skip to main content
10.1145/378583.378672acmconferencesArticle/Chapter ViewAbstractPublication PagessocgConference Proceedingsconference-collections
Article

Algorithms for congruent sphere packing and applications

Authors Info & Claims
Published:01 June 2001Publication History

ABSTRACT

The problem of packing congruent spheres (i.e., copies of the same sph ere) in a bounded domain arises in many applications. In this paper, we present a new pack-and-shake scheme for packing congruent spheres in various bounded 2-D domains. Our packing scheme is based on a number of interesting ideas, such as a trimming and packing approach, optimal lattice packing under translation and/or rotation, shaking procedures, etc. Our packing algorithms have fairly low time complexities. In certain cases, they even run in nearly linear time. Our techniques can be easily generalized to congruent packing of other shapes of objects, and are readily extended to higher dimensional spaces. Applications of our packing algorithms to treatment planning of radiosurgery are discussed. Experimental results suggest that our algorithms produce reasonably dense packings.

References

  1. 1.P.K. Agarwal and M. Sharir, "Pipes, cigars, and kreplach: The union of Minkowski sums in three dimensions," Proc. 15th ACM Symp. on Computational Geometry, 1999, pp. 143-153. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. 2.A. Aggarwal, H. Booth, J. O'Rourke, S. Suri, and C.K. Yap, "Finding minimal convex nested polygons," Proc. 1st Annu. ACM Symp. Comput. Geom., 1985, pp. 296-304. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. 3.N.M. Amato, M.T. Goodrich, and E.A. Ramos, "Computing the arrangement of curve segments: Divide-and-conquer algorithms via sampling," Proc. 11th Annual ACM-SIAM Symp. on Discrete Algorithms, 2000, pp. 705-706. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. 4.B.S. Baker, D.J. Brown and H.K. Katseff, "A 5/4 algorithm for two-dimensional packing," Journal of Algorithms, Vol. 2, 1981, pp. 348-368.Google ScholarGoogle ScholarCross RefCross Ref
  5. 5.B.S. Baker, E.G. Coffman, Jr., and R.L. Rivest, "Orthogonal packing in two dimensions," SIAM J. Comput., Vol. 9, No. 4, 1980, pp. 846-855.Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. 6.B.S. Baker and J.S. Schwarz, "Shelf algorithms for two-dimensional packing problems," SIAM J. Comput., Vol. 12, No. 3, 1983, pp. 508-525.Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. 7.G. B. ar and C. Iturriaga, "Rectangle packing in polynomial time," Proc. 5th Canad. Conf. Comput. Geom., 1993, pp. 455-460.Google ScholarGoogle Scholar
  8. 8.C. Baur and S.P. Fekete, "Approximation of geometric dispersion problems," Proc. APPROX'98, 1998, pp. 63-75. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. 9.M. Bern, S. Mitchell, and J. Ruppert, "Linear-size nonobtuse triangulation of polygons," Proc. 10th Annu. ACM Symp. on Comput. Geom., 1994, pp. 221-230. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. 10.J.D. Bourland and Q.R. Wu, "Use of shape for automated, optimized 3D radiosurgical treatment planning," SPIE Proc. Int. Symp. on Medical Imaging, 1996, pp. 553-558. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. 11.S.-W. Cheng, T.K. Dey, H. Edelsbrunner, M.A. Facello, and S.-H. Teng, "Silver exudation," Proc. 15th Annu. ACM Symp. on CS88 Comput. Geom., 1999, pp. 1-10. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. 12.L.P. Chew, "Guaranteed-quality Delaunay meshing in 3D (short version)," Proc. 13th Annu. ACM Symp. on Comput. Geom., 1997, pp. 391-393. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. 13.E.G. Coffman, Jr., M.R. Garey, D.S. Johnson, and R.E. Tarjan, "Performance bounds for level-oriented two-dimensional packing algorithms," SIAM J. Comput., Vol. 9, 1980, pp. 808-826.Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. 14.E.G. Coffman, Jr., and J.C. Lagarias, "Algorithms for packing squares: a probabilistic analysis," SIAM J. Comput. , Vol. 18, No. 1, 1989, pp. 166-185. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. 15.E.G. Coffman, Jr. and P.W. Shor, "Average-case analysis of cutting and packing in two dimensions," European Journal of Operational Research, Vol. 44, 1990, pp. 134-144.Google ScholarGoogle ScholarCross RefCross Ref
