Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications

SIAM Journal on Computing

Volumn 29, Issue 3, 1999, Pages 912-953

  Agarwal, Pankaj K ^a   Efrat, Alon ^b   Sharir, Micha ^b,c  

^a Duke University   (United States)

^b TEL AVIV UNIVERSITY   (Israel)

^c NEW YORK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; DATA STRUCTURES; FUNCTIONS; OPTIMIZATION;

DIVE AND CONQUER; NEAREST NEIGHBOR SEARCHING; PARAMETRIC SEARCHING; RANGE SEARCHING; SHALLOW LEVELS; VERTICAL DECOMPOSITION;

GEOMETRY;

EID: 0033296344     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/s0097539795295936     Document Type: Article

References (72)

1. P. K. AGARWAL, M. DE BERG, J. MATOUŠEK, AND O. SCHWARZKOPF, Constructing levels in arrangements and higher order Voronoi diagrams, SIAM J. Comput., 27 (1998), pp. 654-667.
2. P. K. AGARWAL, H. EDELSBRUNNER, O. SCHWARZKOPF, AND E. WELZL, Euclidean minimum spanning trees and bichromatic closest pairs, Discrete Comput. Geom., 6 (1991), pp. 407-422.
3. P. K. AGARWAL, A. EFRAT, M. SHARIR, AND S. TOLEDO, Computing a segment-center for a planar point set, J. Algorithms, 15 (1993), pp. 314-323.
4. P. K. AGARWAL AND J. MATOUŠEK, Ray shooting and parametric, search, SIAM J. Comput., 22 (1993), pp. 794-806.
5. P. K. AGARWAL AND J. MATOUŠEK, Range searching with semialgebraic sets, Discrete Comput. Geom., 11 (1994), pp. 393-418.
6. P. K. AGARWAL AND J. MATOUŠEK, Dynamic half-space range searching and its applications, Algorithmica, 14 (1995), pp. 325-345.
7. P. K. AGARWAL, J. MATOUŠEK, AND O. SCHWARZKOPF, Computing many faces in arrangements of lines and segments, SIAM J. Comput., 27 (1998), pp. 491-505.
8. P. K. AGARWAL, O. SCHWARZKOPF, AND M. SHARIR, Overlay of lower envelopes in three dimensions and its applications, Discrete Comput. Geom., 15 (1996), pp. 1-13.
9. P. K. AGARWAL, M. SHARIR, AND P. SHOR, Sharp upper and lower bounds for the length of general Davenport Schinzel sequences, J. Combin. Theory Ser. A, 52 (1989), pp. 228-274.
10. A. AGGARWAL, A. BAR-NOY, S. KHULLER, D. KRAVETS, AND B. SCHIEBER, Efficient minimum cost matching using quadrangle inequality. J. Algorithms, 19 (1995), pp. 116-143.
11. A. AGGARWAL, M. HANSEN, AND T. LEIGHTON, Solving query-retrieval problems by compacting Voronoi diagrams, in Proc. 22nd Annual ACM Symposium on Theory of Computing, 1990, pp. 331-340.
12. D. S. ARNON, G. E. COLLINS, AND S. MCCALLUM, Cylindrical algebraic decomposition I: The basic algorithm, SIAM J. Comput., 13 (1984), pp. 865-877.
13. D. ATKINSON AND P. VAIDYA, Using geometry to solve the transportation problem in the plane, Algorithmica, 13 (1995), pp. 442-461.
14. F. AURENHAMMER, F. HOFFMANN, AND B. ARONOV, Minkowski-type theorems and least square partitioning, in Proc. 8th Annual Symposium on Computational Geometry, 1992, pp. 350-357.
15. J. BENTLEY AND J. SAXE, Decomposable searching problems I: Static-to-dynamic transformation, J. Algorithms, 1 (1980), pp. 301-358.
16. H. BRÖNNIMANN, B. CHAZELLE, AND J. MATOUŠEK, Product range spaces, sensitive sampling, and derandomization, in Proc. 34th Annual IEEE Symposium on Foundations of Computer Science, 1993, pp. 400-409.
17. S. P. BUSS AND P. N. YIANILOS, Linear and O(n log n) time minimum-cost matching algorithms for quasi-convex tours, SIAM J. Comput., 27 (1998), pp. 170-201.
18. B. CHAZELLE, H. EDELSBRUNNER, L. GUIBAS, AND M. SHARIR, A singly exponential stratification scheme for real semialgebraic varieties and its applications, Theoret. Comput. Sci., 84 (1991), pp. 77-105.
19. B. CHAZELLE, H. EDELSBRUNNER, L. GUIBAS, AND M. SHARIR, Diameter, width, closest line pair, and parametric searching, Discrete Comput. Geom., 10 (1993), pp. 183-196.
