article

Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes

Authors:

Micha SharirAuthors Info & Claims

SIAM Journal on Computing, Volume 39, Issue 7

Pages 3248 - 3282

https://doi.org/10.1137/090762968

Published: 01 July 2010 Publication History

Abstract

We show the existence of $\varepsilon$-nets of size $O\left(\frac{1}{\varepsilon}\log\log\frac{1}{\varepsilon}\right)$ for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane and “fat” triangular ranges and for point sets in $\boldsymbol{R}^3$ and axis-parallel boxes; these are the first known nontrivial bounds for these range spaces. Our technique also yields improved bounds on the size of $\varepsilon$-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of $\varepsilon$-nets of size $O\left(\frac{1}{\varepsilon}\log\log\log\frac{1}{\varepsilon}\right)$ for the dual range space of “fat” regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Brönnimann and Goodrich or of Even, Rawitz, and Shahar, we obtain improved approximation factors (computable in expected polynomial time by a randomized algorithm) for the hitting set or the set cover problems associated with the corresponding range spaces.

References

[1]

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.

[2]

P. 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.

[3]

N. Alon and J. Spencer, The Probabilistic Method, Wiley Interscience, New York, 1992.

[4]

M. Bellare, S. Goldwasser, G. Lund, and A. Russell, Efficient probabilistically checkable proofs and applications to approximation, in Proceedings of the 25th Annual ACM Symposium on Theory of Computing, ACM, New York, 1993, pp. 294-304.

[5]

A. Blumer, A. Ehrenfeucht, D. Haussler, and M. Warmuth, Classifying learnable geometric concepts with the Vapnik-Chervonenkis dimension, J. ACM, 36 (1989), pp. 929-965.

[6]

J. D. Boissonnat, M. Sharir, B. Tagansky, and M. Yvinec, Voronoi diagrams in higher dimensions under certain polyhedral distance functions, Discrete Comput. Geom., 19 (1998), pp. 485-519.

[7]

H. Brönnimann and M. T. Goodrich, Almost optimal set covers in finite VC dimensions, Discrete Comput. Geom., 14 (1995), pp. 463-479.

[8]

H. Brönnimann and J. Lenchner, Fast almost-linear-sized nets for boxes in the plane, in Proceedings of the 14th Annual Fall Workshop on Computational Geometrics, Boston, MA, 2004, pp. 36-38.

[9]

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

[10]

C. Chekuri, K. Clarkson, and S. Har-Peled, On the set multi-cover problem in geometric settings, in Proceedings of the 25th Annual ACM Symposium on Computational Geometry, ACM, New York, 2009, pp. 341-350.

[11]

K. L. Clarkson, New applications of random sampling in computational geometry, Discrete Comput. Geom., 2 (1987), pp. 195-222.

[12]

K. L. Clarkson, Algorithms for polytope covering and approximation, in Proceedings of the 3rd Workshop on Algorithms and Data Structures, Montreal, Canada, 1993, pp. 246-252.

[13]

K. L. Clarkson and P. W. Shor, Applications of random sampling in computational geometry, II, Discrete Comput. Geom., 4 (1989), pp. 387-421.

[14]

K. L. Clarkson and K. Varadarajan, Improved approximation algorithms for geometric set cover, Discrete Comput. Geom., 37 (2007), pp. 43-58.

[15]

T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, MIT Press, Cambridge, MA, 2001.

[16]

M. de Berg, Improved bounds on the union complexity of fat objects, Discrete Comput. Geom., 40 (2008), pp. 127-140.

[17]

M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars, Computational Geometry: Algorithms and Applications, 3rd rev. ed., Springer-Verlag, Berlin, 2008.

[18]

M. de Berg, Better bounds on the union complexity of locally fat objects, in Proceedings of the 26th Annual Symposium on Computational Geometry, Snowbird, UT, 2010, pp. 39-47.

[19]

H. Edelsbrunner, L. Guibas, J. Hershberger, J. Pach, R. Pollack, R. Seidel, M. Sharir, and J. Snoeyink, On arrangements of Jordan arcs with three intersections per pair, Discrete Comput. Geom., 4 (1989), pp. 523-539.

[20]

A. Efrat, The complexity of the union of $(\alpha,\beta)$-covered objects, SIAM J. Comput., 34 (2005), pp. 775-787.

[21]

A. Efrat and M. Katz, On the union of $\kappa$-curved objects, Comput. Geom. Theory Appl., 14 (1999), pp. 241-254.

[22]

A. Efrat and M. Sharir, The complexity of the union of fat objects in the plane, Discrete Comput. Geom., 23 (2000), pp. 171-189.

