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ClosestPoint Problems in Computational Geometry
, 1997
"... This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, th ..."
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Cited by 65 (14 self)
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This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, the exact and approximate postoffice problem, and the problem of constructing spanners are discussed in detail. Contents 1 Introduction 1 2 The static closest pair problem 4 2.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Algorithms that are optimal in the algebraic computation tree model . 5 2.2.1 An algorithm based on the Voronoi diagram . . . . . . . . . . . 5 2.2.2 A divideandconquer algorithm . . . . . . . . . . . . . . . . . . 5 2.2.3 A plane sweep algorithm . . . . . . . . . . . . . . . . . . . . . . 6 2.3 A deterministic algorithm that uses indirect addressing . . . . . . . . . 7 2.3.1 The degraded grid . . . . . . . . . . . . . . . . . . ...
Queries with Segments in Voronoi Diagrams
, 1999
"... In this paper we consider proximity problems in which the queries are line segments in the plane. We build a query structure that for a set of n points P can determine the closest point in P to a query segment outside the convex hull of P in O(log n) time. With this we solve the problem of computing ..."
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Cited by 10 (1 self)
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In this paper we consider proximity problems in which the queries are line segments in the plane. We build a query structure that for a set of n points P can determine the closest point in P to a query segment outside the convex hull of P in O(log n) time. With this we solve the problem of computing the closest point to each of n disjoint line segments in O(n log 3 n) time. Nearest foreign neighbors or Hausdorff distance for disjoint, colored segments can be computed in the same time. We explore some connections to Hopcroft's problem. 1 Introduction Since Knuth [13] posed the post office problem preprocess a set of points, or sites, in the plane to quickly report the nearest to a query pointand Shamos and Hoey [17] suggested Voronoi diagrams as a solution, there have been a number of proximity problems in the plane whose solution is to build some type of Voronoi diagram and query with a point. Note: A Voronoi diagram of a set of sites is the partition of the plane into maxim...
Finding a shortest diagonal of a simple polygon in linear time
 Computational Geometry: Theory and Applications
, 1993
"... Abstract A diagonal of a planar, simple polygon P is an open line segment that connects two nonadjacent vertices and lies in the relative interior of P. We present a linear time algorithm for finding a shortest diagonal (in the L2 norm) of a simple polygon, improving the previous best result by a f ..."
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Cited by 3 (1 self)
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Abstract A diagonal of a planar, simple polygon P is an open line segment that connects two nonadjacent vertices and lies in the relative interior of P. We present a linear time algorithm for finding a shortest diagonal (in the L2 norm) of a simple polygon, improving the previous best result by a factor of log n. Our result provides an interesting contrast to a known \Omega (n log n) lower bound for finding a closest pair of vertices in a simple polygonobserve that a shortest diagonal is defined by a closest pair of vertices satisfying an additional visibility constraint. 1 1 Introduction Closest pair problems have been studied in computational geometry for a long time. It is wellknown, for instance, that a closest pair among n points in the plane can be found in O(n log n) time, and that this bound is tight in the worst case in the algebraictree model of computation [7]. The lower bound holds even if the points form the vertices of a simple polygon, given in the boundary order [1].
Computing closest points for segments
 Int. J. Comput. Geom. Appl
"... Abstract We consider the proximity problem of computing for each of n line segments the closest point from a given set of n points in the plane. It generalizes Hopcroft's problem [11] and the nearest foreign neighbors problem [15]. We show that it can be solved in O(n ..."
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Cited by 1 (0 self)
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Abstract We consider the proximity problem of computing for each of n line segments the closest point from a given set of n points in the plane. It generalizes Hopcroft's problem [11] and the nearest foreign neighbors problem [15]. We show that it can be solved in O(n
Lower Bounds for Approximate Polygon Decomposition and Minimum Gap
"... We consider the problem of decomposing polygons (with holes) into various types of simpler polygons. We focus on the problem of partitioning a rectilinear poly gon, with holes, into rectangles, and show an F(n log n) lower bound on the time complexity. The result holds for any decomposition, optim ..."
