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26
Range Searching
, 1996
"... Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , an ..."
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Cited by 74 (1 self)
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Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.
MOTION
"... Motion is ubiquitous in the physical world, yet its study is much less developed than that of another common physical modality, namely shape. While we have several standardized mathematical shape descriptions, and even entire disciplines devoted to that area–such as ComputerAided Geometric Design ( ..."
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Cited by 17 (1 self)
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Motion is ubiquitous in the physical world, yet its study is much less developed than that of another common physical modality, namely shape. While we have several standardized mathematical shape descriptions, and even entire disciplines devoted to that area–such as ComputerAided Geometric Design (CAGD)—the
The Projection Median of a Set of Points in R²
"... We define the projection median of a nonempty and finite multiset of points in R². We show the projection median provides a better approximation of the Euclidean (ℓ2) median than do the rectilinear (ℓ1) median or the centre of mass, both in terms of approximation factor and stability. ..."
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Cited by 8 (1 self)
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We define the projection median of a nonempty and finite multiset of points in R². We show the projection median provides a better approximation of the Euclidean (ℓ2) median than do the rectilinear (ℓ1) median or the centre of mass, both in terms of approximation factor and stability.
Median Trajectories ⋆
"... Abstract. We investigate the concept of a median among a set of trajectories. We establish criteria that a “median trajectory ” should meet, and present two different methods to construct a median for a set of input trajectories. The first method is very simple, while the second method is more compl ..."
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Cited by 7 (0 self)
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Abstract. We investigate the concept of a median among a set of trajectories. We establish criteria that a “median trajectory ” should meet, and present two different methods to construct a median for a set of input trajectories. The first method is very simple, while the second method is more complicated and uses homotopy with respect to sufficiently large faces in the arrangement formed by the trajectories. We give algorithms for both methods, analyze the worstcase running time, and show that under certain assumptions both methods can be implemented efficiently. We empirically compare the output of both methods on randomly generated trajectories, and analyze whether the two methods yield medians that are according to our intuition. Our results suggest that the second method, using homotopy, performs considerably better. 1
Kinetic and Dynamic Data Structures for Closest Pairs and All Nearest Neighbors
, 2008
"... We present simple, fully dynamic and kinetic data structures, which are variants of a dynamic twodimensional range tree, for maintaining the closest pair and all nearest neighbors for a set of n moving points in the plane; insertions and deletions of points are also allowed. If no insertions or del ..."
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Cited by 7 (2 self)
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We present simple, fully dynamic and kinetic data structures, which are variants of a dynamic twodimensional range tree, for maintaining the closest pair and all nearest neighbors for a set of n moving points in the plane; insertions and deletions of points are also allowed. If no insertions or deletions take place, the structure for the closest pair uses O(n log n) space, and processes O(n 2 βs+2(n) log n) critical events, each in O(log 2 n) time. Here s is the maximum number of times where the distances between any two specific pairs of points can become equal, βs(q) = λs(q)/q, and λs(q) is the maximum length of DavenportSchinzel sequences of order s on q symbols. The dynamic version of the problem incurs a slight degradation in performance: If m ≥ n insertions and deletions are performed, the structure still uses O(n log n) space, and processes O(mnβs+2(n) log³ n) events, each in O(log³ n) time. Our kinetic data structure for all nearest neighbors uses O(n log² n) space, and processes O(n 2 β 2 s+2 (n) log3 n) critical events. The expected time to process all events is O(n 2 β 2 s+2 (n) log4 n), though processing a single event may take �(n) expected time in the worst case. If m ≥ n insertions and deletions are performed, then the expected number of events is O(mnβ 2 s+2 (n) log3 n) and processing them all takes O(mnβ 2 s+2 (n) log4 n). An insertion or deletion takes O(n) expected time.
Boundedvelocity approximations of the mobile Euclidean 2centre
 Proc. Annual Fall Workshop on Computational Geometry and Visualization 15
, 2005
"... Given a set P of points (clients) in the plane, a Euclidean 2centre of P is a set of two points (facilities) in the plane such that the maximum distance from any client to its nearest facility is minimized. Geometrically, a Euclidean 2centre of P corresponds to a cover of P by two discs of minimum ..."
