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Convex Polygon Containment: Improving Quadratic to Near Linear Time

Authors Timothy M. Chan , Isaac M. Hair

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  • Theory of computation → Computational geometry
Keywords
  • Polygon containment
  • convex polygons
  • translations
  • rotations

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Abstract

We revisit a standard polygon containment problem: given a convex k-gon P and a convex n-gon Q in the plane, find a placement of P inside Q under translation and rotation (if it exists), or more generally, find the largest copy of P inside Q under translation, rotation, and scaling.
Previous algorithms by Chazelle (1983), Sharir and Toledo (1994), and Agarwal, Amenta, and Sharir (1998) all required Ω(n²) time, even in the simplest k = 3 case. We present a significantly faster new algorithm for k = 3 achieving O(n polylog n) running time. Moreover, we extend the result for general k, achieving O(k^O(1/ε) n^{1+ε}) running time for any ε > 0.
Along the way, we also prove a new O(k^O(1) n polylog n) bound on the number of similar copies of P inside Q that have 4 vertices of P in contact with the boundary of Q (assuming general position input), disproving a conjecture by Agarwal, Amenta, and Sharir (1998).

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Timothy M. Chan and Isaac M. Hair. Convex Polygon Containment: Improving Quadratic to Near Linear Time. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.34

Author Details

Timothy M. Chan
  • Department of Computer 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, Nina Amenta, and Micha Sharir. Largest Placement of One Convex Polygon Inside Another. Discret. Comput. Geom., 19(1):95-104, 1998. URL: https://doi.org/10.1007/PL00009337.
  2. Pankaj K. Agarwal, Boris Aronov, and Micha Sharir. Motion Planning for a Convex Polygon in a Polygonal Environment. Discret. Comput. Geom., 22(2):201-221, 1999. URL: https://doi.org/10.1007/PL00009455.
  3. Pankaj K. Agarwal and Jeff Erickson. Geometric Range Searching and Its Relatives. Contemporary Mathematics, 223:1-56, 1999. URL: https://jeffe.cs.illinois.edu/pubs/pdf/survey.pdf.
  4. Pankaj K. Agarwal and Micha Sharir. Davenport-Schinzel Sequences and Their Geometric Applications. In Jörg-Rüdiger Sack and Jorge Urrutia, editors, Handbook of Computational Geometry, pages 1-47. North Holland / Elsevier, 2000. URL: https://doi.org/10.1016/B978-044482537-7/50002-4.
  5. Boris Aronov, Mark de Berg, Jean Cardinal, Esther Ezra, John Iacono, and Micha Sharir. Subquadratic Algorithms for Some 3SUM-Hard Geometric Problems in the Algebraic Decision-Tree Model. Comput. Geom., 109:101945, 2023. URL: https://doi.org/10.1016/J.COMGEO.2022.101945.
  6. Francis Avnaim and Jean-Daniel Boissonnat. Polygon Placement Under Translation and Rotation. RAIRO Theor. Informatics Appl., 23(1):5-28, 1989. URL: https://doi.org/10.1051/ITA/1989230100051.
  7. Henry Spalding Baird. Model-Based Image Matching Using Location. MIT Press, 1984. Google Scholar
  8. Luis Barba, Jean Cardinal, John Iacono, Stefan Langerman, Aurélien Ooms, and Noam Solomon. Subquadratic Algorithms for Algebraic 3SUM. Discret. Comput. Geom., 61(4):698-734, 2019. Preliminary version in SoCG 2017. URL: https://doi.org/10.1007/S00454-018-0040-Y.
  9. Gill Barequet and Sariel Har-Peled. Polygon Containment and Translational Min-Hausdorff-Distance Between Segment Sets are 3SUM-Hard. Int. J. Comput. Geom. Appl., 11(4):465-474, 2001. URL: https://doi.org/10.1142/S0218195901000596.
  10. Timothy M. Chan. Geometric Applications of a Randomized Optimization Technique. Discret. Comput. Geom., 22(4):547-567, 1999. Preliminary version in SoCG 1998. URL: https://doi.org/10.1007/PL00009478.
