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The NPcompleteness column: an ongoing guide
 JOURNAL OF ALGORITHMS
, 1987
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freem ..."
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Cited by 242 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freeman & Co., New York, 1979 (hereinafter referred to as "[G&J]"; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
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 (13 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 . . . . . . . . . . . . . . . . . . ...
On Enumerating and Selecting Distances
 Int. J. Comput. Geom. Appl
, 1999
"... Given an npoint set, the problems of enumerating the k closest pairs and selecting the kth smallest distance are revisited. For the enumeration problem, we give simpler randomized and deterministic algorithms with O(n log n + k) running time in any fixeddimensional Euclidean space. For the selec ..."
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Given an npoint set, the problems of enumerating the k closest pairs and selecting the kth smallest distance are revisited. For the enumeration problem, we give simpler randomized and deterministic algorithms with O(n log n + k) running time in any fixeddimensional Euclidean space. For the selection problem, we give a randomized algorithm with running time O(n log n + n 2=3 k 1=3 log 5=3 n). We also describe outputsensitive results for halfspace range counting that are of use in more general distance selection problems. None of our algorithms requires parametric search. Keywords: distance enumeration, distance selection, closest pairs, range counting, randomized algorithms. 1 Introduction Finding the closest pair of an npoint set has a long history in computational geometry (see [34] for a nice survey). In the plane, the problem can be solved in O(n log n) time using the Delaunay triangulation. In an arbitrary fixed dimension d, the first O(n log n) algorithm, based on di...
An Algorithm for Fast, Modelfree Tracking Indoors
 SIGMOBILE Mob. Comput. Commun. Rev. 2007
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Pyramid Computer Solutions of the Closest Pair Problem
, 1982
"... Given an N x N array of OS and Is, the closest pair problem is to determine the minimum distance between any pair of ones. Let D be this minimum distance (or D = 2N if there are fewer than two Is). Two solutions to this problem are given, one requiring O(log ( N) + D) time and the other O(log ( N)). ..."
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Given an N x N array of OS and Is, the closest pair problem is to determine the minimum distance between any pair of ones. Let D be this minimum distance (or D = 2N if there are fewer than two Is). Two solutions to this problem are given, one requiring O(log ( N) + D) time and the other O(log ( N)). These solutions are for two types of parallel computers arranged in a pyramid fashion with the base of the pyramid containing the matrix. The results improve upon an algorithm of Dyer that requires o(N) time on a more pOWerfd COmpUter. 0 19135 Academic Press. Inc. 1.
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"... shows that there exists a randomized algebraic algorithm that has worstcase expected complexity O(n) [8], thus beating the best possible deterministic algebraic algorithm. Randomization allows operations such as hashing to be performed in constant expected time, and in fact this is a key element of ..."
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shows that there exists a randomized algebraic algorithm that has worstcase expected complexity O(n) [8], thus beating the best possible deterministic algebraic algorithm. Randomization allows operations such as hashing to be performed in constant expected time, and in fact this is a key element of Rabin’s algorithm. To demonstrate this dependence, Fortune and Hopcroft gave an O(n log log n) time deterministic algorithm by augmenting their model with an operation that is essentially hashing in constant time [3]. In this paper we consider only deterministic algorithms, but assume that input points are represented as fixed point binary values with O(log n) bits. In addition, we augment our model with the floor function; equivalently, we could allow constant time binary shift and mask operations. Both of these assumptions seem reasonable given today’s computing hardware, and in fact seem more realistic than the algebraic model’s assumption that arbitrary precision real numbers can be stored and manipulated. Under such a model, we show that the simple closest pair and knearest neighbors problems can be solved in O(n log log n) time. This is currently the only algorithm to beat the �(n log n) bound
A reliable randomized algorithm for the . . .
, 1997
"... 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.
DOI: 10.1007/s004530010040
 Algorithmica: An International Journal in Computer Science
, 1996
"... In this paper we show that if the input points to the geometric closest pair problem are given with limited precision (each coordinate is specified with O(log n) bits), then we can compute the closest pair in O(n log log n) time. We also apply our spatial decomposition technique to the knearest nei ..."
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In this paper we show that if the input points to the geometric closest pair problem are given with limited precision (each coordinate is specified with O(log n) bits), then we can compute the closest pair in O(n log log n) time. We also apply our spatial decomposition technique to the knearest neighbor and nbody problems, achieving similar improvements.
Limited Precision Input
"... In this paper we show that if the input points to the geometric closest pair problem are given with limited precision (each coordinate is speci ed with O(log n) bits), then we can compute the closest pair in O(n log log n) time. To make use of the limited precision of the input points, we use a reas ..."
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In this paper we show that if the input points to the geometric closest pair problem are given with limited precision (each coordinate is speci ed with O(log n) bits), then we can compute the closest pair in O(n log log n) time. To make use of the limited precision of the input points, we use a reasonable machine model that allows \bit shifting &quot; in constant time  we believe that this model is realistic, and provides an interesting way of beating the (n log n) lower bound that exists for this problem using the more typical algebraic RAM model. 1
An Algorithm for Fast, ModelFree Tracking Indoors
"... Tracking people and objects indoors from signal strength measurements has applications as diverse as security monitoring, selfguided museum tours, and personalization of communications services. Accurate dynamic tracking in realtime has been elusive, though, because signal propagation in buildings ..."
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Tracking people and objects indoors from signal strength measurements has applications as diverse as security monitoring, selfguided museum tours, and personalization of communications services. Accurate dynamic tracking in realtime has been elusive, though, because signal propagation in buildings and the paths that people follow are complex. This paper proposes a simple algorithm for indoor tracking that requires neither a propagation model nor a motion model. It is also simple to compute, requiring only standard tools: Delaunay triangulation, linear interpolation, moving averaging, and local regression. Experiments with real data and simulations based on real data show that the algorithm is not only simple, it is effective. I.