  16. 16.J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, New York, 1988. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. 17.K.M. Daniels and V.J. Milenkovic, "Column-based strip packing using ordered and compliant containment," Proc. of the First ACM Workshop on Applied Computational Geometry (WACG), 1996, pp. 33-38. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. 18.K.M. Daniels and V.J. Milenkovic, "Multiple translational containment, part I: An approximation algorithm," Algorithmica, Vol. 19, 1997, pp. 148-182.Google ScholarGoogle ScholarCross RefCross Ref
  19. 19.K.M. Daniels, V.J. Milenkovic, and D. Roth, "Finding the maximum area axis-parallel rectangle in a simple polygon," Computational Geometry: Theory and Applications, Vol. 7, 1997, pp. 125-148. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. 20.H. Dyckhoff, "A topology of cutting and packing problems," European Journal of Operational Research, Vol. 44, 1990, pp. 145-159.Google ScholarGoogle ScholarCross RefCross Ref
  21. 21.H. Edelsbrunner, L.J. Guibas, J. Pach, R. Pollack, R. Seidel, M. Sharir, "Arrangements of curves in the plane: Topology, combinatorics, and algorithms," Theoretical Computer Science, Vol. 92, 1992, pp. 319-336. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. 22.S.P. Fekete and J. Schepers, "On more-dimensional packing I: Modeling," technical report, ZPR 97-288, Website http://www.zpr.uni-koeln.de/~ paper.Google ScholarGoogle Scholar
  23. 23.S.P. Fekete and J. Schepers, "On more-dimensional packing II: Bounds," technical report, ZPR 97-289, Website http://www.zpr.uni-koeln.de/~ paper.Google ScholarGoogle Scholar
  24. 24.S.P. Fekete and J. Schepers, "On more-dimensional packing III: Exact algorithm," technical report, ZPR 97-290, Website http://www.zpr.uni-koeln.de/~ paper.Google ScholarGoogle Scholar
  25. 25.M. Formann and F. Wagner, "A packing problem with applications to lettering of maps," Proc. 7th Annu. ACM Symp. Comput. Geom., 1991, pp. 281-288. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. 26.R.J. Fowler, M.S. Paterson, and S.L. Tanimoto, "Optimal packing and covering in the plane are NP-complete," Information Processing Letters, Vol. 12, No. 3, 1981, pp. 133-137.Google ScholarGoogle Scholar
  27. 27.E. Friedman, "Packing unit squares in squares: A survey and new results," The Electronic Journal of Combinatorics, Vol. 5, 1998, #DS7.Google ScholarGoogle Scholar
  28. 28.Z. F. uredi, "The densest packing of equal circles into a parallel strip," Discrete & Computational geometry, Vol. 6, 1991, 95-106.Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. 29.M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Company, San Francisco, 1979. Google ScholarGoogle ScholarDigital LibraryDigital Library
  30. 30.I. Golan, "Performance bounds for orthogonal oriented two-dimensional packing algorithms," SIAM J. Comput., Vol. 10, No. 3, 1981, pp. 571-582.Google ScholarGoogle ScholarDigital LibraryDigital Library
  31. 31.R.L. Graham and B.D. Lubachevsky, "Dense packing of equal disks in an equilateral triangle: From 22 to 34 and beyond," The Electronic Journal of Combinatorics, Vol. 2, 1995, #A1.Google ScholarGoogle Scholar
