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Data structures for mobile data
 JOURNAL OF ALGORITHMS
, 1997
"... A kinetic data structure (KDS) maintains an attribute of interest in a system of geometric objects undergoing continuous motion. In this paper we develop a conceptual framework for kinetic data structures, propose a number of criteria for the quality of such structures, and describe a number of fund ..."
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Cited by 257 (53 self)
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A kinetic data structure (KDS) maintains an attribute of interest in a system of geometric objects undergoing continuous motion. In this paper we develop a conceptual framework for kinetic data structures, propose a number of criteria for the quality of such structures, and describe a number of fundamental techniques for their design. We illustrate these general concepts by presenting kinetic data structures for maintaining the convex hull and the closest pair of moving points in the plane; these structures behavewell according to the proposed quality criteria for KDSs.
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 73 (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 . . . . . . . . . . . . . . . . . . ...
Approximate Nearest Neighbor Queries Revisited
, 1998
"... This paper proposes new methods to answer approximate nearest neighbor queries on a set of n points in ddimensional Euclidean space. For any fixed constant d, a data structure with O(" (1\Gammad)=2 n log n) preprocessing time and O(" (1\Gammad)=2 log n) query time achieves approximat ..."
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Cited by 62 (5 self)
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This paper proposes new methods to answer approximate nearest neighbor queries on a set of n points in ddimensional Euclidean space. For any fixed constant d, a data structure with O(" (1\Gammad)=2 n log n) preprocessing time and O(" (1\Gammad)=2 log n) query time achieves approximation factor 1 + " for any given 0 ! " ! 1; a variant reduces the "dependence by a factor of " \Gamma1=2 . For any arbitrary d, a data structure with O(d 2 n log n) preprocessing time and O(d 2 log n) query time achieves approximation factor O(d 3=2 ). Applications to various proximity problems are discussed. 1 Introduction Let P be a set of n point sites in ddimensional space IR d . In the wellknown post office problem, we want to preprocess P into a data structure so that a site closest to a given query point q (called the nearest neighbor of q) can be found efficiently. Distances are measured under the Euclidean metric. The post office problem has many applications within computational...
An Optimal Algorithm for Closest Pair Maintenance
 Discrete Comput. Geom
, 1995
"... Given a set S of n points in kdimensional space, and an L t metric, the dynamic closest pair problem is defined as follows: find a closest pair of S after each update of S (the insertion or the deletion of a point). For fixed dimension k and fixed metric L t , we give a data structure of size O(n) ..."
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Cited by 34 (0 self)
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Given a set S of n points in kdimensional space, and an L t metric, the dynamic closest pair problem is defined as follows: find a closest pair of S after each update of S (the insertion or the deletion of a point). For fixed dimension k and fixed metric L t , we give a data structure of size O(n) that maintains a closest pair of S in O(logn) time per insertion and deletion. The running time of algorithm is optimal up to constant factor because \Omega\Gammaaus n) is a lower bound, in algebraic decisiontree model of computation, on the time complexity of any algorithm that maintains the closest pair (for k = 1). The algorithm is based on the fairsplit tree. The constant factor in the update time is exponential in the dimension. We modify the fairsplit tree to reduce it. 1 Introduction The dynamic closest pair problem is one of the very wellstudied proximity problem in computational geometry [6, 1720, 22, 2426, 2831]. We are given a set S of n points in kdimensional space...
Algorithms for Dynamic Closest Pair and nBody Potential Fields
 In Proc. 6th ACMSIAM Sympos. Discrete Algorithms
, 1995
"... We present a general technique for dynamizing certain problems posed on point sets in Euclidean space for any fixed dimension d. This technique applies to a large class of structurally similar algorithms, presented previously by the authors, that make use of the wellseparated pair decomposition. We ..."
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Cited by 33 (1 self)
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We present a general technique for dynamizing certain problems posed on point sets in Euclidean space for any fixed dimension d. This technique applies to a large class of structurally similar algorithms, presented previously by the authors, that make use of the wellseparated pair decomposition. We prove efficient worstcase complexity for maintaining such computations under point insertions and deletions, and apply the technique to several problems posed on a set P containing n points. In particular, we show how to answer a query for any point x that returns a constantsize set of points, a subset of which consists of all points in P that have x as a nearest neighbor. We then show how to use such queries to maintain the closest pair of points in P . We also show how to dynamize the fast multipole method, a technique for approximating the potential field of a set of point charges. All our algorithms use the algebraic model that is standard in computational geometry, and have worstca...
Computational Geometry
 in optimization 2.5D and 3D NC surface machining. Computers in Industry
, 1996
"... Introduction Computational geometry evolves from the classical discipline of design and analysis of algorithms, and has received a great deal of attention in the last two decades since its inception in 1975 by M. Shamos[108]. It is concerned with the computational complexity of geometric problems t ..."
