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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.
Geometric Range Searching and Its Relatives
 CONTEMPORARY MATHEMATICS
"... ... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems. ..."
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Cited by 256 (40 self)
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... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems.
Similarity Indexing: Algorithms and Performance
 In Proceedings SPIE Storage and Retrieval for Image and Video Databases
, 1996
"... Efficient indexing support is essential to allow contentbased image and video databases using similaritybased retrieval to scale to large databases (tens of thousands up to millions of images). In this paper, we take an in depth look at this problem. One of the major difficulties in solving this pr ..."
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Cited by 111 (1 self)
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Efficient indexing support is essential to allow contentbased image and video databases using similaritybased retrieval to scale to large databases (tens of thousands up to millions of images). In this paper, we take an in depth look at this problem. One of the major difficulties in solving this problem is the high dimension (6100) of the feature vectors that are used to represent objects. We provide an overview of the work in computational geometry on this problem and highlight the results we found are most useful in practice, including the use of approximate nearest neighbor algorithms. We also present a variant of the optimized kd tree we call the VAM kd tree, and provide algorithms to create an optimized Rtree we call the VAMSplit Rtree. We found that the VAMSplit Rtree provided better overall performance than all competing structures we tested for main memory and secondary memory applications. We observed large improvements in performance relative to the R*tree and SStree in secondary memory applications, and modest improvements relative to optimized kd tree variants.Nearest Neighbor Search
A Random Sampling Scheme for Path Planning
 INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH
, 1996
"... Several randomizod path planners have been proposed during the last few years. Their attractiveness stems from their applicability to virtually any type of robots, and their empirically observed success. In this paper we attempt to present a unifying view of these planners and to theoretically expla ..."
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Cited by 82 (26 self)
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Several randomizod path planners have been proposed during the last few years. Their attractiveness stems from their applicability to virtually any type of robots, and their empirically observed success. In this paper we attempt to present a unifying view of these planners and to theoretically explain their success. First, we introduce a general planning scheme that consists of randomly sampling the robot' s configuration space. We then describe two previously developed planners as instances of planners based on this scheme, but applying very different sampling strategies. These planners are probabilistically complete: if a path exists, they will find one with high probability, if we let them run long enough. Next, for one of the planners, we analyze the relation between the probability of failure and the running time. Under assumptions characterizing the "goodness" of the robot's free space, we show that the running time only grows as the absolute value of the logarithm of the probability of failure that we are willing to tolerate. We also show that it increases at a reasonable rate as the space goodness degrades. In the last section we suggest directions for future research.
Range Searching
, 1996
"... Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , an ..."
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Cited by 70 (1 self)
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Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.
On approximating the depth and related problems
 SIAM J. Comput
"... We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear pr ..."
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Cited by 63 (11 self)
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We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear programming with violations approximately. We also use this technique to approximate the disk covering the largest number of red points, while avoiding all the blue points, given two such sets in the plane. Using similar techniques imply that approximate range counting queries have roughly the same time and space complexity as emptiness range queries. 1
Approximate Nearest Neighbor Queries Revisited
, 1998
"... This paper proposes new methods to answer approximate nearest neighbor queries on a set of n points in ddimensional Euclidean space. For any fixed constant d, a data structure with O(" (1\Gammad)=2 n log n) preprocessing time and O(" (1\Gammad)=2 log n) query time achieves approximation factor ..."
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Cited by 57 (3 self)
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This paper proposes new methods to answer approximate nearest neighbor queries on a set of n points in ddimensional Euclidean space. For any fixed constant d, a data structure with O(" (1\Gammad)=2 n log n) preprocessing time and O(" (1\Gammad)=2 log n) query time achieves approximation factor 1 + " for any given 0 ! " ! 1; a variant reduces the "dependence by a factor of " \Gamma1=2 . For any arbitrary d, a data structure with O(d 2 n log n) preprocessing time and O(d 2 log n) query time achieves approximation factor O(d 3=2 ). Applications to various proximity problems are discussed. 1 Introduction Let P be a set of n point sites in ddimensional space IR d . In the wellknown post office problem, we want to preprocess P into a data structure so that a site closest to a given query point q (called the nearest neighbor of q) can be found efficiently. Distances are measured under the Euclidean metric. The post office problem has many applications within computational...
Balanced Aspect Ratio Trees: Combining the Advantages of kd Trees and Octrees
"... Given a set S of n points in R^d, we show, for fixed d, how to construct in O(n log n) time a data structure we call the Balanced Aspect Ratio (BAR) tree. A BAR tree is a binary space partition tree on S that has O(logn) depth and in which every region is convex and “fat ” (that is, has a bounded as ..."
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Cited by 55 (8 self)
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Given a set S of n points in R^d, we show, for fixed d, how to construct in O(n log n) time a data structure we call the Balanced Aspect Ratio (BAR) tree. A BAR tree is a binary space partition tree on S that has O(logn) depth and in which every region is convex and “fat ” (that is, has a bounded aspect ratio). While previous hierarchical data structures, such as kd trees, quadtrees, octrees, fairsplit trees, and balanced box decompositions can guarantee some of these properties, we know of no previous data structure that combines alI of these properties simultaneously. The BAR tree data structure has numerous applications ranging from solving several geometric searching problems in fixed dimensional space to aiding in the visualization of graphs and threedimensional worlds.
Approximate nearest neighbor searching in multimedia databases
 In Proc of 17th IEEE Int. Conf. on Data Engineering (ICDE
, 2001
"... In this paper, we develop a general framework for approximate nearest neighbor queries. We categorize the current approaches for nearest neighbor query processing based on either their ability to reduce the data set that needs to be examined, or their ability to reduce the representation size of eac ..."
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Cited by 48 (11 self)
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In this paper, we develop a general framework for approximate nearest neighbor queries. We categorize the current approaches for nearest neighbor query processing based on either their ability to reduce the data set that needs to be examined, or their ability to reduce the representation size of each data object. We first propose modifications to wellknown techniques to support the progressive processing of approximate nearest neighbor queries. A user may therefore stop the retrieval process once enough information has been returned. We then develop a new technique based on clustering that merges the benefits of the two general classes of approaches. Our clusterbased approach allows a user to progressively explore the approximate results with increasing accuracy. We propose a new metric for evaluation of approximate nearest neighbor searching techniques. Using both the proposed and the traditional metrics, we analyze and compare several techniques with a detailed performance evaluation. We demonstrate the feasibility and efficiency of approximate nearest neighbor searching. We perform experiments on several real data sets and establish the superiority of the proposed clusterbased technique over the existing techniques for approximate nearest neighbor searching. 1
The skip quadtree: a simple dynamic data structure for multidimensional data
 In Proc. 21st ACM Symposium on Computational Geometry
, 2005
"... We present a new multidimensional data structure, which we call the skip quadtree (for point data in R 2) or the skip octree (for point data in R d, with constant d> 2). Our data structure combines the best features of two wellknown data structures, in that it has the welldefined “box”shaped reg ..."
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Cited by 36 (5 self)
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We present a new multidimensional data structure, which we call the skip quadtree (for point data in R 2) or the skip octree (for point data in R d, with constant d> 2). Our data structure combines the best features of two wellknown data structures, in that it has the welldefined “box”shaped regions of region quadtrees and the logarithmicheight search and update hierarchical structure of skip lists. Indeed, the bottom level of our structure is exactly a region quadtree (or octree for higher dimensional data). We describe efficient algorithms for inserting and deleting points in a skip quadtree, as well as fast methods for performing point location and approximate range queries. 1