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31
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 ..."
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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.
Finding Minimum Area kgons
 DISCRETE & COMPUTATIONAL GEOMETRY
, 1992
"... Given a set P of n points in the plane and a number k, we want to find a polygon ~ with vertices in P of minimum area that satisfies one of the following properties: (1) cK is a convex kgon, (2) ~ is an empty convex kgon, or (3) ~ is the convex hull of exactly k points of P. We give algorithms ..."
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Cited by 23 (5 self)
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Given a set P of n points in the plane and a number k, we want to find a polygon ~ with vertices in P of minimum area that satisfies one of the following properties: (1) cK is a convex kgon, (2) ~ is an empty convex kgon, or (3) ~ is the convex hull of exactly k points of P. We give algorithms for solving each of these three problems in time O(kn3). The space complexity is O(n) for k = 4 and O(kn 2) for k> 5. The algorithms are based on a dynamic ptogramming approach. We generalize this approach to polygons with minimum perimeter, polygons with maximum perimeter or area, polygons containing the maximum or minimum number of points, polygons with minimum weight (for some weights added to vertices), etc., in similar time bounds.
Empty Convex Hexagons in Planar Point Sets
, 2007
"... Erdős asked whether every sufficiently large set of points in general position in the plane contains six points that form a convex hexagon without any points from the set in its interior. Such a configuration is called an empty convex hexagon. In this paper, we answer the question in the affirmativ ..."
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Cited by 18 (0 self)
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Erdős asked whether every sufficiently large set of points in general position in the plane contains six points that form a convex hexagon without any points from the set in its interior. Such a configuration is called an empty convex hexagon. In this paper, we answer the question in the affirmative. We show that every set that contains the vertex set of a convex 9gon also contains an empty convex hexagon.
Planar Point Sets With a Small Number of Empty Convex Polygons
, 2004
"... A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P . We construct a set of n points in general position in the plane with only 1:62n² empty triangles, 1:94 ..."
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Cited by 13 (1 self)
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A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P . We construct a set of n points in general position in the plane with only 1:62n² empty triangles, 1:94n² empty quadrilaterals, 1:02n² empty pentagons, and 0:2n² empty hexagons.
Chromatic Variants of the ErdösSzekeres Theorem on Points in Convex Position
, 2002
"... ... In this paper we study several variants on problems related to the ErdösSzekeres Theorem about subsets of S in convex position, when additional chromatic constraints are considered. ..."
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Cited by 10 (3 self)
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... In this paper we study several variants on problems related to the ErdösSzekeres Theorem about subsets of S in convex position, when additional chromatic constraints are considered.
New Algorithms for Minimum Area kgons
, 1991
"... Given a set P of n points in the plane, we wish to find a set Q P of k points for which the convex hull conv(Q) has the minimum area. ..."
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Cited by 9 (1 self)
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Given a set P of n points in the plane, we wish to find a set Q P of k points for which the convex hull conv(Q) has the minimum area.
Finding Sets of Points Without Empty Convex 6Gons
 Discrete Comput. Geom
, 2001
"... Erdos asked whether every large enough set of points in general position in the plane contains six points that form a convex 6gon without any points from the set in its interior. In this note we show how a set of 29 points was found that contains no empty convex 6gon. To this end a fast increme ..."
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Cited by 8 (0 self)
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Erdos asked whether every large enough set of points in general position in the plane contains six points that form a convex 6gon without any points from the set in its interior. In this note we show how a set of 29 points was found that contains no empty convex 6gon. To this end a fast incremental algorithm for finding such 6gons was designed and implemented and a heuristic search approach was used to find promising sets. Also some observations are made that might be useful in proving that large sets always contain an empty convex 6gon. 1 The problem Given a set of points in the plane, no three of which are collinear, we consider subsets of of cardinality that lie in convex position, i.e., they form the vertices of some convex gon, and there is no point of in the interior of this polygon (the polygon is empty). Let t denote the smallest number such that any set of cardinality at least a contains some subset . Erdos [3] proposed the study of...
On the reflexivity of point sets
 Discrete and Computational Geometry: The GoodmanPollack Festschrift, 139–156
, 2003
"... Abstract. We introduce a new measure for planar point sets S. Intuitively, it describes the combinatorial distance from a convex set: The reflexivity ρ(S) ofS is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide effic ..."
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Cited by 8 (1 self)
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Abstract. We introduce a new measure for planar point sets S. Intuitively, it describes the combinatorial distance from a convex set: The reflexivity ρ(S) ofS is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study naturally takes us into the examination of some closely related quantities, such as the convex cover number κ1(S) of a planar point set, which is the smallest number of convex chains that cover S, and the convex partition number κ2(S), which is given by the smallest number of disjoint convex chains that cover S. We prove that it is NPcomplete to determine the convex cover or the convex partition number, and we give logarithmicapproximation algorithms for determining each. 1
Planar Sets With Few Empty Convex Polygons
, 2000
"... A configuration of n points in general position in the plane is described which has less than 1.684n² empty triangles, less than 2.132n² empty convex quadrilaterals, less than 1.229n² empty convex pentagons and less than 0.298n² empty convex hexagons. This improves the constants ..."
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Cited by 7 (0 self)
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A configuration of n points in general position in the plane is described which has less than 1.684n² empty triangles, less than 2.132n² empty convex quadrilaterals, less than 1.229n² empty convex pentagons and less than 0.298n² empty convex hexagons. This improves the constants from a previous construction given by Valtr.