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An Optimal Algorithm for Approximate Nearest Neighbor Searching in Fixed Dimensions
 ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 1994
"... Consider a set S of n data points in real ddimensional space, R d , where distances are measured using any Minkowski metric. In nearest neighbor searching we preprocess S into a data structure, so that given any query point q 2 R d , the closest point of S to q can be reported quickly. Given any po ..."
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Cited by 983 (32 self)
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query point q 2 R d , and ffl ? 0, a (1 + ffl)approximate nearest neighbor of q can be computed in O(c d;ffl log n) time, where c d;ffl d d1 + 6d=ffle d is a factor depending only on dimension and ffl. In general, we show that given an integer k 1, (1 + ffl)approximations to the k nearest neighbors
Matrix positivity preservers in fixed dimension
, 2015
"... A classical theorem of I.J. Schoenberg characterizes functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size. Obtaining similar characterizations in fixed dimension is intricate. In this note, we provide a solution to this problem in the polynomi ..."
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Cited by 1 (1 self)
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A classical theorem of I.J. Schoenberg characterizes functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size. Obtaining similar characterizations in fixed dimension is intricate. In this note, we provide a solution to this problem
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 136 (9 self)
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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
Using Separation Algorithms in Fixed Dimension
 J. ALGORITHMS
, 1989
"... Consider a convex set in d dimensions. Assume that we are given a separation subroutine which, given a point, tells us whether this point is in the set. Moreover, if the point is not in the set, the subroutine separates the point from the set by a hyperplane. We show that if d is fixed and the se ..."
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Cited by 18 (2 self)
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Consider a convex set in d dimensions. Assume that we are given a separation subroutine which, given a point, tells us whether this point is in the set. Moreover, if the point is not in the set, the subroutine separates the point from the set by a hyperplane. We show that if d is fixed
Integer polynomial optimization in fixed dimension
 MATHEMATICS OF OPERATIONS RESEARCH
, 2006
"... We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an integer polynomial over the lattice points of a convex polytope ..."
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Cited by 16 (7 self)
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We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an integer polynomial over the lattice points of a convex
Maximizing Concave Functions in Fixed Dimension
 in: Complexity in Numeric Computation
, 1993
"... In [3, 5, 2] the authors introduced a technique which enabled them to solve the parametric minimum cycle problem with a fixed number of parameters in strongly polynomial time. In the current paper 1 we present this technique as a general tool. In order to allow for an independent reading of this p ..."
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Cited by 14 (0 self)
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In [3, 5, 2] the authors introduced a technique which enabled them to solve the parametric minimum cycle problem with a fixed number of parameters in strongly polynomial time. In the current paper 1 we present this technique as a general tool. In order to allow for an independent reading
Parametric Integer Programming in Fixed Dimension
, 2008
"... We consider the following problem: Given a rational matrix A ∈ Qm×n and a rational polyhedron Q ⊆ Rm+p, decide if for all vectors b ∈ Rm, for which there exists an integral z ∈ Zp such that (b,z) ∈ Q, the system of linear inequalities Ax � b has an integral solution. We show that there exists an al ..."
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an algorithm that solves this problem in polynomial time if p and n are fixed. This extends a result of Kannan (1990) who established such an algorithm for the case when, in addition to p and n, the affine dimension of Q is fixed. As an application of this result, we describe an algorithm to find the maximum
into subcubes of a hypercube of fixed dimension
, 1995
"... We study the problem of scheduling independent jobs in a hypercube where jobs are executed in subcubes of various dimensions. The problem being NPcomplete, several approximation algorithms based on list scheduling have been proposed, having approximation ratio of order of 2. In this paper, a linear ..."
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We study the problem of scheduling independent jobs in a hypercube where jobs are executed in subcubes of various dimensions. The problem being NPcomplete, several approximation algorithms based on list scheduling have been proposed, having approximation ratio of order of 2. In this paper, a
Results 1  10
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1,051,053