20. B. CHAZELLE AND J. FRIEDMAN, A deterministic view of random sampling and its use in geometry, Combinatorica, 10 (1990), pp. 229-249.
21. B. CHAZELLE AND F. P. PREPARATA, Halfspace range searching: An algorithmic application of k-sets, Discrete Comput. Geom., 1 (1986), pp. 83-93.
22. B. CHAZELLE, M. SHARIR, AND E. WELZL, Quasi-optimal upper bounds for simplex range searching and new zone theorems, Algorithmica, 8 (1992), pp. 407-430.
23. K. CLARKSON, New applications of random sampling in computational geometry, Discrete Comput. Geom., 2 (1987), pp. 195-222.
24. K. CLARKSON, H. EDELSBRUNNER, L. GUIBAS, M. SHARIR, AND E. WELZL, Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput. Geom., 5 (1990), pp. 99-160.
25. K. CLARKSON AND P. SHOR, Applications of random sampling in computational geometry, II, Discrete Comput. Geom., 4 (1989), pp. 387-421.
26. G. COLLINS, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, in Second GI Conf. on Automata Theory and Formal Languages, Lecture Notes in Comput. Sci. 33, H. Barkhage, ed., Springer-Verlag, New York, Berlin, Heidelberg, 1975, pp. 134-183.
27. M. DE BERG, K. DOBRINDT, AND O. SCHWARZKOPF, On lazy randomized incremental construction, Discrete Comput. Geom., 14 (1995), pp. 261-286.
28. M. DE BERG, L. GUIBAS, AND D. HALPERIN, Vertical decomposition for triangles in 3-space, Discrete Comput. Geom., 15 (1996), pp. 35-61.
29. Z. DREZNER, The planar two-center and two-median problems, Transportation Science, 18 (1984), pp. 351-361.
30. H. EDELSBRUNNER, Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.
31. H. EDELSBRUNNER AND R. SEIDEL, Voronoi diagrams and arrangements, Discrete Comput. Geom., 1 (1986), pp. 25-44.
32. A. EFRAT, Geometric Location Optimization Problems, Ph.D. dissertation, Tel Aviv University, Tel Aviv, Israel, 1998.
33. A. EFRAT AND A. ITAI, Improvements on bottleneck matching and related problems using geometry, in Proc. 12th Annual Symposium on Computational Geometry, 1996, pp. 301-310.
34. A. EFRAT AND M. SHARIR, A near-linear algorithm for the planar segment center problem, Discrete Comput. Geom., 16 (1996), pp. 239-257.
35. D. EPPSTEIN, Dynamic Euclidean minimum spanning trees and extrema of binary functions, Discrete Comput. Geom., 13 (1995), pp. 111-122.
36. B. GRÜNBAUM, A proof of Vázsonyi's conjecture, Bull. Research Council Israel Sec. A, 6 (1956), pp. 77-78.
37. L. GUIBAS, D. HALPERIN, J. MATOUŠEK, AND M. SHARIR, Vertical decomposition of arrangements of hyperplanes in four dimensions, Discrete Comput. Geom., 14 (1995), pp. 113-122.
38. D. HALPERIN AND M. SHARIR, New bounds for lower envelopes in three dimensions, with applications to visibility of terrains, Discrete Comput. Geom., 12 (1994), pp. 313-326.
39. S. HART AND M. SHARIR, Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes, Combinatorica, 6 (1986), pp. 151-177.
40. D. HAUSSLER AND E. WELZL, ∈-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), pp. 127-151.
41. J. HOLM, K. DE LICHTENBERG, AND M. THORUP, Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity, in Proc. 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 79-89.
42. H. IMAI, D. T. LEE, AND C. YANG, 1-segment center covering problems, ORSA J. Comput., 4 (1992), pp. 426-434.
43. H. KUHN, The Hungarian method for the assignment problem, Naval Research Logistics Quarterly, 2 (1955), pp. 83-97.
44. E. LAWLER, Combinatorial Optimization: Networks and Matroids, Saunders College Publishing, Fort Worth, TX, 1976.
45. p-metric, J. Assoc. Comput. Mach., 27 (1980), pp. 604-618.
