Reporting and counting segment intersections
Journal of Computer and System Sciences
Constructing arrangements of lines and hyperplanes with applications
SIAM Journal on Computing
Applications of random sampling in computational geometry, II
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
A fast planar partition algorithm, II
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
An efficient algorithm for hidden surface removal
SIGGRAPH '89 Proceedings of the 16th annual conference on Computer graphics and interactive techniques
Algorithms for Reporting and Counting Geometric Intersections
IEEE Transactions on Computers
Geometric applications of Davenport-Schinzel sequences
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
An optimal algorithm for intersecting line segments in the plane
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
A fast planar partition algorithm, II
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
An efficient algorithm for hidden surface removal
SIGGRAPH '89 Proceedings of the 16th annual conference on Computer graphics and interactive techniques
Tail estimates for the space complexity of randomized incremental algorithms
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
Computing many faces in arrangements of lines and segments
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Computing envelopes in four dimensions with applications
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Rounding arrangements dynamically
Proceedings of the eleventh annual symposium on Computational geometry
Linear-time triangulation of a simple polygon made easier via randomization
Proceedings of the sixteenth annual symposium on Computational geometry
Entropy-preserving cuttings and space-efficient planar point location
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
A simple entropy-based algorithm for planar point location
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
The design and implementation of panar maps in CGAL
Journal of Experimental Algorithmics (JEA)
Reporting intersecting pairs of convex polytopes in two and three dimensions
Computational Geometry: Theory and Applications
Efficient Expected-Case Algorithms for Planar Point Location
SWAT '00 Proceedings of the 7th Scandinavian Workshop on Algorithm Theory
Reporting Intersecting Pairs of Polytopes in Two and Three Dimensions
WADS '01 Proceedings of the 7th International Workshop on Algorithms and Data Structures
The Design and Implementation of Planar Maps in CGAL
WAE '99 Proceedings of the 3rd International Workshop on Algorithm Engineering
High-Level Filtering for Arrangements of Conic Arcs
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Incremental constructions con BRIO
Proceedings of the nineteenth annual symposium on Computational geometry
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
A simple entropy-based algorithm for planar point location
ACM Transactions on Algorithms (TALG)
Advanced programming techniques applied to Cgal's arrangement package
Computational Geometry: Theory and Applications
Distribution-sensitive point location in convex subdivisions
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
The complexity of the outer face in arrangements of random segments
Proceedings of the twenty-fourth annual symposium on Computational geometry
Markov incremental constructions
Proceedings of the twenty-fourth annual symposium on Computational geometry
Farthest-polygon Voronoi diagrams
ESA'07 Proceedings of the 15th annual European conference on Algorithms
A new point containment test algorithm based on preprocessing and determining triangles
Computer-Aided Design
Farthest-polygon Voronoi diagrams
Computational Geometry: Theory and Applications
A static optimality transformation with applications to planar point location
Proceedings of the twenty-seventh annual symposium on Computational geometry
Guarding art galleries by guarding witnesses
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
Farthest voronoi diagrams under travel time metrics
WALCOM'12 Proceedings of the 6th international conference on Algorithms and computation
Entropy, triangulation, and point location in planar subdivisions
ACM Transactions on Algorithms (TALG)
Higher order city voronoi diagrams
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Improved implementation of point location in general two-dimensional subdivisions
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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In this paper we give a fast randomized algorithm for finding a partition of the plane induced by a given set of line segments. The planar partition problem is interesting because it has important practical implications, especially in computer graphics, which indeed served as an inspiration for our investigation. Our algorithm is ideally suited for a practical use because: (i) it is extremely simple and robust, and (ii) despite this simplicity (or rather because of it) the algorithm is optimal; its expected running time is O(m+n log n), where n is the number of input segments and m is the number of points of intersection. The storage requirement is O(m + n). Though the algorithm itself is simple, the global evolution of the underlying partition is non-trivial, which makes the analysis of the algorithm theoretically interesting in its own right.