. To appear. Google Scholar
  28. Garam Kim, Jongmin Choi, and Hee-Kap Ahn. Maximum overlap of two convex polygons. KIISE Transactions on Computing Practices, 27(8):400-405, 2021. URL: https://doi.org/10.5626/KTCP.2021.27.8.400.
  29. Jirí Matoušek. Approximations and optimal geometric divide-an-conquer. J. Comput. Syst. Sci., 50(2):203-208, 1995. URL: https://doi.org/10.1006/JCSS.1995.1018.
  30. Nimrod Megiddo. Applying parallel computation algorithms in the design of serial algorithms. J. ACM, 30(4):852-865, 1983. URL: https://doi.org/10.1145/2157.322410.
  31. Nimrod Megiddo. Linear-time algorithms for linear programming in R³ and related problems. SIAM J. Comput., 12(4):759-776, 1983. URL: https://doi.org/10.1137/0212052.
  32. Nimrod Megiddo. Linear programming in linear time when the dimension is fixed. J. ACM, 31(1):114-127, 1984. URL: https://doi.org/10.1145/2422.322418.
  33. Nimrod Megiddo. Partitioning with two lines in the plane. J. Algorithms, 6(3):430-433, 1985. URL: https://doi.org/10.1016/0196-6774(85)90011-2.
  34. David M. Mount, Ruth Silverman, and Angela Y. Wu. On the area of overlap of translated polygons. Comput. Vis. Image Underst., 64(1):53-61, 1996. URL: https://doi.org/10.1006/CVIU.1996.0045.
  35. Wlodzimierz Ogryczak and Arie Tamir. Minimizing the sum of the k largest functions in linear time. Inf. Process. Lett., 85(3):117-122, 2003. URL: https://doi.org/10.1016/S0020-0190(02)00370-8.
  36. Joseph O'Rourke, Subhash Suri, and Csaba D. Tóth. Polygons. In Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors, Handbook of Discrete and Computational Geometry, pages 787-810. Chapman and Hall/CRC, 3rd edition, 2017. URL: https://www.csun.edu/~ctoth/Handbook/chap30.pdf.
  37. Mark H. Overmars and Jan van Leeuwen. Maintenance of configurations in the plane. J. Comput. Syst. Sci., 23(2):166-204, 1981. URL: https://doi.org/10.1016/0022-0000(81)90012-X.
  38. Sivan Toledo. Maximizing non-linear concave functions in fixed dimension. In Proc. 33rd Annual Symposium on Foundations of Computer Science (FOCS), pages 676-685, 1992. URL: https://doi.org/10.1109/SFCS.1992.267783.
  39. Eitan Zemel. An O(n) algorithm for the linear multiple choice knapsack problem and related problems. Inf. Process. Lett., 18(3):123-128, 1984. URL: https://doi.org/10.1016/0020-0190(84)90014-0.
  40. Honglin Zhu and Hyuk Jun Kweon. Maximum overlap area of a convex polyhedron and a convex polygon under translation. In Proc. 39th International Symposium on Computational Geometry (SoCG), volume 258 of LIPIcs, pages 61:1-61:16, 2023. URL: https://doi.org/10.4230/LIPICS.SOCG.2023.61.
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