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Smallest colorspanning objects
 IN PROC. 9TH ANNU. EUROPEAN SYMPOS. ALGORITHMS
, 2001
"... Motivated by questions in location planning, we show for a set of colored point sites in the plane how to compute the smallest— by perimeter or area—axisparallel rectangle and the narrowest strip enclosing at least one site of each color. ..."
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Motivated by questions in location planning, we show for a set of colored point sites in the plane how to compute the smallest— by perimeter or area—axisparallel rectangle and the narrowest strip enclosing at least one site of each color.
Quantile Approximation for Robust Statistical Estimation and kEnclosing Problems
, 2000
"... is concerned with finding the smallest shape of some type that encloses all the points of P . Wellknown instances of this problem include finding the smallest enclosing box, minimum volume ball, and minimum volume annulus. In this paper we consider the following variant: Given a set of n points ..."
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is concerned with finding the smallest shape of some type that encloses all the points of P . Wellknown instances of this problem include finding the smallest enclosing box, minimum volume ball, and minimum volume annulus. In this paper we consider the following variant: Given a set of n points in R , find the smallest shape in question that contains at least k points or a certain quantile of the data. This type of problem is known as a kenclosing problem. We present a simple algorithmic framework for computing quantile approximations for the minimum strip, ellipsoid, and annulus containing a given quantile of the points. The algorithms run in O(n log n) time.
Algorithms for Optimal Outlier Removal
, 2009
"... We consider the problem of removing c points from a set S of n points so that the remaining point set is optimal in some sense. Definitions of optimality we consider include having minimum diameter, having minimum area (perimeter) bounding box, having minimum area (perimeter) convex hull. For consta ..."
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We consider the problem of removing c points from a set S of n points so that the remaining point set is optimal in some sense. Definitions of optimality we consider include having minimum diameter, having minimum area (perimeter) bounding box, having minimum area (perimeter) convex hull. For constant values of c, all our algorithms run in O(n log n) time.
On kEnclosing Objects in a Coloured Point Set
"... We introduce the exact coloured kenclosing object problem: given a set P of n points in R2, each of which has an associated colour in {1,..., t}, and a vector c = (c1,..., ct), where ci ∈ Z+ for each 1 ≤ i ≤ t, find a region that contains exactly ci points of P of colour i for each i. We examine t ..."
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We introduce the exact coloured kenclosing object problem: given a set P of n points in R2, each of which has an associated colour in {1,..., t}, and a vector c = (c1,..., ct), where ci ∈ Z+ for each 1 ≤ i ≤ t, find a region that contains exactly ci points of P of colour i for each i. We examine the problems of finding exact coloured kenclosing axisaligned rectangles, squares, discs, and twosided dominating regions in a tcoloured point set. 1
On Piercing Sets of AxisParallel Rectangles and Rings
"... . We consider the ppiercing problem for axisparallel rectangles. We are given a collection of axisparallel rectangles in the plane, and wish to determine whether there exists a set of p points whose union intersects all the given rectangles. We present efficient algorithms for finding a piercing ..."
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. We consider the ppiercing problem for axisparallel rectangles. We are given a collection of axisparallel rectangles in the plane, and wish to determine whether there exists a set of p points whose union intersects all the given rectangles. We present efficient algorithms for finding a piercing set (i.e., a set of p points as above) for values of p = 1; 2; 3; 4; 5. The result for 4 and 5piercing improves an existing result of O(n log 3 n) and O(n log 4 n) to O(n log n) time, and is applied to find a better rectilinear 5center algorithm. We improve the existing algorithm for general (but fixed) p, and we also extend our algorithms to higher dimensional space. We also consider the problem of piercing a set of rectangular rings. 1 Introduction Let R be a set of n axisparallel rectangles in the plane, and let p be a positive integer. R is called ppierceable if there exists a set of p piercing points which intersects every member in R. Our problem, thus, is to determine wheth...
Maximal Covering by Two Isothetic Unit Squares
"... Let P be the point set in two dimensional plane. this paper, we consider the problem of locating two isothetic unit squares such that together they cover maximum number of points of P. In case of overlapping, the points in their common zone are counted once. To solve the problem, we propose an algor ..."
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Let P be the point set in two dimensional plane. this paper, we consider the problem of locating two isothetic unit squares such that together they cover maximum number of points of P. In case of overlapping, the points in their common zone are counted once. To solve the problem, we propose an algorithm that runs in O(n 2 log 2 n) time using O(n log n) space. 1
PRIZECOLLECTING POINT SETS
"... Abstract. Given a set of points P in the plane and profits (or prizes) π: P → R≥0 we want to select a maximum profit set X ⊆ P which maximizes P p∈X π(p) − µ(X) for some particular criterion µ(X). In this paper we consider four such criteria, namely the perimeter and the area of the smallest axisp ..."
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Abstract. Given a set of points P in the plane and profits (or prizes) π: P → R≥0 we want to select a maximum profit set X ⊆ P which maximizes P p∈X π(p) − µ(X) for some particular criterion µ(X). In this paper we consider four such criteria, namely the perimeter and the area of the smallest axisparallel rectangle containing X, and the perimeter and the area of the convex hull conv(X) of X. Our key result is a data structure, called interval heap, that allows us to compute a set of maximum profit with respect to perimeter resp. area of the smallest enclosing axisparallel rectangle in O ` n 2 log n ´ resp. O ` n 3 log n ´ time using O (n) space. In addition, we introduce an O ` n 3 ´ time algorithm for the case that µ(X) measures either the perimeter or the area of conv(X). 1.
Geometric Applications of Posets
, 1998
"... We show the power of posets in computational geometry by solving several problems posed on a set S of n points in the plane: (1) find the n \Gamma k \Gamma 1 rectilinear farthest neighbors (or, equivalently, k nearest neighbors) to every point of S (extendable to higher dimensions), (2) enumerate th ..."
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We show the power of posets in computational geometry by solving several problems posed on a set S of n points in the plane: (1) find the n \Gamma k \Gamma 1 rectilinear farthest neighbors (or, equivalently, k nearest neighbors) to every point of S (extendable to higher dimensions), (2) enumerate the k largest (smallest) rectilinear distances in decreasing (increasing) order among the points of S, (3) given a distance ffi ? 0, report all the pairs of points that belong to S and are of rectilinear distance ffi or more (less), covering k n 2 points of S by rectilinear (4) and circular (5) concentric rings, and (6) given a number k n 2 decide whether a query rectangle contains k points or less.