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Computational geometry for the gourmet old fare and new dishes

  • Complexity And Computational Geometry (Session 17)
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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 510))

Abstract

It is time to bring this quick overview to a close. There are many interesting problems that I have not touched upon here but which deserve intense scrutiny nevertheless. Some concern important combinatorial issues in discrete geometry, such as zone theorems, and the maximum size of k-sets. Others address the practical aspects of implementing and debugging geometric algorithms, and in particular, robustness in the face of round-off errors.

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References

  1. Agarwal, P.K. Partitioning arrangements of lines I: An efficient deterministic algorithm, Disc. Comput. Geom. 5 (1990), 449–483.

    Google Scholar 

  2. Agarwal, P.K., Partitioning arrangements of lines: II. Applications, Disc. Comput. Geom. 5 (1990), 533–573.

    Google Scholar 

  3. Agarwal, P.K., Sharir, M. Applications of a new partitioning scheme, manuscript, 1990.

    Google Scholar 

  4. Aronov, B., Sharir, M. Triangles in space, or building and analyzing castles in the air, Proc. 4th Ann. ACM Sympos. Comput. Geom. (1988), 381–391.

    Google Scholar 

  5. Bern, M. Hidden surface removal for rectangles, J. Comput. Sys. Sci. 40 (1990) 49–69.

    Google Scholar 

  6. Caniglia, L., Galligo, A. and Heintz, J. Some new effectivity bounds in computational geometry, Proc. 6th Internat. Conf. on Applied Algebra, Algorithmic and Error Correcting Codes, Rome, July 1988.

    Google Scholar 

  7. Canny, J.F. A new algebraic method for motion planning and real geometry, Proc. 28th Annu. IEEE Symp. on Foundat. of Computer Science (1987), 39–48.

    Google Scholar 

  8. Chazelle, B. On the convex layers of a planar set, IEEE Trans. Informat. Theory IT-31 (1985), 509–517.

    Google Scholar 

  9. Chazelle, B. An optimal algorithm for intersecting three-dimensional convex polyhedra, Proc. 30th Ann. IEEE Symp. Found. Comp. Sci. (1989).

    Google Scholar 

  10. Chazelle, B. Lower bounds on the complexity of polytope range searching, J. American Math. Soc. 2 (1989), 637–666.

    Google Scholar 

  11. Chazelle, B. Lower bounds for orthogonal range searching: I. The reporting case, J. ACM 37 (1990), 200–212.

    Google Scholar 

  12. Chazelle, B. Lower bounds for orthogonal range searching: II. The arithmetic model, J. ACM 37 (1990), 439–463.

    Google Scholar 

  13. Chazelle, B. Triangulating a simple polygon in linear time, Proc. 31st Annu. IEEE Symp. Foundat. Comput. Sci., (1990), 220–230. To appear in Discrete Comput. Geom. (1991).

    Google Scholar 

  14. Chazelle, B., Edelsbrunner, H. An optimal algorithm for intersecting line segments in the plane, Proc. 29th Ann. IEEE Symp. Found. Comp. Sci. (1988), 590–600.

    Google Scholar 

  15. Chazelle, B., Edelsbrunner, H., Guibas, L.J., Sharir, M. A singly-exponential stratification scheme for real semi-algebraic varieties and its applications, ICALP (1989) 179–193.

    Google Scholar 

  16. Chazelle, B., Friedman, J. A deterministic view of random sampling and its use in geometry, Combinatorica 10 (1990), 229–249.

    Google Scholar 

  17. Chazelle, B., Guibas, L.J. Fractional cascading: I. A data structuring technique, Algorithmica, 1 (1986), 133–162.

    Google Scholar 

  18. Chazelle, B., Guibas, L.J, Lee, D.T. The power of geometric duality, BIT 25 (1985), 76–90.

    Google Scholar 

  19. Chazelle, B., Incerpi, J. Triangulation and shape-complexity, ACM Trans. on Graphics 3 (1984), 135–152.

    Google Scholar 

  20. Cheng, S., Janardan, R. New results on dynamic point location Proc. 31st Ann. IEEE Symp. Foundat. Comput. Sci. (1990), 96–105.

    Google Scholar 

  21. Clarkson, K.L. A randomized algorithm for closest-point queries, SIAM J. Comput. 17 (1988), 830–847.

    Google Scholar 

  22. Clarkson, K.L. New applications of random sampling in computational geometry, Disc. Comp. Geom. 2 (1987), 195–222.

    Google Scholar 

  23. Clarkson, K.L., Edelsbrunner, H., Guibas, L.J., Sharir, M., Welzl, E. Combinatorial complexity bounds for arrangements of curves and surfaces, Disc. Comput. Geom. 5 (1990), 99–160.

    Google Scholar 

  24. Clarkson, K.L., Shor, P.W. Applications of random sampling in computational geometry, II, Disc. Comp. Geom. 4 (1989), 387–421.

