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101
The quadtree and related hierarchical data structures
 ACM Computing Surveys
, 1984
"... A tutorial survey is presented of the quadtree and related hierarchical data structures. They are based on the principle of recursive decomposition. The emphasis is on the representation of data used in applications in image processing, computer graphics, geographic information systems, and robotics ..."
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Cited by 420 (11 self)
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A tutorial survey is presented of the quadtree and related hierarchical data structures. They are based on the principle of recursive decomposition. The emphasis is on the representation of data used in applications in image processing, computer graphics, geographic information systems, and robotics. There is a greater emphasis on region data (i.e., twodimensional shapes) and to a lesser extent on point, curvilinear, and threedimensional data. A number of operations in which such data structures find use are examined in greater detail.
Spatial Data Structures
, 1995
"... An overview is presented of the use of spatial data structures in spatial databases. The focus is on hierarchical data structures, including a number of variants of quadtrees, which sort the data with respect to the space occupied by it. Suchtechniques are known as spatial indexing methods. Hierarch ..."
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Cited by 287 (13 self)
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An overview is presented of the use of spatial data structures in spatial databases. The focus is on hierarchical data structures, including a number of variants of quadtrees, which sort the data with respect to the space occupied by it. Suchtechniques are known as spatial indexing methods. Hierarchical data structures are based on the principle of recursive decomposition. They are attractive because they are compact and depending on the nature of the data they save space as well as time and also facilitate operations such as search. Examples are given of the use of these data structures in the representation of different data types such as regions, points, rectangles, lines, and volumes.
On Packing Rtrees
 In ACM CIKM
, 1993
"... – main idea; file structure – algorithms: insertion/split – deletion – search: range, nn, spatial joins – performance analysis – variations (packed; hilbert;...) 15721 Copyright: C. Faloutsos (2001) 2 Problem • Given a collection of geometric objects (points, lines, polygons,...) • organize them on ..."
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Cited by 219 (16 self)
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– main idea; file structure – algorithms: insertion/split – deletion – search: range, nn, spatial joins – performance analysis – variations (packed; hilbert;...) 15721 Copyright: C. Faloutsos (2001) 2 Problem • Given a collection of geometric objects (points, lines, polygons,...) • organize them on disk, to answer spatial queries (range, nn, etc) 15721 Copyright: C. Faloutsos (2001) 3 1 (Who cares?)
Hilbert Rtree: An improved Rtree using fractals
, 1994
"... We propose a new Rtree structure that outperforms all the older ones. The heart of the idea is to facilitate the deferred splitting approach in Rtrees. This is done by proposing an ordering on the Rtree nodes. This ordering has to be 'good', in the sense that it should group 'similar' data rectan ..."
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Cited by 184 (9 self)
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We propose a new Rtree structure that outperforms all the older ones. The heart of the idea is to facilitate the deferred splitting approach in Rtrees. This is done by proposing an ordering on the Rtree nodes. This ordering has to be 'good', in the sense that it should group 'similar' data rectangles together, to minimize the area and perimeter of the resulting minimum bounding rectangles (MBRs). Following [19] we have chosen the socalled '2Dc' method, which sorts rectangles according to the Hilbert value of the center of the rectangles. Given the ordering, every node has a welldefined set of sibling nodes; thus, we can use deferred splitting. By adjusting the split policy, the Hilbert Rtree can achieve as high utilization as desired. To the contrary, the R tree has no control over the space utilization, typically achieving up to 70%. We designed the manipulation algorithms in detail, and we did a full implementation of the Hilbert Rtree. Our experiments show that the '2to...
Beyond uniformity and independence: Analysis of rtrees using the concept of fractal dimension
 In Proc. PODS
, 1994
"... We propose the concept of fractal dimension of a set of points, in order to quantify the deviation from the uniformity distribution. Using measurements on real data sets (road intersections of U.S. counties, star coordinates from NASA’s InfraredUltraviolet Explorer etc.) we provide evidence that re ..."
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Cited by 154 (19 self)
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We propose the concept of fractal dimension of a set of points, in order to quantify the deviation from the uniformity distribution. Using measurements on real data sets (road intersections of U.S. counties, star coordinates from NASA’s InfraredUltraviolet Explorer etc.) we provide evidence that real data indeed are skewed, and, moreover, we show that they behave as mathematical fractals, with a measurable, noninteger fract al dimension. Armed with this tool, we then show its practical use in predicting the performance of spatial access methods, and specifically of the Rtrees. We provide the jirst analysis of Rtrees for skewed distributions of points: We develop a formula that estimates the number of disk accesses for range queries, given only the fractal dimension of the point set, and its count. Experiments on real data sets show that the formula is very accurate: the relative error is usually below 5%, and it rarely exceeds 10%. We believe that the fractal dimension will help replace the uniformity and independence assumptions, allowing more accurate analysis for any spatial access method, as well as better estimates for query optimization on multiattribute queries. 1
A Simple Algorithm for Nearest Neighbor Search in High Dimensions
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1997
"... Abstract—The problem of finding the closest point in highdimensional spaces is common in pattern recognition. Unfortunately, the complexity of most existing search algorithms, such as kd tree and Rtree, grows exponentially with dimension, making them impractical for dimensionality above 15. In ne ..."
