Results 1  10
of
29
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 ..."
Abstract

Cited by 129 (9 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 113 (13 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 102 (8 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 24 (1 self)
 Add to MetaCart
(Show Context)
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...
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 ..."
Abstract

Cited by 24 (0 self)
 Add to MetaCart
(Show Context)
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 the Hardness of the Shortest Vector Problem
, 1998
"... A lattice is the set of intersection points of an infinite ndimensional grid. One of the most fundamental algorithmic problems on lattices is the shortest vector problem (SVP): given a lattice, find the shortest nonzero vector in it (i.e., the intersection points closest to the origin). We prove t ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
(Show Context)
A lattice is the set of intersection points of an infinite ndimensional grid. One of the most fundamental algorithmic problems on lattices is the shortest vector problem (SVP): given a lattice, find the shortest nonzero vector in it (i.e., the intersection points closest to the origin). We prove that the shortest vector problem is NPhard (for randomized reductions) to approximate within some constant factor greater than 1 in any l p norm (p 1). In particular, we prove the NPhardness of approximating SVP in the Euclidean norm l 2 within any factor less than 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
P 2009, ‘Diameter bounds for equal area partitions of the unit sphere
 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. Another such construction is due to Feige and Schechtman. This paper gives a proof for the bounds on the di ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
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. Another such construction is due to Feige and Schechtman. This paper gives a proof for the bounds on the diameter of regions for each of these partitions.
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 ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
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
 Department of Computer Science, University of Wales
, 2002
"... ..."