Computational geometry: an introduction
Computational geometry: an introduction
Finding the minimum visible vertex distance between two non-intersecting simple polygons
SCG '86 Proceedings of the second annual symposium on Computational geometry
On a multidimensional search technique and its application to the Euclidean one centre problem
SIAM Journal on Computing
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Information and Computation
Finding minimal convex nested polygons
Information and Computation
Introduction to algorithms
Covering convex sets with non-overlapping polygons
Discrete Mathematics
Minimum vertex hulls for polyhedral domains
STACS 90 Proceedings of the seventh annual symposium on Theoretical aspects of computer science
Determining the separation of preprocessed polyhedra: a unified approach
Proceedings of the seventeenth international colloquium on Automata, languages and programming
Linear Programming in Linear Time When the Dimension Is Fixed
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Approximating Polygons and Subdivisions with Minimum Link Paths
ISA '91 Proceedings of the 2nd International Symposium on Algorithms
Approximation schemes in computational geometry
Approximation schemes in computational geometry
Almost optimal set covers in finite VC-dimension: (preliminary version)
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Efficient piecewise-linear function approximation using the uniform metric: (preliminary version)
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Almost optimal polyhedral separators
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
Circular separability of polygon
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Surface approximation and geometric partitions
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Controlled Topology Simplification
IEEE Transactions on Visualization and Computer Graphics
Automatic generation of triangular irregular networks using greedy cuts
VIS '95 Proceedings of the 6th conference on Visualization '95
VC-Dimension of Exterior Visibility
IEEE Transactions on Pattern Analysis and Machine Intelligence
A space-optimal data-stream algorithm for coresets in the plane
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Succinct approximate convex pareto curves
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
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Given a family of disjoint polygons P1, P2,…, Pk in the plane, and an integer parameter m, it is NP-complete to decide if the Pi's can be separated by a polygonal family consisting of m edges, that is, if there exist polygons R1, R2,…, Rk with pairwise-disjoint boundaries such that Pi *** Ri and &Sgr;|Ri| ≤ m. In three dimensions, the problem of separating even two nested convex polyhedra by a k-facet polyhedron is NP-complete. Many other extensions and generalizations of the polyhedral separation problem, either to families of polyhedra or to higher dimensions, are also intractable.In this paper, we present efficient approximation algorithms for constructing separating families of near-optimal size. Our main results are as follows. In two dimensions, we give an O(n log n) time algorithm for constructing a separating family whose size is within a constant factor of an optimal separating family; n is the number of edges in the input family of polygons. In three dimensions, we can separate a convex polyhedron from a nonconvex polyhedron with a convex polyhedral surface whose facet-complexity is O(log n, times the optimal, where n = |P|+|Q| is the complexity of the input polyhedra. Our algorithm runs in O(n4) time, but improves to O (n3) time if the two polyhedra are nested and convex.Our algorithm for separating a convex polyhedron from a nonconvex polyhedron extends to higher dimensions. In d dimensions, for d ≥ 4, the facet-complexity of the approximation polyhedron is O(d log n) times the optimal, and the algorithm runs in O(nd+1)time. Finally, we also obtain results on separating sets of points, a family of convex polyhedra, and separation by non-polyhedral surfaces, such as spherical patches.