research-article

Improved bound for the union of fat triangles

Authors:

Micha SharirAuthors Info & Claims

SODA '11: Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms

Pages 1778 - 1785

Published: 23 January 2011 Publication History

Abstract

We show that, for any fixed δ > 0, the combinatorial complexity of the union of n triangles in the plane, each of whose angles is at least δ, is O(n2^α(n) log* n), with the constant of proportionality depending on δ. This considerably improves the twenty-year-old bound O(n log log n), due to Matoušek et al. [24, 25].

References

[1]

P. K. Agarwal, J. Pach, and M. Sharir, State of the union (of geometric objects), In Proc. Joint Summer Research Conf. on Discrete and Computational Geometry: 20 Years Later, Contemp. Math. 452, Amer. Math. Society, pp. 9--48, 2008.

[2]

B. Aronov, M. de Berg, E. Ezra, and M. Sharir, Improved bound for the union of fat objects in the plane, in preparation.

[3]

B. Aronov, E. Ezra, and M. Sharir, Small-size ε-nets for axis-parallel rectangles and boxes, SIAM J. Comput., 39:3248--3282, 2010.

Digital Library

[4]

M. Bellare, S. Goldwasser, G. Lund, and A. Russell, Efficient probabilistically checkable proofs and applications to approximation, In Proc. 25th Annu. ACM Symp. Theory Comput., pp. 294--304, 1993.

Digital Library

[5]

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

[6]

M. de Berg, Improved bounds on the union complexity of fat objects, Discrete Comput. Geom., 40(1): 127--140, 2008.

Digital Library

[7]

M. de Berg, Better bounds on the union complexity of locally fat objects, Proc. 26th Annu. Sympos. Comput. Geom., pp. 39--47, 2010.

Digital Library

[8]

M. de Berg and O. Schwarzkopf, Cutting and applications, Int. J. Comput. Geom. Appl., 5:343--355, 1995.

[9]

V. Chvátal, A greedy heuristic for the set-covering problem, Math. Oper. Research, 4(3):233--235, 1979.

[10]

K. L. Clarkson, Algorithms for polytope covering and approximation, Proc. 3rd Workshop on Algorithms and Data Structures, pp. 246--252, 1993.

Digital Library

[11]

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

Digital Library

[12]

H. Edelsbrunner, L. J. Guibas, and M. Sharir, The complexity and construction of many faces in arrangements of lines and of segments, Discrete Comput. Geom., 5(2):161--196, 1990.

[13]

A. Efrat, The complexity of the union of (α, β)-covered objects, SIAM J. Comput., 34:775--787, 2005.

Digital Library

[14]

A. Efrat and M. Katz, On the union of k-curved objects, Comput. Geom. Theory Appl. 14:241--254, 1999.

Digital Library

[15]

A. Efrat, G. Rote, and M. Sharir, On the union of fat wedges and separating a collection of segments by a line, Comput. Geom. Theory Appl. 3:277--288, 1993.

Digital Library

[16]

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

[17]

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

Digital Library

[18]

E. Ezra, On the union of cylinders in three dimensions, Proc. 49nd Annu. IEEE Sympos. Found. Comput. Sci., pp. 179--188, 2008.

Digital Library

[19]

E. Ezra and M. Sharir, Almost tight bound for the union of fat tetrahedra in three dimensions, J. ACM 57(1), Article No. 2, 2009.

Digital Library

[20]

U. Feige, A threshold of In n for approximating set cover, J. ACM, 45(4):634--652, 1998.

Digital Library

[21]

D. Haussler and E. Welzl, ε-nets and simplex range queries, Discrete Comput. Geom., 2:127--151, 1987.

[22]

M. J. Katz, M. H. Overmars, and M. Sharir, Efficient hidden surface removal for objects with small union size, Comput. Geom. Theory Appl., 2(4):223--234, 1992.

Digital Library

[23]

K. Kedem, R. Livne, J. Pach, and M. Sharir, On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles, Discrete Comput. Geom., 1:59--71, 1986.

Digital Library

[24]

J. Matoušek, N. Miller, J. Pach, M. Sharir, S. Sifrony, and E. Welzl, Fat triangles determine linearly many holes, Proc. 32nd Annu. IEEE Sympos. Found. Comput. Sci., pp. 49--58, 1991.

Digital Library

[25]

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

Digital Library

[26]

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

Digital Library

[27]

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

Digital Library

[28]

M. van Kreveld, On fat partitioning, fat covering and the union size of polygons, Comput. Geom. Theory Appl., 9(4):197--210, 1998.

Digital Library

[29]

K. Varadarajan, Epsilon nets and union complexity, Proc. 25th Annu. Sympos. Comput. Geom., pp. 11--16, 2009.

Digital Library

[30]

K. Varadarajan, Weighted geometric set cover via quasi-uniform sampling, Proc. 42th Annu. ACM Symp. Theory Comput., pp. 641--648, 2010.