[23]

G. Even, D. Rawitz, and S. Shahar, Hitting sets when the VC-dimension is small, Inform. Process. Lett., 95 (2005), pp. 358-362.

[24]

E. Ezra, Weak $\eps$-Nets for Axis-Parallel Boxes in $d$-Space, manuscript, 2009.

[25]

T. Feder and D. H. Greene, Optimal algorithms for approximate clustering, in Proceedings of the 20th Annual ACM Symposium on Theory of Computing, ACM, New York, 1988, pp. 434-444.

[26]

U. Feige, A threshold of $\ln{n}$ for approximating set cover, J. ACM, 45 (1998), pp. 634-652.

[27]

R. Fowler, M. Paterson, and S. Tanimoto, Optimal packing and covering in the plane are NP-complete, Inform. Process. Lett., 12 (1981), pp. 133-137.

[28]

M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, New York, 1979.

[29]

S. Har-Peled, H. Kaplan, M. Sharir, and S. Smorodinsky, $\eps$-nets for Halfspaces Revisited, manuscript.

[30]

D. Haussler and E. Welzl, $\varepsilon$-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), pp. 127-151.

[31]

H. Kaplan, N. Rubin, M. Sharir, and E. Verbin, Efficient colored orthogonal range counting, SIAM J. Comput., 38 (2008), pp. 982-1011.

[32]

R. M. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations, R. E. Miller and J. W. Thatcher, eds., Plenum Press, New York, 1972, pp. 85-103.

[33]

J. King and D. Kirkpatrick, Improved approximation for guarding simple galleries from the perimeter, available online at http://arxiv.org/abs/1001.4231, 2010.

[34]

J. Komlós, J. Pach, and G. Woeginger, Almost tight bounds for $\epsilon$ nets, Discrete Comput. Geom., 7 (1992), pp. 163-173.

[35]

J. Matoušek, Lectures on Discrete Geometry, Springer-Verlag, New York, 2002.

[36]

J. Matoušek, Efficient partition trees, Discrete Comput. Geom., 8 (1992), pp. 315-334.

[37]

J. Matoušek, Reporting points in halfspaces, Comput. Geom. Theory Appl., 2 (1992), pp. 169-186.

[38]

J. Matoušek, Approximations and optimal geometric divide-and-conquer, J. Comput System Sci., 50 (1995), pp. 203-208.

[39]

J. Matoušek, R. Seidel, and E. Welzl, How to net a lot with little: Small $\eps$-nets for disks and halfspaces, in Proceedings of the 6th Annual ACM Symposium on Computational Geometry, ACM, New York, 1990, pp. 16-22. Revised version available at http://kam.mff.cuni.cz/$\sim$matousek/enets3.ps.gz.

[40]

J. Matoušek, J. Pach, M. Sharir, S. Sifrony, and E. Welzl, Fat triangles determine linearly many holes, SIAM J. Comput., 23 (1994), pp. 154-169.

[41]

A. Naamad, D. T. Lee, and W. L. Hsu, On the maximal empty rectangle problem, Discrete Appl. Math., 8 (1984), pp. 267-277.

[42]

G. Nivasch, Improved bounds and new techniques for Davenport-Schinzel sequences and their generalizations, J. ACM, 57 (2010), pp. 1-44.

[43]

J. Pach and P. K. Agarwal, Combinatorial Geometry, Wiley Interscience, New York, 1995.

[44]

J. Pach and G. Tardos, On the boundary complexity of the union of fat triangles, SIAM J. Comput., 31 (2002), pp. 1745-1760.

[45]

S. A. Plotkin, D. B. Shmoys, and É. Tardos, Fast approximation algorithms for fractional packing and covering problems, Math. Oper. Res., 20 (1995), pp. 257-301.

[46]

E. Pyrga and S. Ray, New existence proofs for $\eps$-nets, in Proceedings of the 24th Annual ACM Symposium on Computational Geometry, ACM, New York, 2008, pp. 199-207.

[47]

M. Sharir and P. K. Agarwal, Davenport-Schinzel Sequences and Their Geometric Applications, Cambridge University Press, New York, 1995.

[48]

K. Varadarajan, Epsilon nets and union complexity, in Proceedings of the 25th Annual ACM Symposium on Computational Geometry, ACM, New York, 2009, pp. 11-16.

[49]

N. E. Young, Randomized rounding without solving the linear program, in Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 1995, pp. 170-178.

