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18
Delaunay Triangulations of Imprecise Points in Linear Time after Preprocessing
, 2008
"... An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one ..."
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Cited by 24 (6 self)
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An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay.
Largest bounding box, smallest diameter, and related problems on imprecise points
 In Proc. 10th Workshop on Algorithms and Data Structures, LNCS 4619
, 2007
"... We model imprecise points as regions in which one point must be located. We study computing the largest and smallest possible values of various basic geometric measures on sets of imprecise points, such as the diameter, width, closest pair, smallest enclosing circle, and smallest enclosing bounding ..."
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Cited by 18 (6 self)
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We model imprecise points as regions in which one point must be located. We study computing the largest and smallest possible values of various basic geometric measures on sets of imprecise points, such as the diameter, width, closest pair, smallest enclosing circle, and smallest enclosing bounding box. We give efficient algorithms for most of these problems, and identify the hardness of others. 1
Closest Pair and the Post Office Problem for Stochastic Points
"... Abstract. Given a (master) set M of n points in ddimensional Euclidean space, consider drawing a random subset that includes each point mi ∈ M with an independent probability pi. How difficult is it to compute elementary statistics about the closest pair of points in such a subset? For instance, wh ..."
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Cited by 7 (2 self)
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Abstract. Given a (master) set M of n points in ddimensional Euclidean space, consider drawing a random subset that includes each point mi ∈ M with an independent probability pi. How difficult is it to compute elementary statistics about the closest pair of points in such a subset? For instance, what is the probability that the distance between the closest pair of points in the random subset is no more than ℓ, for a given value ℓ? Or, can we preprocess the master set M such that given a query point q, we can efficiently estimate the expected distance from q to its nearest neighbor in the random subset? We obtain hardness results and approximation algorithms for stochastic problems of this kind. 1
Approximating Largest Convex Hulls for Imprecise Points
, 2007
"... Assume that a set of imprecise points is given, where each point is specified by a region in which the point will lie. Such a region can be modelled as a circle, square, line segment, etc. We study the problem of maximising the area of the convex hull of such a set. We prove NPhardness when the imp ..."
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Cited by 6 (1 self)
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Assume that a set of imprecise points is given, where each point is specified by a region in which the point will lie. Such a region can be modelled as a circle, square, line segment, etc. We study the problem of maximising the area of the convex hull of such a set. We prove NPhardness when the imprecise points are modelled as line segments, and give linear time approximation schemes for a variety of models, based on the coreset paradigm.
Minimumperimeter intersecting polygons
, 2011
"... Given a set S of segments in the plane, a polygon P is an intersecting polygon of S if every segment in S intersects the interior or the boundary of P. The problem MPIP of computing a minimumperimeter intersecting polygon of a given set of n segments in the plane was first considered by Rappaport i ..."
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Cited by 5 (3 self)
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Given a set S of segments in the plane, a polygon P is an intersecting polygon of S if every segment in S intersects the interior or the boundary of P. The problem MPIP of computing a minimumperimeter intersecting polygon of a given set of n segments in the plane was first considered by Rappaport in 1995. This problem is not known to be polynomial, nor it is known to be NPhard. Rappaport (1995) gave an exponentialtime exact algorithm for MPIP. Hassanzadeh and Rappaport (2009) gave a polynomialtime approximation algorithm with ratio π 2 ≈ 1.57. In this paper, we present two improved approximation algorithms for MPIP: a 1.28approximation algorithm by linear programming, and a polynomialtime approximation scheme by discretization and enumeration. Our algorithms can be generalized for computing an approximate minimumperimeter intersecting polygon of a set of convex polygons in the plane. From the other direction, we show that computing a minimumperimeter intersecting polygon of a set of (not necessarily convex) simple polygons is NPhard.
D.: On minimum and maximumweight minimum spanning trees with neighborhoods
, 2012
"... Abstract. We study optimization problems for the Euclidean minimum spanning tree (MST) on imprecise data. To model imprecision, we accept a set of disjoint disks in the plane as input. From each member of the set, one point must be selected, and the MST is computed over the set of selected points. ..."
