The power of geometric duality
BIT - Ellis Horwood series in artificial intelligence
Constructing arrangements of lines and hyperplanes with applications
SIAM Journal on Computing
Topologically sweeping an arrangement
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
Discrete Mathematics - Topological, algebraical and combinatorial structures; Froli´k's memorial volume
An optimal algorithm for intersecting line segments in the plane
Journal of the ACM (JACM)
An optimal algorithm for finding segments intersections
Proceedings of the eleventh annual symposium on Computational geometry
Topologically sweeping the visibility complex of polygonal scenes
Proceedings of the eleventh annual symposium on Computational geometry
On degeneracy in geometric computations
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Plane-sweep algorithms for intersecting geometric figures
Communications of the ACM
An optimal real-time algorithm for planar convex hulls
Communications of the ACM
A unified scheme for detecting fundamental curves in binary edge images
Computational Geometry: Theory and Applications
Introduction to Algorithms
Lectures on Discrete Geometry
Shattering a set of objects in 2D
Discrete Applied Mathematics
Topological Sweep in Degenerate Cases
ALENEX '02 Revised Papers from the 4th International Workshop on Algorithm Engineering and Experiments
Efficient computation of location depth contours by methods of computational geometry
Statistics and Computing
Algorithms for bivariate medians and a Fermat--Torricelli problem for lines
Computational Geometry: Theory and Applications - Special issue on the thirteenth canadian conference on computational geometry - CCCG'01
Algorithms for Reporting and Counting Geometric Intersections
IEEE Transactions on Computers
Comments on "algorithms for reporting and counting geometric intersections"
IEEE Transactions on Computers
Reliable and efficient geometric computing
CIAC'06 Proceedings of the 6th Italian conference on Algorithms and Complexity
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We present a novel, simple and easily implementable algorithm to report all intersections in an embedding of a complete graph. For graphs with N vertices and complexity K measured as the number of segments of the embedding, the running time of the algorithm is @Q(K+NM), where M is the maximum number of edges cut by any vertical line. Our algorithm handles degeneracies, such as vertical edges or multiply intersecting edges, without requiring numerical perturbations to achieve general position. The algorithm is based on the sweep line technique, one of the most fundamental techniques in computational geometry, where an imaginary line passes through a given set of geometric objects, usually from left to right. The algorithm sweeps the graph using a topological line, borrowing the concept of horizon trees from the topological sweep method [H. Edelsbrunner, L.J. Guibas, Topologically sweeping an arrangement, J. Comput. Syst. Sci. 38 (1989) 165-194; J. Comput. Syst. Sci. 42 (1991) 249-251 (corrigendum)]. The novelty in our approach is to control the topological line through the use of the moving wall that separates at any time the graph into two regions: the region of known structure, in front of the moving wall, and the region that may contain intersections generated by edges-that have not yet been registered in the sweep process-behind the wall. Our method has applications to graph drawing and for depth-based statistical analysis, for computing the simplicial depth median for a set of N data points [G. Aloupis, S. Langerman, M. Soss, G. Toussaint, Algorithms for bivariate medians and a Fermat-Torricelli problem for lines, Comp. Geom. Theory Appl. 26 (1) (2003) 69-79]. We present the algorithm, its analysis, experimental results and extension of the method to general graphs.