Efficient algorithms for geometric optimization
ACM Computing Surveys (CSUR)
Accurate computation of the medial axis of a polyhedron
Proceedings of the fifth ACM symposium on Solid modeling and applications
Approximation and exact algorithms for minimum-width annuli and shells
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Motion planning of a ball amid segments in three dimensions
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus
Proceedings of the sixteenth annual symposium on Computational geometry
Recent Developments in the Theory of Arrangements of Surfaces
Proceedings of the 19th Conference on Foundations of Software Technology and Theoretical Computer Science
Exact computation of the medial axis of a polyhedron
Computer Aided Geometric Design
On lines avoiding unit balls in three dimensions
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Almost tight upper bounds for vertical decompositions in four dimensions
Journal of the ACM (JACM)
Ray shooting amid balls, farthest point from a line, and range emptiness searching
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Finding large sticks and potatoes in polygons
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
On overlays and minimization diagrams
Proceedings of the twenty-second annual symposium on Computational geometry
Voronoi Diagram of Polygonal Chains under the Discrete Fréchet Distance
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Exact Algorithms for the Bottleneck Steiner Tree Problem
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Querying Two Boundary Points for Shortest Paths in a Polygonal Domain
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
FSTTCS'04 Proceedings of the 24th international conference on Foundations of Software Technology and Theoretical Computer Science
Querying two boundary points for shortest paths in a polygonal domain
Computational Geometry: Theory and Applications
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Let ${\cal F}$ 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 ${\cal F}$ in expected time $O(n^{d+\epsilon})$ for any $\epsilon 0$. For d = 3, by combining this algorithm with the point-location technique of Preparata and Tamassia, we can compute, in randomized expected time $O(n^{3+\epsilon})$, for any $\epsilon 0$, a data structure of size $O(n^{3+\epsilon})$ that, for any query point q, can determine in O(log2n) time the function(s) of ${\cal F}$ that attain the lower envelope at q. As a consequence, we obtain improved algorithmic solutions to several 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 randomized expected time $O(n^{17/11+\epsilon})$, for any $\epsilon 0$, improving previous solutions that run in time $O(n^{8/5+\epsilon})$. 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 $O(n^{3+\epsilon})$ storage and preprocessing time, for any $\epsilon 0$, and support polylogarithmic-time queries. These structures improve previous solutions to these problems.