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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 786 (31 self)
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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 positive real ffl, a data point p is a (1 + ffl)approximate nearest neighbor of q if its distance from q is within a factor of (1 + ffl) of the distance to the true nearest neighbor. We show that it is possible to preprocess a set of n points in R d in O(dn log n) time and O(dn) space, so that given a 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 of q can be computed in additional O(kd log n) time.
Voronoi diagrams  a survey of a fundamental geometric data structure
 ACM COMPUTING SURVEYS
, 1991
"... This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. ..."
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Cited by 560 (5 self)
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This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.
Searching in Metric Spaces
, 1999
"... The problem of searching the elements of a set which are close to a given query element under some similarity criterion has a vast number of applications in many branches of computer science, from pattern recognition to textual and multimedia information retrieval. We are interested in the rather ge ..."
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Cited by 321 (34 self)
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The problem of searching the elements of a set which are close to a given query element under some similarity criterion has a vast number of applications in many branches of computer science, from pattern recognition to textual and multimedia information retrieval. We are interested in the rather general case where the similarity criterion defines a metric space, instead of the more restricted case of a vector space. A large number of solutions have been proposed in different areas, in many cases without crossknowledge. Because of this, the same ideas have been reinvented several times, and very different presentations have been given for the same approaches. We
When Is "Nearest Neighbor" Meaningful?
 In Int. Conf. on Database Theory
, 1999
"... . We explore the effect of dimensionality on the "nearest neighbor " problem. We show that under a broad set of conditions (much broader than independent and identically distributed dimensions), as dimensionality increases, the distance to the nearest data point approaches the distance to the fa ..."
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Cited by 292 (1 self)
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. We explore the effect of dimensionality on the "nearest neighbor " problem. We show that under a broad set of conditions (much broader than independent and identically distributed dimensions), as dimensionality increases, the distance to the nearest data point approaches the distance to the farthest data point. To provide a practical perspective, we present empirical results on both real and synthetic data sets that demonstrate that this effect can occur for as few as 1015 dimensions. These results should not be interpreted to mean that highdimensional indexing is never meaningful; we illustrate this point by identifying some highdimensional workloads for which this effect does not occur. However, our results do emphasize that the methodology used almost universally in the database literature to evaluate highdimensional indexing techniques is flawed, and should be modified. In particular, most such techniques proposed in the literature are not evaluated versus simple...
Voronoi Diagrams and Delaunay Triangulations
 Computing in Euclidean Geometry
, 1992
"... The Voronoi diagram is a fundamental structure in computationalgeometry and arises naturally in many different fields. This chapter surveys properties of the Voronoi diagram and its geometric dual, the Delaunay triangulation. The emphasis is on practical algorithms for the construction of Voronoi ..."
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Cited by 198 (3 self)
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The Voronoi diagram is a fundamental structure in computationalgeometry and arises naturally in many different fields. This chapter surveys properties of the Voronoi diagram and its geometric dual, the Delaunay triangulation. The emphasis is on practical algorithms for the construction of Voronoi diagrams. 1 Introduction Let S be a set of n points in ddimensional euclidean space E d . The points of S are called sites. The Voronoi diagram of S splits E d into regions with one region for each site, so that the points in the region for site s2S are closer to s than to any other site in S. The Delaunay triangulation of S is the unique triangulation of S so that there are no elements of S inside the circumsphere of any triangle. Here `triangulation' is extended from the planar usage to arbitrary dimension: a triangulation decomposes the convex hull of S into simplices using elements of S as vertices. The existence and uniqueness of the Delaunay triangulation are perhaps not obvio...
The TVtree  an index structure for highdimensional data
 VLDB Journal
, 1994
"... We propose a file structure to index highdimensionality data, typically, points in some feature space. The idea is to use only a few of the features, utilizing additional features whenever the additional discriminatory power is absolutely necessary. We present in detail the design of our tree struc ..."
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Cited by 193 (7 self)
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We propose a file structure to index highdimensionality data, typically, points in some feature space. The idea is to use only a few of the features, utilizing additional features whenever the additional discriminatory power is absolutely necessary. We present in detail the design of our tree structure and the associated algorithms that handle such `varying length' feature vectors. Finally we report simulation results, comparing the proposed structure with the R tree, which is one of the most successful methods for lowdimensionality spaces. The results illustrate the superiority of our method, with up to 80% savings in disk accesses. Type of Contribution: New Index Structure, for highdimensionality feature spaces. Algorithms and performance measurements. Keywords: Spatial Index, Similarity Retrieval, Query by Content 1 Introduction Many applications require enhanced indexing, capable of performing similarity searching on several, nontraditional (`exotic') data types. The targ...
StatStream: Statistical Monitoring of Thousands of Data Streams in Real Time
 In VLDB
, 2002
"... Consider the problem of monitoring tens of thousands of time series data streams in an online fashion and making decisions based on them. In addition to single stream statistics such as average and standard deviation, we also want to find high correlations among all pairs of streams. A stock market ..."
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Cited by 167 (10 self)
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Consider the problem of monitoring tens of thousands of time series data streams in an online fashion and making decisions based on them. In addition to single stream statistics such as average and standard deviation, we also want to find high correlations among all pairs of streams. A stock market trader might use such a tool to spot arbitrage opportunities.
Geometric structures for threedimensional shape representation
 ACM Trans. Graph
, 1984
"... Different geometric structures are investigated in the context of discrete surface representation. It is shown that minimal representations (i.e., polyhedra) can be provided by a surfacebased method using nearest neighbors structures or by a volumebased method using the Delaunay triangulation. Bot ..."
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Cited by 166 (3 self)
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Different geometric structures are investigated in the context of discrete surface representation. It is shown that minimal representations (i.e., polyhedra) can be provided by a surfacebased method using nearest neighbors structures or by a volumebased method using the Delaunay triangulation. Both approaches are compared with respect to various criteria, such as space requirements, computation time, constraints on the distribution of the points, facilities for further calculations, and agreement with the actual shape of the object.
Finding Nearest Neighbors in Growthrestricted Metrics
 In 34th Annual ACM Symposium on the Theory of Computing
, 2002
"... Most research on nearest neighbor algorithms in the literature has been focused on the Euclidean case. In many practical search problems however, the underlying metric is nonEuclidean. Nearest neighbor algorithms for general metric spaces are quite weak, which motivates a search for other classes o ..."
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Cited by 150 (0 self)
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Most research on nearest neighbor algorithms in the literature has been focused on the Euclidean case. In many practical search problems however, the underlying metric is nonEuclidean. Nearest neighbor algorithms for general metric spaces are quite weak, which motivates a search for other classes of metric spaces that can be tractably searched.