Results 1  10
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17
Approximate Nearest Neighbor Queries in Fixed Dimensions
, 1993
"... Given a set of n points in ddimensional Euclidean space, S ae E d , and a query point q 2 E d , we wish to determine the nearest neighbor of q, that is, the point of S whose Euclidean distance to q is minimum. The goal is to preprocess the point set S, such that queries can be answered as effic ..."
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Cited by 105 (10 self)
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Given a set of n points in ddimensional Euclidean space, S ae E d , and a query point q 2 E d , we wish to determine the nearest neighbor of q, that is, the point of S whose Euclidean distance to q is minimum. The goal is to preprocess the point set S, such that queries can be answered as efficiently as possible. We assume that the dimension d is a constant independent of n. Although reasonably good solutions to this problem exist when d is small, as d increases the performance of these algorithms degrades rapidly. We present a randomized algorithm for approximate nearest neighbor searching. Given any set of n points S ae E d , and a constant ffl ? 0, we produce a data structure, such that given any query point, a point of S will be reported whose distance from the query point is at most a factor of (1 + ffl) from that of the true nearest neighbor. Our algorithm runs in O(log 3 n) expected time and requires O(n log n) space. The data structure can be built in O(n 2 ) expe...
On the complexity of computing minimum energy consumption broadcast subgraphs
 in Symposium on Theoretical Aspects of Computer Science
, 2001
"... Abstract. We consider the problem of computing an optimal range assignment in a wireless network which allows a specified source station to perform a broadcast operation. In particular, we consider this problem as a special case of the following more general combinatorial optimization problem, calle ..."
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Cited by 96 (11 self)
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Abstract. We consider the problem of computing an optimal range assignment in a wireless network which allows a specified source station to perform a broadcast operation. In particular, we consider this problem as a special case of the following more general combinatorial optimization problem, called Minimum Energy Consumption Broadcast Subgraph (in short, MECBS): Given a weighted directed graph and a specified source node, find a minimum cost range assignment to the nodes, whose corresponding transmission graph contains a spanning tree rooted at the source node. We first prove that MECBS is not approximable within a constant factor (unless P=NP). We then consider the restriction of MECBS to wireless networks and we prove several positive and negative results, depending on the geometric space dimension and on the distancepower gradient. The main result is a polynomialtime approximation algorithm for the NPhard case in which both the dimension and the gradient are equal to 2: This algorithm can be generalized to the case in which the gradient is greater than or equal to the dimension. 1
Separators for spherepackings and nearest neighbor graphs
 J. ACM
, 1997
"... Abstract. A collection of n balls in d dimensions forms a kply system if no point in the space is covered by more than k balls. We show that for every kply system �, there is a sphere S that intersects at most O(k 1/d n 1�1/d) balls of � and divides the remainder of � into two parts: those in the ..."
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Cited by 71 (7 self)
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Abstract. A collection of n balls in d dimensions forms a kply system if no point in the space is covered by more than k balls. We show that for every kply system �, there is a sphere S that intersects at most O(k 1/d n 1�1/d) balls of � and divides the remainder of � into two parts: those in the interior and those in the exterior of the sphere S, respectively, so that the larger part contains at most (1 � 1/(d � 2))n balls. This bound of O(k 1/d n 1�1/d) is the best possible in both n and k. We also present a simple randomized algorithm to find such a sphere in O(n) time. Our result implies that every knearest neighbor graphs of n points in d dimensions has a separator of size O(k 1/d n 1�1/d). In conjunction with a result of Koebe that every triangulated planar graph is isomorphic to the intersection graph of a diskpacking, our result not only gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. The separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space.
LowDegree Minimum Spanning Trees
 Discrete Comput. Geom
, 1999
"... Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that under the Lp norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where ..."
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Cited by 20 (1 self)
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Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that under the Lp norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where the maximum degree is minimized. We give the bestknown bounds for the maximum MST degree for arbitrary Lp metrics in all dimensions, with a focus on the rectilinear metric in two and three dimensions. We show that for any finite set of points in the rectilinear plane there exists an MST with maximum degree of at most 4, and for threedimensional rectilinear space the maximum possible degree of a minimumdegree MST is either 13 or 14. 1 Introduction Minimum spanning tree (MST) construction is a classic optimization problem for which several efficient algorithms are known [9] [15] [19]. Solutions of many other problems hinge on the construction of an MST as an intermediary step [4], with th...