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Cited by 12 (0 self)
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Introduction Computational geometry evolves from the classical discipline of design and analysis of algorithms, and has received a great deal of attention in the last two decades since its inception in 1975 by M. Shamos[108]. It is concerned with the computational complexity of geometric problems that arise in various disciplines such as pattern recognition, computer graphics, computer vision, robotics, VLSI layout, operations research, statistics, etc. In contrast with the classical approach to proving mathematical theorems about geometryrelated problems, this discipline emphasizes the computational aspect of these problems and attempts to exploit the underlying geometric properties possible, e.g., the metric space, to derive efficient algorithmic solutions. The classical theorem, for instance, that a set S is convex if and only if for any 0 ff 1 the convex combination ffp + (1 \Gamma<F
Analysis of Heuristics for the FreezeTag Problem
 In Proc. Scandinavian Workshop on Algorithms, Vol. 2368 of SpringerVerlag LNCS
, 2002
"... In the Freeze Tag Problem (FTP) we are given a swarm of n asleep (frozen or inactive) robots and a single awake (active) robot, and we want to awaken all robots in the shortest possible time. A robot is awakened when an active robot "touches" it. The goal is to compute an optimal awakening ..."
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Cited by 10 (2 self)
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In the Freeze Tag Problem (FTP) we are given a swarm of n asleep (frozen or inactive) robots and a single awake (active) robot, and we want to awaken all robots in the shortest possible time. A robot is awakened when an active robot "touches" it. The goal is to compute an optimal awakening schedule such that all robots are awake by time t , for the smallest possible value of t . We devise and test a variety of heuristic strategies on geometric and network datasets. Our experiments show that all of the strategies perform well, with the simple greedy strategy performing particularly well. A theoretical analysis of the greedy strategy gives a tight approximation bound of ( log n) for points in the plane. We show more generally that the (tight) performance bound is ((log n) ) in d dimensions. This is in contrast with the case of general metric spaces, where greedy is known to have a (log n) approximation factor, and no method is known to achieve an approximation bound of o(log n).
Randomized Data Structures for the Dynamic ClosestPair Problem
, 1993
"... We describe a new randomized data structure, the sparse partition, for solving the dynamic closestpair problem. Using this data structure the closest pair of a set of n points in Ddimensional space, for any fixed D, can be found in constant time. If a frame containing all the points is known in adv ..."
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Cited by 10 (2 self)
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We describe a new randomized data structure, the sparse partition, for solving the dynamic closestpair problem. Using this data structure the closest pair of a set of n points in Ddimensional space, for any fixed D, can be found in constant time. If a frame containing all the points is known in advance, and if the floor function is available at unitcost, then the data structure supports insertions into and deletions from the set in expected O(log n) time and requires expected O(n) space. Here, it is assumed that the updates are chosen by an adversary who does not know the random choices made by the data structure. This method is more efficient than any deterministic algorithm for solving the problem in dimension D ? 1. The data structure can be modified to run in O(log 2 n) expected time per update in the algebraic computation tree model of computation. Even this version is more efficient than the currently best known deterministic algorithm for D ? 2. 1 Introduction We ...
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"... On bounded leg shortest paths problems Let V be a set of points in a ddimensional lpmetric space. Let s, t ∈ V and let L be any real number. An Lbounded leg path from s to t is an ordered set of points which connects s to t such that the leg between any two consecutive points in the set is at mos ..."
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On bounded leg shortest paths problems Let V be a set of points in a ddimensional lpmetric space. Let s, t ∈ V and let L be any real number. An Lbounded leg path from s to t is an ordered set of points which connects s to t such that the leg between any two consecutive points in the set is at most L. The minimal path among all these paths is the Lbounded leg shortest path from s to t. In the st Bounded Leg Shortest Path (stBLSP) problem we are given two points s and t and a real number L, and are required to compute an Lbounded leg shortest path from s to t. In the AllPairs Bounded Leg Shortest Path (apBLSP) problem we are required to build a data structure that, given any two query points from V and any real number L, outputs the length of the Lbounded leg shortest path (a distance
Computational Geometry II
"... Introduction This is a follow up on the previous Chapter dealing with geometric problems and their efficient solutions. The classes of problems that we address in this Chapter include proximity, optimization, intersection, searching, point location, and some discussions of geometric software that i ..."
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Introduction This is a follow up on the previous Chapter dealing with geometric problems and their efficient solutions. The classes of problems that we address in this Chapter include proximity, optimization, intersection, searching, point location, and some discussions of geometric software that is under development. 2 Proximity Geometric problems pertaining to the questions of how close two geometric entities are among a collection of objects or how similar two geometric patterns match each other abound. For example, in pattern classification and clustering, features that are similar according to some metric, are to be clustered in a group. The two aircrafts that are closest at any time instant in the air space will have the largest likelihood of collision with each other. In some cases one may be interested in how far apart or how dissimilar the objects are. Some of