46. D. T. LEE AND R. L. DRYSDALE, III, Generalization of Voronoi diagrams in the plane, SIAM J. Comput., 10 (1981), pp. 73-87.
47. D. LEVEN AND M. SHARIR, Intersection and proximity problems and Voronoi diagrams, in Advances in Robotics 1: Algorithmic and Geometric Aspects of Robotics, J. T. Schwartz and C.-K. Yap, eds., Lawrence Erlbaum Associates, Hillsdale, NJ, 1987, pp. 187-228.
48. D. LEVEN AND M. SHARIR, Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams, Discrete Comput. Geom., 2 (1987), pp. 9-31.
49. O. MARCOTTE AND S. SURI, Fast matching algorithms for points on a polygon, SIAM J. Comput., 20 (1991), pp. 405-422.
50. J. MATOUŠEK, Approximations and optimal geometric divide-and-conquer, J. Comput. Systems Sci., 50 (1995), pp. 203-208.
51. J. MATOUŠEK, On geometric optimization with few violated constraints, Discrete Comput. Geom., 14 (1995), pp. 365-384.
52. J. MATOUŠEK, Reporting points in halfspaces, Comput. Geom., 2 (1992), pp. 169-186.
53. J. MATOUŠEK AND O. SCHWARZKOPF, On ray shooting in convex polytopes, Discrete Comput. Geom., 10 (1993), pp. 215-232.
54. J. MATOUŠEK, M. SHARIR, AND E. WELZL, A subexponential bound for linear programming and related problems, Algorithmica. 16 (1996), pp. 498-516.
55. N. MEGIDDO, Applying parallel computation algorithms in the design of serial algorithms, J. Assoc. Comput. Mach., 30 (1983), pp. 852-865.
56. K. MEHLHORN, S. MEISER, AND R. RASCH, Furthest Site Abstract Voronoi Diagrams, Tech. report, Max-Planck Institut für Informatik, Saarbrücken, 1992.
57. R. MOTWANI AND P. RAGHAVAN, Randomized Algorithms, Cambridge University Press, New York, 1995.
58. K. MULMULEY, Computational Geometry: An Introduction Through Randomized Algorithms, Prentice-Hall, Englewood Cliffs, NJ, 1993.
59. K. MULMULEY, On levels in arrangements and Voronoi diagrams, Discrete Comput. Geom., 6 (1991), pp. 307-338.
60. M. OVERMARS, The Design of Dynamic Data Structures, Lecture Notes in Comput. Sci. 156, Springer-Verlag, Berlin, 1983.
61. J. SCHWARTZ AND M. SHARIR, On the "piano movers" problem II: General technique for computing topological properties of real algebraic manifolds, Adv. in Appl. Math., 4 (1983), pp. 298-351.
62. O. SCHWARZKOPF AND M. SHARIR, Vertical decomposition of a single cell in a three-dimensional arrangement of surfaces and its applications, Discrete Comput. Geom., 18 (1997), pp. 269-288.
63. M. SHARIR, Intersection and closest-pair problems for a set of planar discs, SIAM J. Comput., 14 (1985), pp. 448-468.
64. M. SHARIR, On k-sets in arrangements of curves and surfaces, Discrete Comput. Geom., 6 (1991), pp. 593-613.
65. M. SHARIR, Almost tight upper bounds for lower envelopes in higher dimensions, Discrete Comput. Geom., 12 (1994), pp. 327-345.
66. M. SHARIR AND P. K. AGARWAL, Davenport-Schinzel Sequences and Their Geometric Applications, Cambridge University Press, New York, 1995.
67. M. SHARIR AND E. WELZL, A combinatorial bound for linear programming and related problems, in Proc. 9th Annual Symposium on Theoretical Aspects of Computer Science, 1992, pp. 569-579.
68. M. SHARIR AND E. WELZL, Circular and Spherical Separability, manuscript.
69. S. TOLEDO, Maximizing non-linear concave functions in fixed dimensions, in Complexity in Numerical Computations, P. M. Pardalos, ed., World Scientific, Singapore, 1993, pp. 429-447.
70. P. M. VAIDYA, Geometry helps in matching, SIAM J. Comput., 18 (1989), pp. 1201-1225.
71. L. G. VALIANT, Parallelism in comparison problems, SIAM J. Comput., 4 (1975), pp. 348-355.
72. K. R. VARADARAJAN, A divide-and-conquer algorithm for rain-cost perfect matching in the plane, in Proc. 39th Annual IEEE Symposium on Foundations of Computer Science, 1998, pp. 320-329.