The power of geometric duality
BIT - Ellis Horwood series in artificial intelligence
Computational geometry: an introduction
Computational geometry: an introduction
Planar point location using persistent search trees
Communications of the ACM
Halfplanar range search in linear space and O(n0.695) query time
Information Processing Letters
On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Optimal point location in a monotone subdivision
SIAM Journal on Computing
Constructing arrangements of lines and hyperplanes with applications
SIAM Journal on Computing
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Optimal shortest path queries in a simple polygon
SCG '87 Proceedings of the third annual symposium on Computational geometry
Partition trees for triangle counting and other range searching problems
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Visibility and intersectin problems in plane geometry
SCG '85 Proceedings of the first annual symposium on Computational geometry
Geometric transforms for fast geometric algorithms
Geometric transforms for fast geometric algorithms
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
Good splitters for counting points in triangles
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
Quasi-optimal upper bounds for simplex range searching and new zone theorems
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Minimum-link paths among obstacles in the plane
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Space-efficient ray-shooting and intersection searching: algorithms, dynamization, and applications
SODA '91 Proceedings of the second annual ACM-SIAM symposium on Discrete algorithms
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An arrangement of n lines (or line segments) in the plane is the partition of the plane defined by these objects. Such an arrangement consists of &Ogr;(n2) regions, called faces. In this paper we study the problem of calculating and storing arrangements implicitly, using subquadratic space and preprocessing, so that, given any query point p, we can calculate efficiently the face containing p. First, we consider the case of lines and show that with &Lgr;(n) space1 and &Lgr;(n3/2) preprocessing time, we can answer face queries in &Lgr;(√n) + &Ogr;(K) time, where K is the output size. (The query time is achieved with high probability.) In the process, we solve three interesting subproblems: 1) given a set of n points, find a straight-edge spanning tree of these points such that any line intersects only a few edges of the tree, 2) given a simple polygonal path &Ggr;, form a data structure from which we can find the convex hull of any subpath of &Ggr; quickly, and 3) given a set of points, organize them so that the convex hull of their subset lying above a query line can be found quickly. Second, using random sampling, we give a trade-off between increasing space and decreasing query time. Third, we extend our structure to report faces in an arrangement of line segments in &Lgr;(n1/3) time, given &Lgr;(n4/3) space and &Lgr;(n5/3) preprocessing time. Lastly, we note that our techniques allow us to compute m faces in an arrangement of n lines in time &Lgr;(m2/3n2/3 + n), which is nearly optimal.