Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Reporting points in halfspaces
Computational Geometry: Theory and Applications
Discrete & Computational Geometry - Special issue on ACM symposium on computational geometry, North Conway
Ray shooting and parametric search
SIAM Journal on Computing
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Ray Shooting Amidst Spheres in Three Dimensions and Related Problems
SIAM Journal on Computing
Computing Envelopes in Four Dimensions with Applications
SIAM Journal on Computing
Voronoi diagrams of lines in 3-space under polyhedral convex distance functions
Journal of Algorithms - Special issue on SODA '95 papers
Vertical Decomposition of Shallow Levels in 3-Dimensional Arrangements and Its Applications
SIAM Journal on Computing
3-Dimensional Euclidean Voronoi Diagrams of Lines with a Fixed Number of Orientations
SIAM Journal on Computing
A Singly-Expenential Stratification Scheme for Real Semi-Algebraic Varieties and Its Applications
ICALP '89 Proceedings of the 16th International Colloquium on Automata, Languages and Programming
Almost Tight Upper Bounds for Vertical Decompositions in Four Dimensions
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Ray shooting and stone throwing with near-linear storage
Computational Geometry: Theory and Applications
Ray shooting and intersection searching amidst fat convex polyhedra in 3-space
Proceedings of the twenty-second annual symposium on Computational geometry
Ray shooting and intersection searching amidst fat convex polyhedra in 3-space
Computational Geometry: Theory and Applications
Extremal point queries with lines and line segments and related problems
Computational Geometry: Theory and Applications
Semialgebraic Range Reporting and Emptiness Searching with Applications
SIAM Journal on Computing
A simple framework for the generalized nearest neighbor problem
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
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In the range emptiness searching problem, we are given a set P of n points in Rd, and wish to preprocess them into a data structure that supports efficient range emptiness queries, in which we specify a range σ, which is a semi-algebraic set in Rd of some fixed kind, and wish to determine whether P ∩ σ = θ. Range emptiness searching arises in many applications, and has been treated by Matoušek [15] in the special case where the ranges are halfspaces bounded by hyperplanes. In this paper we extend the analysis to the general semi-algebraic case, and show how to adapt Matoušek's technique to this case, without the need to linearize the ranges into a higher-dimensional space. This yields more efficient solutions to several interesting problems, and we demonstrate the new technique in two applications:(i) An algorithm for ray shooting amid balls in R3, which uses O* (n) storage and preprocessing, and answers a query in O* (n2/3) time, improving the previous bound of O* (n3/47).(ii) An algorithm that preprocesses, in O*(n) time, a set P of n points in R3 into a data structure with O*(n) storage, so that, for any query line l or segment e, the point of P farthest from l or from e can be computed in O* (n1/2) time.Our technique is closely related to the notions of nearest- or farthest-neighbor generalized Voronoi diagrams, and of the union or intersection of geometric objects, where sharper bounds on the combinatorial complexity of these structures yield faster range emptiness searching algorithms. For example, in the case of ray shooting amid balls, the structure that arises in our algorithm is the Euclidean Voronoi diagram of lines in 3-space, and the performance of the algorithm depends on the complexity of such diagrams (for which tight bounds are still unknown).