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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.
Optimal Placement of Convex Polygons to Maximize Point Containment
, 1996
"... Given a convex polygon P with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P that contains the maximum number of points in S. We allow both translation and rotation. Our algorithm is selfcontained and utilizes the geometric properties of the co ..."
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Cited by 6 (1 self)
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Given a convex polygon P with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P that contains the maximum number of points in S. We allow both translation and rotation. Our algorithm is selfcontained and utilizes the geometric properties of the containing regions in the parameter space of transformations. The algorithm requires O(nk 2 m 2 log(mk)) time and O(n +m) space, where k is the maximum number of points contained. This provides a linear improvement over the best previously known algorithm when k is large (\Theta(n)) and a cubic improvement when k is small. We also show that the algorithm can be extended to solve bichromatic and general weighted variants of the problem. 1 Introduction A planar rigid motion ae is an affine transformation of the plane that preserves distance (and therefore angles and area also). We say that a polygon P contains a set S of points if every point in S lies on P or in the interior of P . In th...
Randomized Algorithms for Geometric Optimization Problems
, 2000
"... This chapter reviews randomization algorithms developed in the last few years to solve a wide range of geometric optimization problems. We review a number of general techniques, including randomized binary search, randomized linearprogramming algorithms, and random sampling. Next, we describe sever ..."
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This chapter reviews randomization algorithms developed in the last few years to solve a wide range of geometric optimization problems. We review a number of general techniques, including randomized binary search, randomized linearprogramming algorithms, and random sampling. Next, we describe several applications of these techniques, including facility location, proximity problems, nearest neighbor searching, statistical estimators, and Euclidean TSP.