Computational geometry: an introduction
Computational geometry: an introduction
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SIAM Journal on Computing
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Journal of the ACM (JACM)
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Discrete & Computational Geometry
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Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
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Discrete & Computational Geometry
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STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
On the general motion planning problem with two degrees of freedom
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Triangles in space or building (and analyzing) castles in the Air
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
A deterministic algorithm for partitioning arrangements of lines and its application
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Ray shooting and other applications of spanning trees with low stabbing number
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Minimum-link paths among obstacles in the plane
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Stabbing and ray shooting in 3 dimensional space
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
An algorithm for generalized point location and its applications
Journal of Symbolic Computation
Efficient NC algorithms for set cover with applications to learning and geometry
Proceedings of the 30th IEEE symposium on Foundations of computer science
Region Extraction and Verification for Spatial and Spatio-temporal Databases
SSDBM 2009 Proceedings of the 21st International Conference on Scientific and Statistical Database Management
Ensuring the semantic correctness of complex regions
ER'07 Proceedings of the 2007 conference on Advances in conceptual modeling: foundations and applications
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We show that the total number of edges of m faces of an arrangement of n lines in the plane is &Ogr;(m2/3-&dgr; n2/3+2&dgr; + n), for any &dgr; 0. The proof takes an algorithmic approach, that is, we describe an algorithm for the calculation of these m faces and derive the upper bound from the analysis of the algorithm. The algorithm uses randomization and, with high probability, its time complexity is within a log2n factor of the above bound. If instead of lines we have an arrangement of n line segments, then the maximum number of edges of m faces is Ogr;(m2/3-&dgr;n2/3+2&dgr; + n&agr;(n)logm), for any &dgr; 0, where &agr;(n) is the functional inverse of Ackermann's function. We give a (randomized) algorithm that produces these faces and, with high probability, takes time that is within a log2n factor of the combinatorial bound.