Cited By

Acharyya AKeikha VMajumdar DPandit S(2024)Constrained hitting set problem with intervalsTheoretical Computer Science10.1016/j.tcs.2024.114402990:COnline publication date: 1-Apr-2024
https://dl.acm.org/doi/10.1016/j.tcs.2024.114402
Khan ASubramanian AWiese A(2024)A PTAS for the horizontal rectangle stabbing problemMathematical Programming: Series A and B10.1007/s10107-024-02106-y206:1-2(607-630)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1007/s10107-024-02106-y
Bertschinger DEl Maalouly NKleist LMiltzow TWeber S(2023)The Complexity of Recognizing Geometric HypergraphsGraph Drawing and Network Visualization10.1007/978-3-031-49272-3_12(163-179)Online publication date: 20-Sep-2023
https://dl.acm.org/doi/10.1007/978-3-031-49272-3_12
Show More Cited By

Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes
1. Mathematics of computing
2. Theory of computation
  1. Randomness, geometry and discrete structures

Recommendations

Small-size ε-nets for axis-parallel rectangles and boxes
STOC '09: Proceedings of the forty-first annual ACM symposium on Theory of computing

We show the existence of ε-nets of size O(1/ε log log 1/ε) for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane with "fat" triangular ranges, and for point sets in reals³ and axis-parallel boxes; these ...
Epsilon nets and union complexity
SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry

We consider the following combinatorial problem: given a set of n objects (for example, disks in the plane, triangles), and an integer L ≥ 1, what is the size of the smallest subset of these n objects that covers all points that are in at least L of the ...
Output-Sensitive Construction of the Union of Triangles

We present an efficient algorithm for the following problem: Given a collection $T =\{\Delta_1, \ldots, \Delta_n\}$ of $n$ triangles in the plane, such that there exists a subset $S \subset T$ (unknown to us) of $\xi \ll n$ triangles, such that $\bigcup_...

Comments

Information & Contributors

Information

Published In

cover image SIAM Journal on Computing

SIAM Journal on Computing Volume 39, Issue 7

May 2010

757 pages

ISSN:0097-5397

Issue’s Table of Contents

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 July 2010

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

38
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 13 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Acharyya AKeikha VMajumdar DPandit S(2024)Constrained hitting set problem with intervalsTheoretical Computer Science10.1016/j.tcs.2024.114402990:COnline publication date: 1-Apr-2024
https://dl.acm.org/doi/10.1016/j.tcs.2024.114402
Khan ASubramanian AWiese A(2024)A PTAS for the horizontal rectangle stabbing problemMathematical Programming: Series A and B10.1007/s10107-024-02106-y206:1-2(607-630)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1007/s10107-024-02106-y
Bertschinger DEl Maalouly NKleist LMiltzow TWeber S(2023)The Complexity of Recognizing Geometric HypergraphsGraph Drawing and Network Visualization10.1007/978-3-031-49272-3_12(163-179)Online publication date: 20-Sep-2023
https://dl.acm.org/doi/10.1007/978-3-031-49272-3_12
Rubin N(2022)An Improved Bound for Weak Epsilon-nets in the PlaneJournal of the ACM10.1145/355598569:5(1-35)Online publication date: 27-Oct-2022
https://dl.acm.org/doi/10.1145/3555985
Chekuri CInamdar TQuanrud KVaradarajan KZhang Z(2022)Algorithms for covering multiple submodular constraints and applicationsJournal of Combinatorial Optimization10.1007/s10878-022-00874-x44:2(979-1010)Online publication date: 1-Sep-2022
https://dl.acm.org/doi/10.1007/s10878-022-00874-x
Alon NJartoux BKeller CSmorodinsky SYuditsky Y(2022)The -t-Net ProblemDiscrete & Computational Geometry10.1007/s00454-022-00376-x68:2(618-644)Online publication date: 1-Sep-2022
https://dl.acm.org/doi/10.1007/s00454-022-00376-x
Huang ZFeng QWang JXu J(2022)Small Candidate Set for Translational Pattern SearchAlgorithmica10.1007/s00453-022-00997-x84:10(3034-3053)Online publication date: 1-Oct-2022
https://dl.acm.org/doi/10.1007/s00453-022-00997-x
Khan ASubramanian AWiese A(2022)A PTAS for the Horizontal Rectangle Stabbing ProblemInteger Programming and Combinatorial Optimization10.1007/978-3-031-06901-7_27(361-374)Online publication date: 27-Jun-2022
https://dl.acm.org/doi/10.1007/978-3-031-06901-7_27
Bansal NBatra JMarx D(2021)Non-uniform geometric set cover and scheduling on multiple machinesProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458243(3011-3021)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458243
Soto JTelha C(2021)Independent Sets and Hitting Sets of Bicolored Rectangular FamiliesAlgorithmica10.1007/s00453-021-00810-183:6(1918-1952)Online publication date: 1-Jun-2021
https://dl.acm.org/doi/10.1007/s00453-021-00810-1
Show More Cited By

View Options

View options

Media

Figures

Other

Tables

View Issue’s Table of Contents