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Iterated Nearest Neighbors and Finding Minimal Polytopes
, 1994
"... Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based o ..."
Abstract

Cited by 57 (6 self)
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Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the first time to efficiently maintain minimal sets as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convex kvertex polygons and polytopes, and to improvemany previous results. Weachievemany of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in time O(n log n+kn). We also demonstrate a related technique for finding minimum area kpoint sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.
Algorithms for Some Intersection Searching Problems Involving Curved Objects
, 1995
"... Two classes of geometric intersection searching problems are considered, i.e., problems in which a set S of geometric objects is to be preprocessed into a data structure so that for any query object q, the objects of S that are intersected by q can be counted or reported efficiently. In the first ..."
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Cited by 6 (1 self)
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Two classes of geometric intersection searching problems are considered, i.e., problems in which a set S of geometric objects is to be preprocessed into a data structure so that for any query object q, the objects of S that are intersected by q can be counted or reported efficiently. In the first class, S is a set of curved objects, such as dballs, dspheres, circles, or circular arcs, and q is also a curved object. In the second class, the objects in S are curved or linear and each is assigned a color. Given a query q, such as a disk or an annulus, the goal is to count or report the distinct colors in the set of objects intersected by q. Efficient algorithms are presented for several problems from these classes. The solution techniques are based on geometric transforms, on compositions of known solutions for simplex range searching, on the locus approach, and on persistent data structures. Keywords: Computational geometry, data structures, intersection searching. 1 Introdu...