Low-dimensional linear programming with violations

SIAM Journal on Computing

Volumn 34, Issue 4, 2005, Pages 879-893

  Chan, Timothy M ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Algorithms; Computational geometry; LP

Indexed keywords

ALGORITHMS; PROBLEM SOLVING;

BICHROMATIC POINTS; COMPUTATIONAL GEOMETRY; LP;

LINEAR PROGRAMMING;

EID: 24344437297     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539703439404     Document Type: Article

References (60)

1. P. K. AGARWAL, B. ARONOV, T. M. CHAN, AND M. SHARIR, On levels in arrangements of lines, segments, planes, and triangles, Discrete Comput. Geom., 19 (1998), pp. 315-331.
2. 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.
3. P. K. AGARWAL AND J. ERICKSON, Geometric range searching and its relatives, in Discrete and Computational Geometry: Ten Years Later, B. Chazelle, J. E. Goodman, and R. Pollack, eds., AMS, Providence, RI, 1999, pp. 1-56.
4. P. K. AGARWAL AND J. MATOUŠEK, Dynamic half-space range reporting and its applications, Algorithmica, 13 (1995), pp. 325-345.
5. P. K. AGARWAL AND M. SHARIR, Efficient algorithms for geometric optimization, ACM Comput. Surv., 30 (1998), pp. 412-458.
6. N. ALON AND E. GYORI, The number of small semispaces of a finite set of points in the plane, J. Combin. Theory Ser. A, 41 (1986), pp. 154-157.
7. E. AMALDI AND V. KANN, The complexity and approximability of finding maximum feasible subsystems of linear relations, Theoret. Comput. Sci., 147 (1995), pp. 181-210.
8. E. AMALDI AND V. KANN, On the approximability of minimizing nonzero variables or unsatisfied relations in linear systems, Theoret. Comput. Sci., 209 (1998), pp. 237-260.
9. J. BASCH, L. J. GUIBAS, AND J. HERSHBERGER, Data structures for mobile data, J. Algorithms, 31 (1999), pp. 1-28.
10. G. S. BRODAL AND R. JACOB, Dynamic planar convex hull, in Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2002, pp. 617-626.
11. T. M. CHAN, Fixed-dimensional linear programming queries made easy, in Proceedings of the 12th ACM Symposium on Computational Geometry, ACM, New York, 1996, pp. 284-290.
12. T. M. CHAN, Output-sensitive results on convex hulls, extreme points, and related problems, Discrete Comput. Geom., 16 (1996), pp. 369-387.
13. T. M. CHAN, Deterministic algorithms for 2-d convex programming and 3-d online linear programming, J. Algorithms, 27 (1998), pp. 147-166.
14. T. M. CHAN, Geometric applications of a randomized optimization technique, Discrete Comput. Geom., 22 (1999), pp. 547-567.
15. T. M. CHAN, Remarks on κ-Level Algorithms in the Plane, 1999, manuscript.
16. T. M. CHAN, Dynamic planar convex hull operations in near-logarithmic amortized time, J. ACM, 48 (2001), pp. 1-12.
17. T. M. CHAN, Random sampling, halfspace range reporting, and construction of (≤ κ)-levels in three dimensions, SIAM J. Comput., 30 (2000), pp. 561-575.
18. T. M. CHAN, On enumerating and selecting distances, Internat. J. Comput. Geom. Appl., 11 (2001), pp. 291-304.
19. B. CHAZELLE, Cutting hyperplanes for divide-and-conquer, Discrete Comput. Geom., 9 (1993), pp. 145-158.
20. B. CHAZELLE AND D. P. DOBKIN, Intersection of convex objects in two and three dimensions, J. ACM, 34 (1987), pp. 1-27.
21. B. CHAZELLE, L. GUIBAS, AND D. T. LEE, The power of geometric duality, BIT, 25 (1985), pp. 76-90.
22. K. L. CLARKSON, Las Vegas algorithms for linear and integer programming when the dimension is small, J. ACM, 42 (1995), pp. 488-499.
23. K. L. CLARKSON, A bound on local minima of arrangements that implies the upper bound theorem, Discrete Comput. Geom., 10 (1993), pp. 427-233.
24. K. L. CLARKSON AND P. W. SHOR, Applications of random sampling in computational geometry, II, Discrete Comput. Geom., 4 (1989), pp. 387-421.
25. R. COLE, Slowing down sorting networks to obtain faster sorting algorithms, J. ACM, 34, (1987), pp. 200-208.
26. A. DATTA, H.-P. LENHOF, C. SCHWARZ, AND M. SMID, Static and dynamic algorithms for κ-point clustering problems, J. Algorithms, 19 (1995), pp. 474-503.
27. T. K. DEY, Improved bounds on planar κ-sets and κ-levels, Discrete Comput. Geom., 19 (1998), pp. 373-382.
28. D. P. DOBKIN AND D. G. KIRKPATRICK, Fast detection of polyhedral intersection, Theoret. Comput. Sci., 27 (1983), pp. 241-253.
29. D. P. DOBKIN AND D. G. KIRKPATRICK, Determining the separation of preprocessed polyhedra: A unified approach, in Proceedings of the 17th International Colloquium of Automata, Languages, and Programming, Lecture Notes in Comput. Sci. 443, Springer-Verlag, New York, 1990, pp. 400-413.