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Cited by 4 (2 self)
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Abstract. We study optimization problems for the Euclidean minimum spanning tree (MST) on imprecise data. To model imprecision, we accept a set of disjoint disks in the plane as input. From each member of the set, one point must be selected, and the MST is computed over the set of selected points. We consider both minimizing and maximizing the weight of the MST over the input. The minimum weight version of the problem is known as the minimum spanning tree with neighborhoods (MSTN) problem, and the maximum weight version (maxMSTN) has not been studied previously to our knowledge. We provide deterministic and parameterized approximation algorithms for the maxMSTN problem, and a parameterized algorithm for the MSTN problem. Additionally, we present hardness of approximation proofs for both settings. 1
On the Most Likely Convex Hull of Uncertain Points
"... Abstract. Consider a set of ddimensional points where the existence or the location of each point is determined by a probability distribution. The convex hull of this set is a random variable distributed over exponentially many choices. We are interested in finding the most likely convex hull, nam ..."
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Cited by 4 (1 self)
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Abstract. Consider a set of ddimensional points where the existence or the location of each point is determined by a probability distribution. The convex hull of this set is a random variable distributed over exponentially many choices. We are interested in finding the most likely convex hull, namely, the one with the maximum probability of occurrence. We investigate this problem under two natural models of uncertainty: the point (also called the tuple) model where each point (site) has a fixed position si but only exists with some probability pii, for 0 < pii ≤ 1, and the multipoint model where each point has multiple possible locations or it may not appear at all. We show that the most likely hull under the point model can be computed in O(n3) time for n points in d = 2 dimensions, but it is NP–hard for d ≥ 3 dimensions. On the other hand, we show that the problem is NP–hard under the multipoint model even for d = 2 dimensions. We also present hardness results for approximating the probability of the most likely hull. While we focus on the most likely hull for concreteness, our results hold for other natural definitions of a probabilistic hull. 1
An Algorithm for Computing the Convex Hull of a Set of Imprecise Line Segment Intersection
"... Abstract Data imprecision constitutes an important gap between theory and practice in computational geometry. A lot of research about imprecision in computational geometry is directed at computing the convex hull of imprecise points rather than imprecise line segment intersection. In this paper we ..."
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Abstract Data imprecision constitutes an important gap between theory and practice in computational geometry. A lot of research about imprecision in computational geometry is directed at computing the convex hull of imprecise points rather than imprecise line segment intersection. In this paper we introduce an algorithm to construct the convex hull for a set of imprecise line segment intersection in time.
Competitive Query Strategies for Minimising the Ply of the Potential Locations of Moving Points
"... ABSTRACT We study the problem of maintaining the locations of a collection of n entities that are moving with some fixed upper bound on their speed. We assume a setting where we may query the current location of entities, but handling this query takes a certain unit of time, during which we cannot ..."
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ABSTRACT We study the problem of maintaining the locations of a collection of n entities that are moving with some fixed upper bound on their speed. We assume a setting where we may query the current location of entities, but handling this query takes a certain unit of time, during which we cannot query any other entities. In this model, we can never know the exact locations of all entities at any one time. Instead, we maintain a representation of the potential locations of all entities. We measure the quality of this representation by its ply: the maximum number of entities that could potentially be at the same location. Since the ply could be large for every query strategy, we analyse the performance of our algorithms in a competitive framework: we consider the worst case ratio of the ply achieved by our algorithms to the intrinsic ply (the smallest ply achievable by any algorithm, even one that knows in advance the full trajectories of all entities). We show that, if our goal is to mimimise the ply at some number τ of time units in the future, an O(1)competitive algorithm exists, provided τ is sufficiently large. If τ is small and the n entities move in any constant dimension d, our algorithm is competitive, where is the average time since the last query over all entities. We also provide matching lower bounds, and we show that computing the intrinsic ply exactly is NPhard, even when the trajectories are known in advance.