    Google Scholar 

  25. Clarkson, K.L., Tarjan, R.E., Van Wyk, C.J. A fast Las Vegas algorithm for triangulating a simple polygon, Disc. and Comput. Geom. 4 (1989), 432–432.

    Google Scholar 

  26. Collins, G.E. Quantifier elimination for real closed fields by cylindric algebraic decomposition, Proc. 2nd GI Conf. on Automata Theory and Formal Languages, Springer-Verlag, LNCS 35, Berlin (1975), 134–183.

    Google Scholar 

  27. Davenport, J., Heintz, J. Real quantifier elimination is doubly exponetial, J. Symbolic Comput. 5 (1988), 29–35.

    Google Scholar 

  28. Dobkin, D.P., Kirkpatrick, D.G. Fast detection of polyhedral intersection, Theoret. Comput. Sci. 27 (1983), 241–253.

    Google Scholar 

  29. Edelsbrunner, H. Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, Germany, 1987.

    Google Scholar 

  30. Edelsbrunner, H., O'Rourke, J., Seidel, R. Constructing arrangements of lines and hyperplanes with applications, SIAM J. Comput. 15 (1986), 341–363.

    Google Scholar 

  31. Fredman, M.L. How good is the information theory bound in sorting?, Theoret. Comput. Sci, 1, pp. 355–361, 1976.

    Google Scholar 

  32. Fredman, M.L. A lower bound on the complexity of orthogonal range queries, J. ACM, 28 (1981), 696–705.

    Google Scholar 

  33. Fredman, M.L. Lower bounds on the complexity of some optimal data structures, SIAM J. Comput. 10 (1981), 1–10.

    Google Scholar 

  34. Fuchs, H., Kedem, Z., Naylor, B. On visible surface generation by a priori tree structures, Computer Graphics (SIGGRAPH'80), 124–133.

    Google Scholar 

  35. Garey, M.R., Johnson, D.S., Preparata, F.P., Tarjan, R.E. Triangulating a simple polygon, Inform. Process. Lett. 7 (1978), 175–180.

    Google Scholar 

  36. Goodrich, M.F., Atallah, M., Overmars, M. An input-size/output-size trade-off in the time complexity of rectilinear hidden surface removal, Proc. 16th ICALP.

    Google Scholar 

  37. Goodrich, M.F., Tamassia, R. Dynamic trees and dynamic point location, Johns Hopkins Univ. Tech. Rep., 1990.

    Google Scholar 

  38. Grigor'ev, D. and Vorobjov, N. Solving systems of polynomial inequalities in subexponential time, J. Symbolic Comput. 5 (1988), 37–64.

    Google Scholar 

  39. Guibas, L.J., Seidel, R. Computing convolutions using reciprocal search, Proc. 2nd Ann. ACM Symp. Comput. Geom. (1986), 90–99.

    Google Scholar 

  40. Haussler, D., Welzl, E. Epsilon-nets and simplex range queries, Disc. Comp. Geom. 2, (1987), 127–151.

    Google Scholar 

  41. Hershberger, J., Suri, S. Finding tailored partitions, Proc. 5th Ann. ACM Symp. Comput. Geom. (1989), 255–265.

    Google Scholar 

  42. Hertel, S., Mehlhorn, K. Fast triangulation of a simple polygon, Proc. Conf. Found. Comput. Theory, New York, Lecture Notes on Computer Science 158 (1983), 207–218.

    Google Scholar 

  43. Kirkpatrick, D.G., Klawe, M.M., Tarjan, R.E. O(n log log n) polygon triangulation with simple data structures, Proc. 6th Ann. ACM Symp. Comput. Geom. (1990), 34–43.

    Google Scholar 

  44. Mairson, H.G., Stolfi, J. Reporting and counting intersections between two sets of line segments, Proc. NATO Advanced Study Inst. Theoret. Found. Comput. Graphics and CAD, Il Ciocco, Castelvecchio Pascoli, Italy, Springer-Verlag, 1987.

    Google Scholar 

  45. Matoušek, J. Construction of ɛ-nets, Disc. Comput. Geom. 5 (1990), 427–448.

    Google Scholar 

  46. Matoušek, J. Approximations and optimal geometric divide-and-conquer, KAM Series (tech. report) 90–174, Charles University, 1990. Also to appear in Proc. 23rd ACM Symp. Theory of Comput., 1991.

    Google Scholar 

  47. Matoušek, J. Cutting hyperplane arrangements, to appear in Disc. Comput. Geom., 1991. Also, in Proc. 6th ACM Symp. Comput. Geom. (1990), 1–9.

    Google Scholar 

  48. Mehlhorn, K. Data Structures and Algorithms 3: Multidimensional Searching and Computational Geometry, Springer-Verlag, Heidelberg, Germany, 1984.