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Cited by 4 (2 self)
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Given a set P of points (clients) in the plane, a Euclidean 2centre of P is a set of two points (facilities) in the plane such that the maximum distance from any client to its nearest facility is minimized. Geometrically, a Euclidean 2centre of P corresponds to a cover of P by two discs of minimum radius r (the Euclidean 2radius). Given a set of mobile clients, where each client follows a continuous trajectory in the plane with bounded velocity, the motion of the corresponding mobile Euclidean 2centre is not necessarily continuous. Consequently, we consider strategies for defining the trajectories of a pair of mobile facilities that guarantee a fixeddegree approximation of the Euclidean 2centre while maintaining bounded relative velocity. In an attempt to balance the conflicting goals of closeness of approximation and a low maximum relative velocity, we introduce reflectionbased 2centre functions by reflecting the position of a mobile client across the mobile Steiner centre and the mobile rectilinear 1centre, respectively.
Efficient tradeoff schemes in data structures for querying moving objects
 In Proc. European Symposium on Algorithms, LNCS 3221
, 2004
"... The ability to represent and query continuously moving objects is important in many applications of spatiotemporal database systems. In this paper we develop data structures for answering various queries on moving objects, including range and proximity queries, and study tradeoffs between various p ..."
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Cited by 4 (1 self)
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The ability to represent and query continuously moving objects is important in many applications of spatiotemporal database systems. In this paper we develop data structures for answering various queries on moving objects, including range and proximity queries, and study tradeoffs between various performance measures—query time, data structure size, and accuracy of results. 1
Statistical regular pavings to analyze massive data of aircraft trajectories
 J. Aeros. Comput. Info. Comm. 9
, 2012
"... A variety of tasks conducted by aviation system decision makers and researchers requires analyzing aircraft trajectory data. Datasets containing high frequency aircraft position information collected over large geographic areas and long periods of time are too large to store in the primary memory of ..."
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Cited by 2 (2 self)
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A variety of tasks conducted by aviation system decision makers and researchers requires analyzing aircraft trajectory data. Datasets containing high frequency aircraft position information collected over large geographic areas and long periods of time are too large to store in the primary memory of personal computers. This paper introduces the use of statistical regular pavingsasdatastructurescapableofsummarizing very large aircraft trajectory datasets. Recursively computable statistics can be stored for variablesized regions of airspace. The regions themselves can be created automatically to reflect the varying density of aircraft observations, dedicating more computational resources and providing more detailed information in areas with more air traffic. In particular, statistical regular pavings are able to very quickly aggregate or separate data with different characteristics so that data describing individual aircraft or collected using different technologies (reflecting different levels of precision) can be stored separately and yet also very quickly combined using standard arithmetic operations.
Staying in the middle: Exact and approximate medians in R¹ and R² for moving points
 IN PROC. OF THE CANADIAN CONFERENCE ON COMPUTATIONAL GEOMETRY
, 2003
"... Many divideandconquer based geometric algorithms and orderstatistics problems ask for a point that lies “in the middle” of a given point set. We study several fundamental problems of this type for moving points in one and two dimensions. In particular, we show how to kinetically maintain the medi ..."
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Cited by 1 (0 self)
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Many divideandconquer based geometric algorithms and orderstatistics problems ask for a point that lies “in the middle” of a given point set. We study several fundamental problems of this type for moving points in one and two dimensions. In particular, we show how to kinetically maintain the median of a set of n points moving on the real line, and a center point of a set of n points moving in the plane, that is, a point such that any line through it has at most 2n/3 on either side of it. Since the maintenance of exact medians and center points can be quite expensive, we also show how to maintain εapproximate medians and center points and argue that the latter can be made to be much more stable under motion. These results are based on a new algorithm to maintain an εapproximation of a range space under insertions and deletions, which is of independent interest. All our approximation algorithms run in nearlinear time.