On the Hardness of the Shortest Vector Problem
, 1998
"... An ndimensional lattice is the set of all integral linear combinations of n linearly independent vectors in R^m. One of the most studied algorithmic problems on lattices is the shortest vector problem (SVP): given a lattice, find the shortest nonzero vector in it. We prove that the shortest vector ..."
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Cited by 12 (1 self)
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An ndimensional lattice is the set of all integral linear combinations of n linearly independent vectors in R^m. One of the most studied algorithmic problems on lattices is the shortest vector problem (SVP): given a lattice, find the shortest nonzero vector in it. We prove that the shortest vector problem is NPhard (for randomized reductions) to approximate within some constant factor greater than 1 in any norm l_p (p>1). In particular, we prove the NPhardness of approximating SVP in the Euclidean norm within any factor less than sqrt 2. The same NPhardness results hold for deterministic nonuniform reductions. A deterministic uniform reduction is also given under a reasonable number theoretic conjecture concerning the distribution of smooth numbers. In proving the NPhardness of SVP we develop a number of technical tools that might be of independent interest. In particular, a lattice packing is constructed with the property that the number of unit spheres contained in an ndimensional ball of radius greater then 1 + (sqrt 2) grows exponentially in n, a new constructive version of Sauer's lemma(a combinatorial result somehow related to the notion of VCdimension) is presented, considerably simplifying all previously known constructions.
A partition of the unit sphere into regions of equal area and small diameter
 Electronic Transactions on Numerical Analysis
"... Abstract. The recursive zonal equal area (EQ) sphere partitioning algorithm is a practical algorithm for partitioning higher dimensional spheres into regions of equal area and small diameter. This paper describes the partition algorithm and its implementation in Matlab, provides numerical results an ..."
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Cited by 7 (0 self)
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Abstract. The recursive zonal equal area (EQ) sphere partitioning algorithm is a practical algorithm for partitioning higher dimensional spheres into regions of equal area and small diameter. This paper describes the partition algorithm and its implementation in Matlab, provides numerical results and gives a sketch of the proof of the bounds on the diameter of regions. A companion paper [13] gives details of the proof.
On lower bounds of the density of Delone sets and holes in sequences of sphere packings
 Exp. Math
"... We study lower bounds of the packing density of a system of nonoverlapping equal spheres in Rn, n ≥ 2, as a function of the maximal circumradius of its Voronoi cells. Our viewpoint is that of Delone sets which allows to investigate the gap between the upper bounds of Rogers or KabatjanskiĭLeven˘ste ..."
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Cited by 4 (2 self)
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We study lower bounds of the packing density of a system of nonoverlapping equal spheres in Rn, n ≥ 2, as a function of the maximal circumradius of its Voronoi cells. Our viewpoint is that of Delone sets which allows to investigate the gap between the upper bounds of Rogers or KabatjanskiĭLeven˘stein and the MinkowskiHlawka type lower bounds for the density of latticepackings, without entering the fundamental problem of constructing Delone sets with Delone constants between 2−0.401 and 1. As a consequence we provide explicit asymptotic lower bounds of the covering radii (holes) of the BarnesWall, Craig and MordellWeil lattices, respectively BWn, A (r) n and MWn, and of the Delone constants of the BCH packings, when n goes to infinity.
The Gamma Test: Data derived estimates of noise for unknown smooth models using near neighbour asymptotics
, 2002
"... The Gamma test is a simple technique for assessing the extent to which a given set of M data points can be modelled by an unknown smooth nonlinear function f.Ifthe underlying model is of the form y = f(x)+r where r is a random variable representing that part of the data which cannot be accounted f ..."
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Cited by 2 (1 self)
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The Gamma test is a simple technique for assessing the extent to which a given set of M data points can be modelled by an unknown smooth nonlinear function f.Ifthe underlying model is of the form y = f(x)+r where r is a random variable representing that part of the data which cannot be accounted for by the smooth function f,the Gamma test produces an estimate #M for the variance Var(r). This estimate is rapidly computed directly from the data, and since its introduction in 1995 has been used extensively for a variety of di#erent applications in several theses and papers. Thus it is of some interest to provide a formal basis for the method, and this is precisely the problem addressed in this thesis.
Testing the Congruence of dDimensional Point Sets
 J. Comput. Geom. Appl
, 2000
"... This paper presents an algorithm that tests the congruence of two sets of n points in ddimensional space in O(n d 1 3 de log n) time. This improves the previous best algorithm for dimensions d 6. ..."
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Cited by 1 (1 self)
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This paper presents an algorithm that tests the congruence of two sets of n points in ddimensional space in O(n d 1 3 de log n) time. This improves the previous best algorithm for dimensions d 6.