  32. 32.R.L. Graham and B.D. Lubachevsky, "Repeated patterns of dense packings of equal disks in a square," The Electronic Journal of Combinatorics, Vol. 3, 1996, #R16.Google ScholarGoogle Scholar
  33. 33.R.L. Graham and N.J.A. Sloane, "Penny-packing and two-dimensional codes," Discrete & Computational Geometry, Vol. 5, 1990, pp. 1-11. Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. 34.J. Heistermann and T. Lengauer, "Efficient automatic part nesting on irregular and inhomogeneous surfaces," Proc. 4th ACM-SIAM Symp. Discrete Algorithms, 1993, pp. 251-259. Google ScholarGoogle ScholarDigital LibraryDigital Library
  35. 35.D. Hilbert, Mathematische Probleme, Archiv. Math. Phys. 1 (1901), 44-63 and 213-237 = Gesamm. Abh., III, 290-329. English translation in BAMS 8 (1902), 437-479 = Proc. of Symposia in Pure Mathematics, 28 (1976), 1-34.Google ScholarGoogle Scholar
  36. 36.D.S. Hochbaum and W. Maass, "Approximation schemes for covering and packing problems in image processing and VLSI," Journal of ACM, Vol. 32, No. 1, 1985, pp. 130-136. Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. 37.G.A. Kabatiansky and V.I. Levenshtein, "Bounds for packings on a sphere and in space," Problemy Peredachi Informatsii, Vol. 14, No. 1, 1978, pp. 3-25.Google ScholarGoogle Scholar
  38. 38.R.M. Karp, M. Luby, and A. Marchetti-Spaccamela, "A probabilistic analysis of multidimensional bin-packing problems," Proc. 16th Annu. ACM Symp. Theory Comput., 1984, pp. 289-299. Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. 39.M. Keil and J. Snoeyink, "On the time bound for convex decomposition of simple polygons," to appear in International J. of Computational Geometry and Applications.Google ScholarGoogle Scholar
  40. 40.G. Kyperberg and W. Kuperberg, "Double-lattice packings of convex bodies in the plane," Discrete & Computational Geometry, Vol. 5, 1990, pp. 389-397.Google ScholarGoogle ScholarDigital LibraryDigital Library
  41. 41.J. Leech, 'Sphere packing and error-correcting codes," Canadian Journal of Mathematics, Vol. 23, 1971, pp. 718-745.Google ScholarGoogle Scholar
  42. 42.K. Li and K.H. Cheng, "On three-dimensional packing," SIAM J. Comput., Vol. 19, No. 5, 1990, pp. 847-867. Google ScholarGoogle ScholarDigital LibraryDigital Library
  43. 43.X.-Y. Li, S.-H. Teng, and A. . Ung.or, "Biting: Advancing front meets sphere packing," to appear in a special issue of International Journal of Numerical Methods in Engineering, 1999.Google ScholarGoogle Scholar
  44. 44.Z. Li and V.J. Milenkovic, "Compaction and separation algorithms for nonconvex polygons and their applications," European Journal of Operational Research, Vol. 84, 1995, pp. 539-561.Google ScholarGoogle ScholarCross RefCross Ref
  45. 45.C.D. Maranas, C.A. Floudas, and P.M. Pardalos, "New results in the packing of equal circles in a square," Discrete Mathematics, Vol. 142, 1995, pp. 287-293. Google ScholarGoogle ScholarDigital LibraryDigital Library
  46. 46.M. Mckenna, J. O'Rourke, and S. Suri, "Finding maximal rectangles inscribed in an orthogonal polygon," Proc. 23rd Allerton Conf. Commun. Control Comput., 1985, pp. 486-495.Google ScholarGoogle Scholar
  47. 47.N. Megiddo and K.J. Supowit, "On the complexity of some common geometric location problems," SIAM J. Comput., Vol. 13, No. 1, 1984, pp. 182-196.Google ScholarGoogle Scholar
  48. 48.V.J. Milenkovic, "Position-based physics: Animating and packing spheres inside polyhedra," Proc. 7th Canad. Conf. Comput. Geom., 1995, pp. 79-84.Google ScholarGoogle Scholar
  49. 49.V.J. Milenkovic, "Translational polygon containment and minimal enclosure using linear programming based restriction," Proc. of the 1996 ACM Symp. on the Theory of Computing (STOC), 1996, pp. 109-118. Google ScholarGoogle ScholarDigital LibraryDigital Library