    Google Scholar 

  49. Mehlhorn, K., Näher, S. Dynamic fractional cascading, Algorithmica 5 (1990), 215–242.

    Google Scholar 

  50. Mehlhorn, K., Simon, K. Intersecting two polyhedra one of which is convex, Univ. Saarland, Tech. Report, Saarbrücken, West Germany, 1986.

    Google Scholar 

  51. Mulmuley, K. A fast planar partition algorithm, Proc. 29th Ann. IEEE Symp. Found. Comp. Sci. (1988).

    Google Scholar 

  52. Mulmuley, K. A fast planar partition algorithm, II, Proc. 5th Ann. ACM Symp. Comp. Geo. (1989), 33–43.

    Google Scholar 

  53. Nievergelt, J., Preparata, F.P. Plane-sweep algorithms for intersecting geometric figures, Comm. ACM, 25 (1982), 739–747.

    Google Scholar 

  54. Overmars, M., Sharir, M. Output-sensitive hidden surface removal algorithms, Proc. 30th Ann. IEEE Symp. Foundat. Comput. Sci. (1989), 598–603.

    Google Scholar 

  55. Overmars, M., Sharir, M. Merging visibility maps, Proc. 6th Ann. ACM Symp. Comput. Geom. (1990), 168–176.

    Google Scholar 

  56. Overmars, M.H., van Leeuwen, J. Maintenance of configurations in the plane, Journal of Computer and System Sciences 23 (1981), 166–204.

    Google Scholar 

  57. Paterson, M.S., Yao, F.F. Binary partitions with applications to hidden-surface removal and solid modelling, Proc. 5th Ann. ACM Symp. Comput. Geom. (1989), 23–32.

    Google Scholar 

  58. Preparata, F.P., Shamos, M.I. Computational Geometry, Springer-Verlag, New York, 1985.

    Google Scholar 

  59. Preparata, F.P., Tamassia, R. Fully dynamic techniques for point location and transitive closure in planar structures, Proc. 29th Ann. IEEE Symp. Found. Comp. Sci. (1988), 558–567.

    Google Scholar 

  60. Preparata, F.P., Tamassia, R. Efficient spatial point location, Proc. 1989 Workshop on Algorithms and Data Structures.

    Google Scholar 

  61. Preparata, F.P., Vitter, J., Yvinec, M. Computation of the axial view of a set of isothetic parallelepipeds, ACM Trans. on Graphics 5 (1990), 278–300.

    Google Scholar 

  62. Prill, D. On approximations and incidence in cylindrical algebraic decompositions, SIAM J. Comput. 15 (1986), 972–993.

    Google Scholar 

  63. Reif, J., Sen, S. An efficient output-sensitive hidden surface removal algorithm and its parallelization, Proc. 4th Ann. ACM Symp. Comput. Geom. (1988), 193–200.

    Google Scholar 

  64. Renegar, J. A faster PSPACE algorithm for deciding the existential theory of the reals, Proc. 29th Annu. IEEE Symp. on Foundat. of Computer Science (1988), 291–295.

    Google Scholar 

  65. Schwartz, J.T., Sharir, M. On the “piano movers” problem. II: General techniques for computing topological properties of real algebraic manifolds, Adv. in Appl. Math. 4 (1983), 298–351.

    Google Scholar 

  66. Seidel, R. A convex hull algorithm optimal for point sets in even dimensions, Univ. British Columbia, tech. Rep. 81–14, 1981.

    Google Scholar 

  67. Seidel, R. Constructing higher-dimensional convex hulls at logarithmic cost per face, Proc. 18th Ann. ACM Symp. Theory Comput. (1986), 404–413.

    Google Scholar 

  68. Seidel, R. Linear programming and convex hulls made easy, Proc. 6th Ann. ACM Symp. Comput. Geom. (1990), 211–215.

    Google Scholar 

  69. Seidel, R. A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons, manuscript, 1990.

    Google Scholar 

  70. Sutherland, I.E., Sproull, R.F., Schumaker, R.A. A characterization of ten hidden surface algorithms, Computing Surveys 6 (1974), 1–55.

    Google Scholar 

  71. Tarjan, R.E., Van Wyk, C.J. An O(n log log n)-time algorithm for triangulating a simple polygon, SIAM J. Comput. 17 (1988), 143–178.

    Google Scholar 

  72. Whitney, H. Elementary structure of real algebraic varieties, Annals of Math. 66 (1957).

    Google Scholar 

  73. Willard, D.E. Lower bounds for dynamic range query problems that permit subtraction, Proc. 13th ICALP, 1986.

    Google Scholar 

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Javier Leach Albert Burkhard Monien Mario Rodríguez Artalejo

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© 1991 Springer-Verlag Berlin Heidelberg

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Chazelle, B. (1991). Computational geometry for the gourmet old fare and new dishes. In: Albert, J.L., Monien, B., Artalejo, M.R. (eds) Automata, Languages and Programming. ICALP 1991. Lecture Notes in Computer Science, vol 510. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54233-7_174

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  • DOI: https://doi.org/10.1007/3-540-54233-7_174

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