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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 56 (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.
A SemiDynamic Construction of Higher Order Voronoï Diagrams and its Randomized Analysis
, 1993
"... The kDelaunay tree extends the Delaunay tree introduced in [1,2]. It is a hierarchical data structure that allows the semidynamic construction of the higher order Vorono diagrams of a finite set of n points in any dimension. In this paper, we prove that a randomized construction of the kDelaunay ..."
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Cited by 6 (3 self)
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The kDelaunay tree extends the Delaunay tree introduced in [1,2]. It is a hierarchical data structure that allows the semidynamic construction of the higher order Vorono diagrams of a finite set of n points in any dimension. In this paper, we prove that a randomized construction of the kDelaunay tree, and thus, of all the order < k Vorono diagrams, can be done in O(n log n1 k3n) expected time and O(k2t expected storage in the plane, which is asymptotically optimal for fixed k. Our algorithm extends to d dimensional space with expected time complexity o and space complexity O The algorithn is simple and experimental results are given.