Voronoi diagrams and arrangements
Discrete & 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
A fast planar partition algorithm, I
Journal of Symbolic Computation
An algorithm for generalized point location and its applications
Journal of Symbolic Computation
A fast planar partition algorithm, II
Journal of the ACM (JACM)
Efficient point location in a convex spatial cell-complex
SIAM Journal on Computing
Applications of random sampling to on-line algorithms in computational geometry
Discrete & Computational Geometry
New bounds for lower envelopes in three dimensions, with applications to visibility in terrains
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Ray shooting and parametric search
SIAM Journal on Computing
Computing a face in an arrangement of line segments and related problems
SIAM Journal on Computing
Applications of parametric searching in geometric optimization
SODA selected papers from the third annual ACM-SIAM symposium on Discrete algorithms
On lazy randomized incremental construction
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Applying Parallel Computation Algorithms in the Design of Serial Algorithms
Journal of the ACM (JACM)
Computating the width of a set
SCG '85 Proceedings of the first annual symposium on Computational geometry
On Range Searching with Semialgebraic Sets
MFCS '92 Proceedings of the 17th International Symposium on Mathematical Foundations of Computer Science
Efficient randomized algorithms for some geometric optimization problems
Proceedings of the eleventh annual symposium on Computational geometry
Efficient approximation and optimization algorithms for computational metrology
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Line transversals of balls and smallest enclosing cylinders in three dimensions
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Anisotropic voronoi diagrams and guaranteed-quality anisotropic mesh generation
Proceedings of the nineteenth annual symposium on Computational geometry
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Let F be a collection of n d-variate, possibly partially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2-faces of the lower envelope (i.e., pointwise minimum) of F in expected time O(nd+&egr;), for any &egr;0. For d=3, by combining this algorithm with the point location technique of Preparata and Tamassia, we can compute, in randomized expected time O(n3+&egr;) for any &egr;0, a data structure of size O(n3+&egr;) that, given any query point q, can determine in O(log2n) time whether q lies above, below or on the envelope. As a consequence, we obtain improved algorithmic solutions to many problems in computational geometry, including (a) computing the width of a point set in 3-space, (b) computing the biggest stick in a simple polygon in the plane, and (c) computing the smallest-width annulus covering a planar point set. The solutions to these problems run in time O(n17/11+&egr;), for any &egr;0 improving previous solutions that run in time O(n8/5+&egr;). We also present data structures for (i) performing nearest-neighbor and related queries for fairly general collections of objects in 3-space and for collections of moving objects in the plane, and (ii) performing ray-shooting and related queries among n spheres or more general objects in 3-space. Both of these data structures require O(n3+&egr;) storage and preprocessing time, for any &egr;0, and support polylogarithmic-time queries. These structures improve previous solutions to these problems.