Plane-sweep algorithms for intersecting geometric figures
Communications of the ACM
Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Worst-case optimal algorithms for constructing visibility polygons with holes
SCG '86 Proceedings of the second annual symposium on Computational geometry
Efficient plane sweeping in parallel
SCG '86 Proceedings of the second annual symposium on Computational geometry
Parallel geometric algorithms on mesh-connected computers
ACM '87 Proceedings of the 1987 Fall Joint Computer Conference on Exploring technology: today and tomorrow
Intersecting line segments in parallel with an output-sensitive number of processors
SPAA '89 Proceedings of the first annual ACM symposium on Parallel algorithms and architectures
Parallel algorithms for arrangements
SPAA '90 Proceedings of the second annual ACM symposium on Parallel algorithms and architectures
Parallel methods for visibility and shortest path problems in simple polygons (preliminary version)
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Constructing arrangements optimally in parallel (preliminary version)
SPAA '91 Proceedings of the third annual ACM symposium on Parallel algorithms and architectures
Line-segment intersection made in-place
Computational Geometry: Theory and Applications
Computational Geometry on a Systolic Chip
IEEE Transactions on Computers
New algorithms for computer graphics
EGGH'87 Proceedings of the Second Eurographics conference on Advances in Computer Graphics Hardware
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This paper settles a long-standing open question of computational geometry: Is it possible to compute all k intersections between n arbitrary line segments in time linear in k? We answer this question affirmatively by presenting the first algorithm with a running time of the form O(k + f(n)), where f is a subquadratic function of n. The function f we achieve is actually quasi-linear in n, which makes our algorithm the most efficient to date for each value of k. To obtain this result we must turn away from traditional, sweep-line-based schemes. Instead, we introduce a new hierarchical strategy for dealing with segments without ever reducing the dimensionality of the problem. This framework is used to solve other related problems. In particular, we are able to present the first subquadratic algorithm for counting intersections (as opposed to reporting each of them explicitly), and we give the first optimal algorithm for computing the intersections of a line arrangement with a query segment. Using duality arguments we also present an improved algorithm for a point enclosure problem.