A simple probablistic algorithm for approximating two and three-dimensional objects
نویسنده
چکیده
Approximating complex geometric objects with simple ones is an important problem in geometric computing (i.e. in GIS, graphics, image processing). Usually, given an error bound d, to approximate a geometric object O we make two \copies" of O (O1; O2) such that the Hausdro distance between them and O is bounded by d and then we simply compute the minimum size polyhedron (polygon in 2D) between O1; O2. (Recall that the Hausdro distance between two objects is the maxmindistance, or more formally the sup inf-distance, between all the points in the two objects.) In two dimension (2D), given a small error bound, optimal linear time algorithms are known to approximate simple polygonal objects [II86; II88; HS91]. All these algorithms are very similar to the result of Suri [Su86]. If the error bound is large such that O1; O2 become non-simple, Guibas et al. also present O(n log n) time algorithm to solve the problem [GHMS93]. In three dimension (3D), the situation is a little di erent. Even computing the minimum size convex polyhedron between a pair of convex polyhedra is NP-complete [DJ90; DJ92; DG97]. (We call a pair of nested convex polyhedra convex annulus throughout this paper.) Nevertheless, several approximation algorithms have been proposed to solve this problem and among them Mitchell and Suri rst proposed an O(n) time O(log n)factor approximate solution [MS95]. Later, Clarkson presented a simple randomized algorithm with the same approximate ratio [Cl93] and most recently Br onnimann and Goodrich obtained a constant factor approximate solution for this problem using a beautiful combination of set covers in nite VC-dimension and randomized \natural selection" algorithms [BG95]. However, their algorithm is very slow (O(n log n) time) when the optimal solution has size n since it uses the algorithm of Matous ek et al. as a subroutine which spends O(n) time to compute -nets in a set system [MSW90].
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