30. M. E. DYER, Linear time algorithms for two- and three-variable linear programs, SIAM J. Comput., 13 (1984), pp. 31-45.
31. H. EDELSBRUNNER AND E. P. MÜCKE, Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms, ACM Trans. Graphics, 9 (1990), pp. 66-104.
32. I. EMIRIS AND J. CANNY, A general approach to removing degeneracies, SIAM J. Comput., 24 (1995), pp. 650-664.
33. D. EPPSTEIN, Dynamic three-dimensional linear programming, ORSA J. Comput., 4 (1992), pp. 360-368.
34. A. EFRAT, M. LINDENBAUM, AND M. SHARIR, Finding maximally consistent sets of halfspaces, in Proceedings of the 5th Canadian Conference on Computational Geometry, University of Waterloo, Waterloo, Ontario, Canada, 1993, pp. 432-436.
35. A. EFRAT, M. SHARIR, AND A. ZIV, Computing the smallest κ-enclosing circle and related problems, Comput. Geom., 4 (1994), pp. 119-136.
36. J. ERICKSON, New lower bounds for convex hull problems in odd dimensions, SIAM J. Comput., 28 (1999), pp. 1198-1214.
37. H. EVERETT, J.-M. ROBERT, AND M. VAN KREVELD, An optimal algorithm for the (≤ κ)-levels, with applications to separation and transversal problems, Internat. J. Comput. Geom. Appl., 6 (1996), pp. 247-261.
38. 2) problems in computational geometry, Comput. Geom., 5 (1995), pp. 165-185.
39. M. C. GOLUMBIC, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
40. J. E. GOODMAN AND R. POLLACK, On the number of κ-subsets of a set of n points in the plane, J. Combin. Theory Ser. A, 36 (1984), pp. 101-104.
41. S. HAR-PELED, Taking a walk in a planar arrangement, SIAM J. Sci. Comput., 30 (2000), pp. 1341-1367.
42. S. HAR-PELED AND S. MAZUMDAR, Fast algorithms for computing the smallest κ-enclosing disc, in Proceedings of the 11th European Symposium on Algorithms, Lecture Notes in Comput. Sci. 2832, Springer-Verlag, New York, 2003, pp. 278-288.
43. N. KATOH AND T. TOKUYAMA, Lovász's lemma for the three-dimensional κ-level of concave surfaces and its applications, Discrete Comput. Geom., 27 (2002), pp. 567-584.
44. N. KATOH AND T. TOKUYAMA, Notes on computing peaks in κ-levels and parametric spanning trees, in Proceedings of the 17th ACM Symposium on Computational Geometry, ACM, New York, 2001, pp. 241-248.
45. J. MATOUŠEK, Reporting points in halfspaces, Comput. Geom., 2 (1992), pp. 169-186.
46. J. MATOUŠEK, On enclosing k points by a circle, Inform. Process. Lett., 53 (1995), pp. 217-221.
47. J. MATOUŠEK, On geometric optimization with few violated constraints, Discrete Comput. Geom., 14 (1995), pp. 365-384.
48. N. MEGIDDO, Applying parallel computation algorithms in the design of serial algorithms, J. ACM, 30 (1983), pp. 852-865.
49. 3 and related problems, SIAM J. Comput., 12 (1983), pp. 759-776.
50. N. MEGIDDO, Linear programming in linear time when the dimension is fixed, J. ACM, 31 (1984), pp. 114-127.
51. d of order 1 to κ, in Proceedings of the 22nd ACM Symposium on Theory of Computing, ACM, New York, 1990, pp. 322-330.
52. K. MULMULEY, Dehn-Sommerville relations, upper bound theorem, and levels in arrangements, in Proceedings of the 9th ACM Symposium on Computational Geometry, ACM, New York, 1993, pp. 240-246.
53. M. H. OVERMARS AND J. VAN LEEUWEN, Maintenance of configurations in the plane, J. Comput. System Sci., 23 (1981), pp. 166-204.
54. E. RAMOS, On range reporting, ray shooting, and κ-level construction, in Proceedings of the 15th ACM Symposium on Computational Geometry, ACM, New York, 1999, pp. 390-399.
55. M. REICHLING, On the detection of a common intersection of κ convex polyhedra, in Computational Geometry and Its Applications, Lecture Notes in Comput. Sci. 333, Springer-Verlag, New York, 1988, pp. 180-186.
56. T. ROOS AND P. WIDMAYER, κ-violation linear programming, Inform. Process Lett., 52 (1994), pp. 109-114.
57. R. SEIDEL, Small-dimensional linear programming and convex hulls made easy, Discrete Comput. Geom., 6 (1991), pp. 423-434.
58. M. SHARIR, On κ-sets in arrangements of curves and surfaces, Discrete Comput. Geom., 6 (1991), pp. 593-613.
59. M. SHARIR, S. SMORODINSKY, AND G. TARDOS, An improved bound for κ-sets in three dimensions, Discrete Comput. Geom., 26 (2001), pp. 195-204.
60. M. SHARIR AND E. WELZL, A combinatorial bound for linear programming and related problems, in Proceedings of the 9th Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Comput. Sci. 577, Springer-Verlag, New York, 1992, pp. 569-579.