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Cited by 125 (1 self)
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Abstract—The problem of finding the closest point in highdimensional spaces is common in pattern recognition. Unfortunately, the complexity of most existing search algorithms, such as kd tree and Rtree, grows exponentially with dimension, making them impractical for dimensionality above 15. In nearly all applications, the closest point is of interest only if it lies within a userspecified distance e. We present a simple and practical algorithm to efficiently search for the nearest neighbor within Euclidean distance e. The use of projection search combined with a novel data structure dramatically improves performance in high dimensions. A complexity analysis is presented which helps to automatically determine e in structured problems. A comprehensive set of benchmarks clearly shows the superiority of the proposed algorithm for a variety of structured and unstructured search problems. Object recognition is demonstrated as an example application. The simplicity of the algorithm makes it possible to construct an inexpensive hardware search engine which can be 100 times faster than its software equivalent. A C++ implementation of our algorithm is available upon request to search@cs.columbia.edu/CAVE/.
Parallel Rtrees
, 1992
"... We consider the problem of exploiting parallelism to accelerate the performance of spatial access methods and specifically, Rtrees [11]. Our goal is to design a server for spatial data, so that to maximize the throughput of range queries. This can be achieved by (a) maximizing parallelism for large ..."
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Cited by 69 (1 self)
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We consider the problem of exploiting parallelism to accelerate the performance of spatial access methods and specifically, Rtrees [11]. Our goal is to design a server for spatial data, so that to maximize the throughput of range queries. This can be achieved by (a) maximizing parallelism for large range queries, and (b) by engaging as few disks as possible on point queries [22]. We propose a simple hardware architecture consisting of one processor with several disks attached to it. On this architecture, we propose to distribute the nodes of a traditional Rtree, with crossdisk pointers (`Multiplexed' Rtree). The Rtree code is identical to the one for a singledisk Rtree, with the only addition that we have to decide which disk a newly created Rtree node should be stored in. We propose and examine several criteria to choose a disk for a new node. The most successful one, termed `proximity index' or PI, estimates the similarity of the new node with the other Rtree nodes already o...
STRIPES: An Efficient Index for Predicted Trajectories
 in SIGMOD
, 2004
"... Moving object databases are required to support queries on a large number of continuously moving objects. A key requirement for indexing methods in this domain is to efficiently support both update and query operations. Previous work on indexing such databases can be broadly divided into two categor ..."
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Cited by 66 (0 self)
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Moving object databases are required to support queries on a large number of continuously moving objects. A key requirement for indexing methods in this domain is to efficiently support both update and query operations. Previous work on indexing such databases can be broadly divided into two categories: indexing the past positions and indexing the future predicted positions. In this paper we focus on an efficient indexing method for indexing the future positions of moving objects. In this paper we propose an indexing method, called STRIPES, which indexes predicted trajectories in a dual transformed space. Trajectories for objects in ddimensional space become points in a higherdimensional 2dspace. This dual transformed space is then indexed using a regular hierarchical grid decomposition indexing structure. STRIPES can evaluate a range of queries including timeslice, window, and moving queries. We have carried out extensive experimental evaluation comparing the performance of STRIPES with the best known existing predicted trajectory index (the TPR*tree), and show that our approach is significantly faster than TPR*tree for both updates and search queries. 1.
Fast and effective retrieval of medical tumor shapes
 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
, 1998
"... We investigate the problem of retrieving similar shapes from a large database; in particular, we focus on medical tumor shapes (“Find tumors that are similar to a given pattern.”). We use a natural similarity function for shapematching, based on concepts from mathematical morphology, and we show h ..."
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Cited by 47 (0 self)
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We investigate the problem of retrieving similar shapes from a large database; in particular, we focus on medical tumor shapes (“Find tumors that are similar to a given pattern.”). We use a natural similarity function for shapematching, based on concepts from mathematical morphology, and we show how it can be lowerbounded by a set of shape features for safely pruning candidates, thus giving fast and correct output. These features can be organized in a spatial access method, leading to fast indexing for range queries and nearestneighbor queries. In addition to the lowerbounding, our second contribution is the design of a fast algorithm for nearestneighbor search, achieving significant speedup while provably guaranteeing correctness. Our experiments demonstrate that roughly 90 percent of the candidates can be pruned using these techniques, resulting in up to 27 times better performance compared to sequential scan.
Scalable Network Distance Browsing in Spatial Databases
, 2008
"... An algorithm is presented for finding the k nearest neighbors in a spatial network in a bestfirst manner using network distance. The algorithm is based on precomputing the shortest paths between all possible vertices in the network and then making use of an encoding that takes advantage of the fact ..."
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Cited by 46 (8 self)
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An algorithm is presented for finding the k nearest neighbors in a spatial network in a bestfirst manner using network distance. The algorithm is based on precomputing the shortest paths between all possible vertices in the network and then making use of an encoding that takes advantage of the fact that the shortest paths from vertex u to all of the remaining vertices can be decomposed into subsets based on the first edges on the shortest paths to them from u. Thus, in the worst case, the amount of work depends on the number of objects that are examined and the number of links on the shortest paths to them from q, rather than depending on the number of vertices in the network. The amount of storage required to keep track of the subsets is reduced by taking advantage of their spatial coherence which is captured by the aid of a shortest path quadtree. In particular, experiments on a number of large road networks as