Planar point location using persistent search trees
Communications of the ACM
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Journal of Algorithms
Halfplanar range search in linear space and O(n0.695) query time
Information Processing Letters
Worst-case optimal algorithms for constructing visibility polygons with holes
SCG '86 Proceedings of the second annual symposium on Computational geometry
Quadratic bounds for hidden line elimination
SCG '86 Proceedings of the second annual symposium on 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
Worst-case optimal hidden-surface removal
ACM Transactions on Graphics (TOG)
Space searching for intersecting objects
Journal of Algorithms
Separating two simple polygons by a sequence of translations
Discrete & Computational Geometry
Making data structures persistent
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
Topologically sweeping an arrangement
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
Partition trees for triangle counting and other range searching problems
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
The complexity of many faces in arrangements of lines of segments
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Implicitly representing arrangements of lines or segments
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
A deterministic algorithm for partitioning arrangements of lines and its application
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Visibility and intersectin problems in plane geometry
SCG '85 Proceedings of the first annual symposium on Computational geometry
Arrangements of Curves in the Plane - Topology, Combinatorics, and Algorithms
ICALP '88 Proceedings of the 15th International Colloquium on Automata, Languages and Programming
A deterministic algorithm for partitioning arrangements of lines and its application
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Cutting hyperplane arrangements
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
Intersection queries for curved objects (extended abstract)
SCG '91 Proceedings of the seventh 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
Shortest paths among obstacles in the plane
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Processing an Offline Insertion-Query Sequence with Applications
FAW '09 Proceedings of the 3d International Workshop on Frontiers in Algorithmics
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In this paper we consider the following problem: Given a set @@@@ of n (possibly intersecting) line segments in the plane, preprocess them so that, given a query ray &rgr; emanating from a point p, we can quickly compute the intersection point &PHgr;(@@@@, &rgr;) of &rgr; with a segment of @@@@ that lies nearest to p. We present an algorithm that preprocesses @@@@, in time &Ogr;(n3/2 log&ohgr; n), into a data structure of size &Ogr;(n&agr;(n) log4 n), so that for a query ray &rgr;, &PHgr;(@@@@, &rgr;) can be computed in &Ogr;(√nlog3 n), where &ohgr; is a constant n) is a functional inverse of Ackermann's function. If the given segments are non-intersecting, the storage goes down to &Ogr;(nlog3 n) and the query time is only &Ogr;(√n log2 n). The main tool that we use is spanning trees with low stabbing number, i.e. with the property that no line intersects more than &Ogr;(√n) edges of the tree. Using such trees we obtain faster algorithms for several other problems, including implicit point location, polygon containment and implicit hidden surface removal.