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Cited by 1 (1 self)
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We consider the problem of decomposing polygons (with holes) into various types of simpler polygons. We focus on the problem of partitioning a rectilinear poly gon, with holes, into rectangles, and show an F(n log n) lower bound on the time complexity. The result holds for any decomposition, optimal or approximative. The bound matches the complexity of a number of algorithms in the literature, proving their optimality and settling the complexity of approximate polygon decomposition in these cases.
An Optimal Parallel Algorithm for the AllNearestForeignNeighbors Problem in Arbitrary Dimensions
"... Given a set S of n points in IR D , D 2. Each point p 2 S is assigned a color c(p) chosen from a fixed color set. The AllNearestForeignNeighbors Problem (ANFNP ) is to find for each point p 2 S its nearest foreign neighbors, i.e. the set of all points in Snfpg that are closest to p among the ..."
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Given a set S of n points in IR D , D 2. Each point p 2 S is assigned a color c(p) chosen from a fixed color set. The AllNearestForeignNeighbors Problem (ANFNP ) is to find for each point p 2 S its nearest foreign neighbors, i.e. the set of all points in Snfpg that are closest to p among the points in S with a color different from c(p). We introduce the Well Separated Color Decomposition (WSCD) which gives an optimal O(log n) parallel algorithm to solve the ANFNP, for fixed dimension D 2 and fixed L t metric d t , 1 t 1. The WSCD is based upon the Well Separated Pair Decomposition ([5]). The ANFNP finds extensive applications in VLSI design and verification ([11]) for two dimensions, and in trafficcontrol systems and Geographic Information Systems (GIS) ([7, 12, 13, 14, 15]) for D ? 2 dimensions. To the best of our knowledge, this is the only known optimal parallel algorithm for the ANFNP . Keywords: Closest Pair, Closest Foreign Pair, Computational Geometry, Paralle...
The Colored Quadrant Priority Search Tree with an Application to the AllNearestForeignNeighbors Problem
"... Consider a dynamic set of points in the plane having different colors. Let ]C and 1C2 be two keys according to which points may be sorted, e.g. the x and the ycoordinate. A point p C S is called proper with respect to a query point po C S iff ]C(po) < ]C(p), ]C2(po ) < ]C2(p) , and if p has a d ..."
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Consider a dynamic set of points in the plane having different colors. Let ]C and 1C2 be two keys according to which points may be sorted, e.g. the x and the ycoordinate. A point p C S is called proper with respect to a query point po C S iff ]C(po) < ]C(p), ]C2(po ) < ]C2(p) , and if p has a different color than Po.
Extreme Distances in Multicolored Point Sets
"... Given a set of n colored points in some ddimensional Euclidean space, a bichromatic closest (resp. farthest) pair is a closest (resp. farthest) pair of points of dierent colors. We present ecient algorithms to maintain both a bichromatic closest pair and a bichromatic farthest pair when the the ..."
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Given a set of n colored points in some ddimensional Euclidean space, a bichromatic closest (resp. farthest) pair is a closest (resp. farthest) pair of points of dierent colors. We present ecient algorithms to maintain both a bichromatic closest pair and a bichromatic farthest pair when the the points are xed but they dynamically change color. We do this by solving the more general problem of maintaining a bichromatic edge of minimum (resp. maximum) weight in an undirected weighted graph with colored vertices, when vertices dynamically change color.
and
, 1993
"... The following two computational problems are studied: Duplicate grouping: Assume that n items are given, each of which is labeled by an integer key from the set 0,..., U � 1 4. Store the items in an array of size n such that items with the same key occupy a contiguous segment of the array. Closest p ..."
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The following two computational problems are studied: Duplicate grouping: Assume that n items are given, each of which is labeled by an integer key from the set 0,..., U � 1 4. Store the items in an array of size n such that items with the same key occupy a contiguous segment of the array. Closest pair: Assume that a multiset of n points in the ddimensional Euclidean space is given, where d � 1 is a fixed integer. Each point is represented as a dtuple of integers in the range 0,..., U � 14 Ž or of arbitrary real numbers.. Find a closest pair, i.e., a pair of points whose distance is minimal over all such pairs.