  11. Timothy M. Chan. More Logarithmic-Factor Speedups for 3SUM, (median, +)-Convolution, and Some Geometric 3SUM-Hard Problems. ACM Trans. Algorithms, 16(1):7:1-7:23, 2020. URL: https://doi.org/10.1145/3363541.
  12. 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.
  13. L. Paul Chew and Klara Kedem. A Convex Polygon Among Polygonal Obstacles: Placement and High-Clearance Motion. Comput. Geom., 3:59-89, 1993. URL: https://doi.org/10.1016/0925-7721(93)90001-M.
  14. Karen L. Daniels and Victor Milenkovic. Multiple Translational Containment. Part I: An Approximate Algorithm. Algorithmica, 19(1/2):148-182, 1997. URL: https://doi.org/10.1007/PL00014415.
  15. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational Geometry: Algorithms and Applications. Springer, 3rd edition, 2008. URL: https://www.worldcat.org/oclc/227584184.
  16. Matthew Dickerson and Daniel Scharstein. Optimal Placement of Convex Polygons to Maximize Point Containment. In Proc. 7th ACM-SIAM Symposium on Discrete Algorithm (SODA), pages 114-121, 1996. URL: http://dl.acm.org/citation.cfm?id=313852.313899.
  17. Taekang Eom, Seungjun Lee, and Hee-Kap Ahn. Largest Similar Copies of Convex Polygons Amidst Polygonal Obstacles. CoRR, abs/2012.06978, 2020. URL: https://arxiv.org/abs/2012.06978.
  18. Allan Grønlund and Seth Pettie. Threesomes, Degenerates, and Love Triangles. J. ACM, 65(4):22:1-22:25, 2018. URL: https://doi.org/10.1145/3185378.
  19. Daniel M. Kane, Shachar Lovett, and Shay Moran. Near-Optimal Linear Decision Trees for k-SUM and Related Problems. J. ACM, 66(3):16:1-16:18, 2019. URL: https://doi.org/10.1145/3285953.
  20. Marvin Künnemann and André Nusser. Polygon Placement Revisited: (Degree of Freedom + 1)-SUM Hardness and an Improvement via Offline Dynamic Rectangle Union. In Proc. ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3181-3201, 2022. URL: https://doi.org/10.1137/1.9781611977073.124.
  21. D. T. Lee. Interval, Segment, Range, and Priority Search Trees. 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.CH18.
  22. Seungjun Lee, Taekang Eom, and Hee-Kap Ahn. Largest Triangles in a Polygon. Comput. Geom., 98:101792, 2021. URL: https://doi.org/10.1016/J.COMGEO.2021.101792.
  23. 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.
  24. Victor Milenkovic. Translational Polygon Containment and Minimal Enclosure Using Linear Programming Based Restriction. In Proc. 28th ACM Symposium on Theory of Computing (STOC), pages 109-118, 1996. URL: https://doi.org/10.1145/237814.237840.
  25. Victor Milenkovic. Multiple Translational Containment. Part II: Exact Algorithms. Algorithmica, 19(1/2):183-218, 1997. URL: https://doi.org/10.1007/PL00014416.
  26. Victor Milenkovic. Rotational Polygon Containment and Minimum Enclosure Using Only Robust 2D Constructions. Comput. Geom., 13(1):3-19, 1999. URL: https://doi.org/10.1016/S0925-7721(99)00006-1.
  27. Joseph O’Rourke, Subhash Suri, and Csaba D Tóth. Polygons. In Handbook of Discrete and Computational Geometry, pages 787-810. CRC Press, 2017. URL: http://www.csun.edu/~ctoth/Handbook/chap30.pdf.
  28. Franco P. Preparata and Michael Ian Shamos. Computational Geometry: An Introduction. Texts and Monographs in Computer Science. Springer, 1985. URL: https://doi.org/10.1007/978-1-4612-1098-6.
  29. Micha Sharir. A Near-Linear Algorithm for the Planar 2-Center Problem. Discret. Comput. Geom., 18(2):125-134, 1997. Preliminary version in SoCG 1996. URL: https://doi.org/10.1007/PL00009311.
  30. Micha Sharir and Sivan Toledo. External Polygon Containment Problems. Computational Geometry, 4(2):99-118, 1994. Preliminary version in SoCG 1991. URL: https://doi.org/10.1016/0925-7721(94)90011-6.
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