On approximate halfspace range counting and relative epsilon-approximations

Proceedings of the Annual Symposium on Computational Geometry

Volumn , Issue , 2007, Pages 327-336

  Aronov, Boris ^a   Har Peled, Sariel ^b   Sharir, Micha ^c,d  

^a POLYTECHNIC UNIVERSITY   (United States)

^b University of Illinois at Urbana Champaign   (United States)

^c TEL AVIV UNIVERSITY   (Israel)

^d NEW YORK UNIVERSITY   (United States)

Author keywords

Approximate range queries; Discrepancy; Epsilon approximations; Halfspaces; Partition trees; Queries; Range; Range spaces; VC dimension

Indexed keywords

COMPUTATIONAL GEOMETRY; NUMERICAL METHODS; PROBLEM SOLVING; QUERY LANGUAGES; THEOREM PROVING;

APPROXIMATE RANGE QUERIES; EPSILON-APPROXIMATIONS; HALFSPACES; PARTITION TREES; RANGE SPACES;

APPROXIMATION THEORY;

EID: 35348843977     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/1247069.1247128     Document Type: Conference Paper

References (27)

1. P. Afshani and T. M. Chan, On approximate range counting and depth, These Proceedings.
2. B. Aronov and S. Har-Peled, On approximating the depth and related problems, Proc. 16th Annu. ACM-SIAM Sympos. Discrete Algo., 2005, 886-894.
3. B. Aronov and S. Har-Peled, On approximating the depth and related problems, manuscript, 2006. Full version of [2], available from http://www.uiuc.edu/~sariel/papers/04/depth.
4. B. Aronov and M. Sharir, Approximate halfspace range counting, in preparation.
5. S. Arya, T. Malamatos, and D. Mount, Space-time tradeoffs for approximate spherical range counting, Proc. 16th Annu. ACM-SIAM Sympos. Discrete Algo., 2005, 535-544.
6. S. Arya, T. Malamatos, and D. Mount, The effect of corners on the complexity of approximate range searching, Proc. 22th Annu. A CM Sympos. Comput. Geom., 2006, 11-20.
7. B. Chazelle and E. Welzl, Quasi-optimal range searching in spaces with finite VC dimension, Discrete Comput. Geom. 4 (1989), 467-490.
8. B. Chazelle, The Discrepancy Method: Randomness and Complexity, Cambridge University Press, New York, 2001.
9. B. Chazelle, The discrepancy method in computational geometry, chapter 44, in Handbook of Discrete and Computational Geometry, 2nd Edition, J.E. Goodman and J. O'Rourke, Eds., CRC Press, Boca Raton, 2004, 983-996.
10. E. Cohen, Size-estimation framework with applications to transitive closure and reachability, J. Comput. Syst. Sci. 55 (1997), 441-453.
11. E. Cohen and H. Kaplan, Spatially-decaying aggregation over a network: model and algorithms, SIGMOD '04: Proc. 2004 ACM SIGMOD Internat. Conf. on Management of Data, 2004, 707-718.
12. E. Cohen, H. Kaplan, Y. Mansour and M. Sharir, Approximations with relative errors in range spaces of finite VC dimension, manuscript, 2006.
13. D. Haussler, Decision theoretic generalizations of the PAC model for neural nets and other learning applications, Inf. Comput. 100 (1992), 78-150.
14. D. Haussler and E. Welzl, Epsilon nets and simplex range queries, Discrete Comput. Geom. 2 (1987), 127-151.
15. S. Har-Peled and M. Sharir, Relative ε-approximations in geometry, Manuscript, 2006. Available from http://www.uiuc.edu/~sariel/papers/06/relative
16. H. Kaplan, E. Ramos and M. Sharir, Randomized incremental construction of convex hulls and Voronoi diagrams, and approximate range counting, in preparation.
17. H. Kaplan and M. Sharir, Randomized incremental construction of three-dimensional convex hulls and planar Voronoi diagrams, and approximate range counting, Proc. 17th ACM-SIAM Sympos. Discrete Algorithms (2006), 484-493.
18. Y. Li, P.M. Long, and A. Srinivasan, Improved bounds on the sample complexity of learning, J. Comput. Syst. Sci. 62 (2001), 516-527.
19. J. Matoušek, Reporting points in halfspaces, Comput. Geom. Theory Appl. 2 (1991), 169-186.
20. J. Matoušek, Efficient partition trees, Discrete Comput. Geom. 8 (1992), 315-334.
21. J. Matoušek, Geometric Discrepancy, Algorithms and Combinatorics, Vol. 18, Springer Verlag, Heidelberg, 1999.
22. J. Matoušek, Using the Borsuk-Ulam Theorem, Universitext, Springer-Verlag, Berlin, 2003, Lectures on topological methods in combinatorics and geometry, Written in cooperation with Anders Björner and Günter M. Ziegler.
23. J. Matoušek, E. Welzl and L. Wernisch, Discrepancy and approximations for bounded VC-dimension, Combinatorica 13 (1993), 455-466.
24. J. Pach and P.K. Agarwal, Combinatorial Geometry, Wiley Interscience, New York, 1995.
25. D. Pollard, Rates of uniform almost-sure convergence for empirical processes indexed by unbounded classes of functions, Manuscript, 1986.
26. V.N. Vapnik and A. Ya. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory of Probability and its Applications 16 (1971), 264-280.
27. E. Welzl, On spanning trees with low crossing numbers, In Data Structures and Efficient Algorithms, Final Report on the DFG Special Joint Initiative, volume 594 of Lect Notes in Comp. Sci., pages 233-249, Springer-Verlag, 1992.