  50. 50.V.J. Milenkovic, "Multiple translational containment, part II: Exact algorithms," Algorithmica, Vol. 19, 1997, pp. 183-218.Google ScholarGoogle ScholarCross RefCross Ref
  51. 51.V.J. Milenkovic, "Densest translational lattice packing of non-convex polygons," Proc. 16th ACM Annual Symp. on Computational Geometry (SCG), 2000, pp. 280-289. Google ScholarGoogle ScholarDigital LibraryDigital Library
  52. 52.G.L. Miller, D. Talmor, S.-H. Teng, and N. Walkington, "A Delaunay based numerical method for three dimensions: Generation, formulation and partition," Proc. 27th Annu. ACM Symp. Theory Comput., 1995, pp. 683-692. Google ScholarGoogle ScholarDigital LibraryDigital Library
  53. 53.H. Minkowski, "Diskontinuit. atsbereich fur arithmetische Aequivalenz," J. reine angew. Math. 129, 1905, pp. 220-274.Google ScholarGoogle ScholarCross RefCross Ref
  54. 54.D.M. Mount and R. Silverman, "Packing and covering the plane with translates of a convex polygon," Journal of Algorithms, Vol. 11, 1990, pp. 564-580. Google ScholarGoogle ScholarDigital LibraryDigital Library
  55. 55.K.J. Nurmela and P. R. J. Ostergard, "Packing up to 50 equal circles in a square," Discrete & Computational Geometry, Vol. 18, 1997, pp. 111-120.Google ScholarGoogle ScholarCross RefCross Ref
  56. 56.C.A. Rogers, Packing and Covering, Cambridge Univ. Press, Cambridge, 1964.Google ScholarGoogle Scholar
  57. 57.J. Ruppert, "A new and simple algorithm for quality 2-dimensional mesh generation," Proc. 3rd Annu. ACM-SIAM Symp. on Discrete Algorithms, 1992, pp. 83-92. Google ScholarGoogle ScholarDigital LibraryDigital Library
  58. 58.J.R. Shewchuk, "Tetrahedral mesh generation by Delaunay refinement," Proc. 14th Annu. ACM Symp. on Comput. Geom., 1998, pp. 86-95. Google ScholarGoogle ScholarDigital LibraryDigital Library
  59. 59.A. Steinberg, "A strip-packing algorithm with absolute performance bound 2," SIAM J. Comput., Vol. 26, No. 2, 1997, pp. 401-409. Google ScholarGoogle ScholarDigital LibraryDigital Library
  60. 60.G.F. Toth and W. Kuperberg, "A survey of recent results in the theory of packing and covering," Chapter in New Trends in Discrete and Computational Geometry, Algorithms and Combinatorics, Vol. 10, Springer-Verlag, 1993, pp. 251-279.Google ScholarGoogle Scholar
  61. 61.J. Wang, "Packing of unequal spheres and automated radiosurgical treatment planning," Journal of Combinatorial Optimization, Vol. 3, 1999, pp. 453-463Google ScholarGoogle ScholarCross RefCross Ref
  62. 62.J. Wang, "Medial axis and optimal locations for min-max sphere packing," to appear in Journal of Combinatorial Optimization.Google ScholarGoogle Scholar
  63. 63.Q.R. Wu, "Treatment planning optimization for Gamma unit radiosurgery," Ph.D. Thesis, The Mayo Graduate School, 1996.Google ScholarGoogle Scholar
  64. 64.C.K. Yap, "An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments," Discrete Comput. Geom., Vol. 2, 1987, pp. 365-393.Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Algorithms for congruent sphere packing and applications

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in
      • Published in

        cover image ACM Conferences
        SCG '01: Proceedings of the seventeenth annual symposium on Computational geometry
        June 2001
        326 pages
        ISBN:158113357X
        DOI:10.1145/378583

        Copyright © 2001 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 1 June 2001

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • Article

        Acceptance Rates

        SCG '01 Paper Acceptance Rate39of106submissions,37%Overall Acceptance Rate625of1,685submissions,37%

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader
      About Cookies On This Site

      We use cookies to ensure that we give you the best experience on our website.

      Learn more

      Got it!