Digital Library

Cited By

Dutta KGhosh AJartoux BMustafa N(2019)Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial PartitioningDiscrete & Computational Geometry10.1007/s00454-019-00075-061:4(756-777)Online publication date: 1-Jun-2019
https://dl.acm.org/doi/10.1007/s00454-019-00075-0
Ezra Eda Fonseca GLewiner TPeñaranda LChan TKlein R(2013)Small-size relative (p,ε)-approximations for well-behaved range spacesProceedings of the twenty-ninth annual symposium on Computational geometry10.1145/2462356.2462363(233-242)Online publication date: 17-Jun-2013
https://dl.acm.org/doi/10.1145/2462356.2462363
Schulz ATóTh C(2013)The union of colorful simplices spanned by a colored point setComputational Geometry: Theory and Applications10.1016/j.comgeo.2012.01.00646:5(574-590)Online publication date: 1-Jul-2013
https://dl.acm.org/doi/10.1016/j.comgeo.2012.01.006
Show More Cited By

Improved bound for the union of fat triangles
1. Theory of computation
  1. Randomness, geometry and discrete structures

Recommendations

On the Boundary Complexity of the Union of Fat Triangles

A triangle is said to be {\it $\delta$-fat\/} if its smallest angle is at least $\delta>0$. A connected component of the complement of the union of a family of triangles is called a {\it hole}. It is shown that any family of n $\delta$-fat triangles in ...
Fat triangles determine linearly many holes
SFCS '91: Proceedings of the 32nd annual symposium on Foundations of computer science

It is shown that for every fixed delta >0 the following holds: if F is a union of n triangles, all of whose angles are at least delta , then the complement of F has O(n) connected components, and the boundary of F consists of O(n log log n) segments. ...
Threesomes, Degenerates, and Love Triangles
Distributed Computing, Cryptography, Distributed Computing, Cryptography, Coding Theory, Automata Theory, Complexity Theory, Programming Languages, Algorithms, Invited Paper Foreword and Databases

The 3SUM problem is to decide, given a set of n real numbers, whether any three sum to zero. It is widely conjectured that a trivial O(n²)-time algorithm is optimal on the Real RAM, and optimal even in the nonuniform linear decision tree model. Over the ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

SODA '11: Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms

January 2011

1785 pages

Program Chair:
Dana Randall
Georgia Institute of Technology

Sponsors

SIAM Activity Group on Discrete Mathematics
SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 23 January 2011

Check for updates

Qualifiers

Research-article

Conference

SODA '11

Sponsor:

SIGACT

SODA '11: 22nd ACM-SIAM Symposium on Discrete Algorithms

January 23 - 25, 2011

California, San Francisco

Acceptance Rates

Overall Acceptance Rate 411 of 1,322 submissions, 31%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

7
Total Citations
View Citations
71
Total Downloads

Downloads (Last 12 months)1
Downloads (Last 6 weeks)1

Reflects downloads up to 14 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Dutta KGhosh AJartoux BMustafa N(2019)Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial PartitioningDiscrete & Computational Geometry10.1007/s00454-019-00075-061:4(756-777)Online publication date: 1-Jun-2019
https://dl.acm.org/doi/10.1007/s00454-019-00075-0
Ezra Eda Fonseca GLewiner TPeñaranda LChan TKlein R(2013)Small-size relative (p,ε)-approximations for well-behaved range spacesProceedings of the twenty-ninth annual symposium on Computational geometry10.1145/2462356.2462363(233-242)Online publication date: 17-Jun-2013
https://dl.acm.org/doi/10.1145/2462356.2462363
Schulz ATóTh C(2013)The union of colorful simplices spanned by a colored point setComputational Geometry: Theory and Applications10.1016/j.comgeo.2012.01.00646:5(574-590)Online publication date: 1-Jul-2013
https://dl.acm.org/doi/10.1016/j.comgeo.2012.01.006
Chan TGrant EKönemann JSharpe M(2012)Weighted capacitated, priority, and geometric set cover via improved quasi-uniform samplingProceedings of the twenty-third annual ACM-SIAM symposium on Discrete algorithms10.5555/2095116.2095241(1576-1585)Online publication date: 17-Jan-2012
https://dl.acm.org/doi/10.5555/2095116.2095241
Ene AHar-Peled SRaichel BDey TWhitesides S(2012)Geometric packing under non-uniform constraintsProceedings of the twenty-eighth annual symposium on Computational geometry10.1145/2261250.2261253(11-20)Online publication date: 17-Jun-2012
https://dl.acm.org/doi/10.1145/2261250.2261253
Pach JTardos GHurtado Fvan Kreveld M(2011)Tight lower bounds for the size of epsilon-netsProceedings of the twenty-seventh annual symposium on Computational geometry10.1145/1998196.1998271(458-463)Online publication date: 13-Jun-2011
https://dl.acm.org/doi/10.1145/1998196.1998271
Pettie SHurtado Fvan Kreveld M(2011)On the structure and composition of forbidden sequences, with geometric applicationsProceedings of the twenty-seventh annual symposium on Computational geometry10.1145/1998196.1998258(370-379)Online publication date: 13-Jun-2011
https://dl.acm.org/doi/10.1145/1998196.1998258

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents