H.V. Jagadish, "Spatial Search with Polyhedra," Proc. Sixth IEEE Int'l Conf. Data Eng., Feb. 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....potential of improving the e#ciency. Such approximations include circles, ellipses, rectilinear polygons, rotated mbbs, trapezoids, k corner bounding convex polygons, and convex hull. Indexing techniques including sphere trees [14] and SS Trees [20] for circles, cell trees [7] and polyh edratree [10] for convex polygons, and extended R # trees for di#erent approximation methods [3] were developed for some of these approximations. It is not hard to believe that finer approximations than mbbs can also be used for trajectories in order to improve query performance. The main technical ....

....are more e#cient than that for mbbs. Previous study in spatial databases focused on approximating 2 d spatial objects. Di#erent approximations have been proposed and data structures were developed for them: e.g. sphere trees [14] and SS Trees [20] for circles, Cell trees [7] and polyhedra tree [10] for convex polygons. 3] compared mbb, rotated mbb, minimum bounding circle, minimum bounding ellipse, convex hull (ch) and minimum bounding k corner convex polygon (k cn) based on point location region queries for 2 d region data. R # trees augmented by approximations in the leaf nodes were ....

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H. V. Jagadish. Spatial search with polyhedra. In Proc. ICDE, 1990.

....data collections. The Database people, on the other hand, are mainly concerned with the efficiency aspect of a multimedia system. For this purpose, they have investigated a broad range of techniques regarding storage (e.g. 3] data modeling (e.g. 11] and multi dimensional indexing (e.g. [6][12] for multimedia, all of which unanimously suggest the use of simple, uniform, and low dimensional features that allow economical and efficient storage, indexing and access. The feature effectiveness in terms of retrieval accuracy, however, is in most cases taken for granted due to the ....

Jagadish, H. V., "Spatial search with polyhedra", Proc. of Sixth IEEE Int. Conf. on Data Engineering, pp. 311-319, 1990.

....The R tree was empirically shown to improve search performance by up to 50 , compared to the basic R tree. Also, space utilization was improved. The P tree In many R tree variants, the predicates describing tree nodes are (hyper) rectangles. A notable exception is the P tree of Jagadish [Jag90] In the P tree, a node s predicate can be an arbitrary polyhedron. This approach can substantially improve the approximation of the contents of a node. In the experiments reported in [Jag90] it was found that 10 gons are a good choice for two dimensional data. A drawback of the approach is that ....

....the predicates describing tree nodes are (hyper) rectangles. A notable exception is the P tree of Jagadish [Jag90] In the P tree, a node s predicate can be an arbitrary polyhedron. This approach can substantially improve the approximation of the contents of a node. In the experiments reported in [Jag90] it was found that 10 gons are a good choice for two dimensional data. A drawback of the approach is that 10 gons require more storage space than rectangles, and thus the degree of the tree nodes is reduced, leading to P trees that are higher than corresponding R trees. R tree variants for ....

H.V. Jagadish. Spatial search with polyhedra. In Proc. IEEE Intl. Conf. Data Engineering, pages 311--319, 1990.

....in the n dimensional space. The linear quadtree is based on the two dimensional Peano curve (also termed the Z or N order [28] Many studies have been conducted to compare the performance of spatial operations for spatial databases adopting certain space filling orders. The reader is referred to [2, 10, 14, 20]. a b c d e f g h i Figure 1: An example quadtree for storing line segments. The linear quadtree can be used to store a variety of spatial object types. For example, a quadtree data structure for storing line segments [25] subdivides the feature space successively into four equal sized ....

H. V. Jagadish. Spatial search with polyhedra. In Proc. of the 6th IEEE Intl. Conf. on Data Engr., pp. 311--319, Los Angeles, Feb. 1990.

....and also introduce optimized algorithms for query processing using striped trees. Note that tree striping as defined so far is independent of the multidimensional index structure used. Any multidimensional index structure such as the R tree [7] and its variants (R tree [17] R tree [2] P tree [9]) Buddy tree [16] linear quadtrees [6] z ordering [14] or other space filling curves [8] and gridfile based methods [13, 5] may be used for this purpose. Before we describe our theoretical model, we first provide a simple algorithm for processing queries using striped trees. As the single ....

Jagadish H. V.: `Spatial Search with Polyhedra', Proc. 6th Int. Conf. on Data Engineering, Los Angeles, CA, 1990, pp. 311-319.

....interval. This includes priority search trees [McC] which have optimal worst case bounds for (1 dimensional) intervals, optimal interval index in secondary storage [KRVV93] R trees [Gut84, RL84, RL85] R trees [SRF87] R trees [BKSS90] for 2 dimensional intervals (i.e. rectangles) P trees [J90] which extend Rtrees to higher dimensions, and indices based on combination of interval and range trees [Ed83, SiW82] To facilitate the above indexing structures for arbitrary objects, the idea of filtering using minimum bounding boxes (MBB) also called minimum bounding rectangles, is widely ....

....2, whatever the query object, the filtering provides no help as all MBBs intersect. In MBBs, the hyper planes used to bound the objects can only have 2 directions in a 2 dimensional space and, in general, d directions in d dimensional space. To improve the selectivity (or filtering) of MBBs, [J90] suggested increasing the number of directions of hyper planes beyond the dimension of the space. Each facet of the proposed bounding polyhedron of an object 2 These methods are outside the scope of this paper. They include uniform grid method [Fr84] quadtree based methods [Ta82, SaW85, NS87, ....

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H.V. Jagadish, Spatial Search with Polyhedra, Proc. IEEE International Conference on Data Engineering, 311--319, 1990.

....solvers. The degree of incompleteness of the constraint solver determines how well indexed a structure is. O trees are certainly also definable in the GiST framework, but to the best of our knowledge they have not been previously empirically studied. They can be seen as a form of P tree [5]. P trees are defined by mapping an object to the lower and upper bounds on any fixed number k of dimensions (in our case x, y, v and w) and then storing the result in a k dimensional R tree. This approach immediately loses the connection between the dimensions that is vital for strong constraint ....

....by mapping an object to the lower and upper bounds on any fixed number k of dimensions (in our case x, y, v and w) and then storing the result in a k dimensional R tree. This approach immediately loses the connection between the dimensions that is vital for strong constraint solving behaviour. [5] also only gives a theoretical discussion of P trees, and gives no empirical comparison. 7. Conclusion and Future Work Height balanced constraint search trees are the natural form of constraint search tree for storing large amounts of constraint data. The constraint viewpoint makes the ....

H. V. Jagadish. Spatial search with polyhedra. In Proceedings of the IEEE Int. Conf on Data Engineering, pages 311--319, 1990.

....methods based on trees (R tree [23] k dtrees [6] k d B trees [49] hB trees [35] cell trees [22] etc. One of the most promising approaches in the last class is the R tree [23] and its numerous variants (Greene s variation [21] the R tree [51] R trees using Minimum Bounding Polygons [28], the R tree [5] the Hilbert R tree [32] etc. We use R trees, because they have already been used successfully for highdimensionality spaces (10 20 dimensions [15] in contrast, grid files and linear quadtrees may suffer from the dimensionality curse . 2.3 Tumor Growth Models Our target ....

H. V. Jagadish. Spatial search with polyhedra. Proc. Sixth IEEE Int'l Conf. on Data Engineering, February 1990.

....proposed to retrieve objects which are n dimensional points or solids. The spatial access methods optimize queries to retrieve all points or solids enclosed in or overlapping with a given search region. The proposed access methods include R tree [5, 6] Grid Files [7 11] and other search trees [12, 13, 13 17]. However, these access methods were designed with the assumption of static world with no moving objects. The update rates were assumed to be much smaller than the search query rate. The object shape, size and location distributions are assumed to be random rectangles with a high degree of ....

....query is directed to the regions which overlap the area of the given query. Typically the query examines one of the sub regions at the leaf 4 level of the R tree to identify the objects intersecting the area. R tree [5] R tree[6] k D B tree [17] hB tree [13] and searching with polyhedra [12] are relevant access methods. R tree is a a multilevel balanced indexing tree structure allowing overlapping and has more than 50 space utilization. R trees have the advantage of efficient secondary storage usage while suffers the problem of complex page split algorithm and excessive overlap of ....

H.V. Jagadish, Spatial Search with Polyhedra, 6th International Conference on Data Engineering, (1990).

....either or both assumptions do not hold. Apart from these problems, which we show how to solve, our application needs a multidimensional indexing method that works for large, disk based databases. The prevailing methods form three classes: a) R 3 trees [4] and the rest of the R tree family [18, 23]; b) linear quadtrees [41] and (c) grid files [35] Most multidimensional indexing methods explode exponentially with the dimensionality, eventually reducing to sequential scanning. For linear quadtrees, the effort is proportional to the hypersurface of the query region [20] the hypersurface ....

H. V. Jagadish. Spatial search with polyhedra. Proc. Sixth IEEE Int'l Conf. on Data Engineering, February 1990.

....revesz cse.unl.edu 1 Introduction Suppose that we are presented with a new picture and want to find the most similar pictures that are already present in a spatial database representing in some way a large number of pictures. The above task is a similarity retrieval problem on image databases [1, 3, 4, 5, 8, 10]. It has many applications. For example, an art gallery may want to find paintings similar to the paintings of a favorite artist, police may want to search a fingerprints database when they find new fingerprints at a crime scene, or a real estate agent may want to find a home with a blueprint ....

....sense because the longer the lines the more significant visually the angle difference becomes. Hence we cannot talk about some simple linear relationship of angle difference plus length difference. Previously proposed similarity or distance measures do not consider this non linear relationship [1, 3, 4, 5, 8, 10]. Next we give a few simple examples assuming for simplicity that ff = fi = fl = 1. Example 2.1 As an simple example shown in figure 1, the two squares have the DistCP nt equals to p (20 Gamma 10) 2 (17 Gamma 10) 2 which equals to 12.2. There is no rotation between these two polygons, ....

H.V.Jagadish, Spatial Search with Polyhedra, Sixth International Conference on Data Engineering, IEEE 1990, pp311-319.

....taken into account. In addition to using one filter, several filters of different type may be combined into a filter sequence. Each filter in such a sequence will reduce the operand in size. Optimization techniques developed to access huge amounts of geometric data concentrate on spatial indices [8, 9, 18]. Emphasis is on efficient retrieval of sets of geometrical objects. Using a hierarchically organized structure a spatial index tries to retrieve from disk a requested set of disk pages, containing the set of objects, with minimal cost for disk I O. In [14] an overview and classification is given ....

....bounding circle (MBC) 20] the minimum bounding ellipse (MBE) 21] the convex hull (CH) 1] 17] and the n corner. An n corner is defined as the optimal n sided polygon circumscribing a convex polygon [6] In addition to these approximations we use a variant of the MBB, derived from ideas in [9], that is very well suited for solving n way joins. This is the g degrees rotated x range (gRxR) A gRxR is a pair of two x values [x min ; xmax ] x origin O y y origin O x 90RxR 45RxR 0RxR 90RxR 45RxR 0RxR 0RxR, 45RxR, 90RxR Fig. 3. View of the single filters 0RxR, 45RxR, 90RxR, and ....

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Jagadish, H.V.: Spatial search with Polyhedra. Proc. 6th. Int Conf. on Data Engineering, Los Angeles, California, USA, 1990.

....Access Method (SAM) which, by definition, is a method that can handle k dimensional points, rectangles, or even more complicated shapes. The most popular methods form three classes: a) tree based methods like the R tree [Gut84] and its variants (R tree [SRF87] hB tree [LS90] P tree [Jag90a] R tree [BKSS90] Hilbert R trees [KF94] Symbols Definitions. N Number of objects in database n dimensionality of original space ( features case only) k dimensionality of target space D( the distance function between two objects jj xjj 2 the length ( L 2 norm) of vector x ....

H. V. Jagadish. Spatial search with polyhedra. Proc. Sixth IEEE Int'l Conf. on Data Engineering, February 1990.

....be a Spatial Access Method (SAM) which, by definition, is a method that can handle k dimensional points, rectangles, or even more complicated shapes. The most popular methods form three classes: a) tree based methods like the R tree [20] and its variants (R tree [45] hB tree [31] P tree [24], R tree [7] Hilbert Rtrees [27] etc. b) methods using linear quadtrees [18] or, equivalently, the z ordering [37, 38] or other spacefilling curves [14, 23] and finally (c) methods that use grid files [36, 22] There are also retrieval methods for the case where only the triangular ....

H.V. Jagadish. Spatial search with polyhedra. Proc. Sixth IEEE Int'l Conf. on Data Engineering, February 1990.

....it is not going to change and the one time preprocessing step is worth the effort. We have previously mentioned that R trees are used for performing collision detection. These structures are also frequently used for searching and retrieving information from large (multidimensional) databases [41, 82, 88, 14, 55, 98, 57]. In this application, the databases are often subject to many insertions and deletions, so being able to efficiently maintain a dynamic data structure is very important. From this viewpoint, assuming full knowledge of the set of primitive objects is not realistic and therefore, not often done. ....

H. V. Jagadish. Spatial Search with Polyhedra In Proc. 6 th Int. Conf. Data Engineering, Los Angeles, USA, pp. 311--319, Feb. 1990.

....black) organized in an R tree with fanout 3. 0 1 0 1 4 5 6 8 10 11 12 1 2 3 Figure 2: Data (dark rectangles) organized in an R tree with fanout=3 Subsequent work on R trees includes the packed [19] and Hilbert packed R trees [12] the R tree [23] R trees using Minimum Bounding Polygons [11] and the R tree [4] The latter seems to give the best search times, mainly thanks to the idea of deferring the splits by force reinserting some of the entries of the overflowing nodes. The last part in this section is a summary of previous attempts to analyze the R trees. In [6] we assumed ....

H. V. Jagadish. Spatial search with polyhedra. Proc. Sixth IEEE Int'l Conf. on Data Engineering, February 1990.

....names come from their complexity; among the three, the quadratic split algorithm is the one that achieves the best trade off between splitting time and search performance. Subsequent work on R trees includes the work by Greene [11] the R tree [27] R trees using Minimum Bounding Polygons [17], and finally, the R tree [3] which seems to have the best performance among the R tree variants. The main idea in the R tree is the concept of forced re insert. When a node overflows, some of its children are carefully chosen; they are deleted and re inserted, usually resulting in a ....

H. V. Jagadish. Spatial search with polyhedra. Proc. Sixth IEEE Int'l Conf. on Data Engineering, February 1990.

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H.V. Jagadish, "Spatial Search with Polyhedra," Proc. Sixth IEEE Int'l Conf. Data Eng., Feb. 1990.

....which can be stored in some multiresolution, hierarchical fashion, clustering related (i.e. nearby) points together. Whenever a transformation is used (e.g. a twodimensional rectangle corresponds to a fourdimensional point [9] 12] a polyhedron is mapped to a high dimensionality point [15]) We focus on rectilinear hyperrectangles, that is, n d rectangles with sides aligned with the axis. The problem we examine here is the following: 1041 4347 97 10.00 1997 IEEE . C. Faloutsos is with the Department of Computer Science, University ....

H.V. Jagadish, "Spatial Search with Polyhedra," Proc. Sixth IEEE Int'l Conf. Data Eng., Feb. 1990.

....12 = 0:9166 Delta Delta Delta and 1248 6 Theta32 2 = 0:8125, respectively. This indicates that, in a d dimensional space (d 3) accessing the minimum bounding hyper rectangle of a given query region may incur additional non consecutive disk accesses, and hence supports the argument made in [15] that the minimum bounding rectangle may not be a good approximation of a non rectangular object. 5.3 Comparison with the Gray coded and z curves It may be argued that it is not convincing to make a definitive conclusion that the Hilbert curve is better or worse than others solely on the basis of ....

H. V. Jagadish. Spatial search with polyhedra. In Proceedings of the 6th Inter. Conference on Data Engineering, pages 311--319, Los Angeles, LA, February 1990.

....(s) and f c (s) are fairly close to 11 12 and 4872 6 Theta32 2 , respectively. This indicates that, in a d dimensional space (d 3) accessing the minimum bounding hyper rectangle of a given query region may incur additional non consecutive disk accesses, and hence supports the argument made in [13] that the minimum bounding rectangle may not be a good approximation to a non rectangular object. The main conclusions from our experiments are: ffl The exact solution given in Theorem 2 matches exactly the experimental results for square queries of size 2 k Theta 2 k . ffl The asymptotic ....

H.V. Jagadish. Spatial search with polyhedra. Proc. Sixth IEEE Int'l Conf. on Data Engineering, February 1990.

....which can be stored in some multi resolution, hierarchical fashion, clustering related (i.e. nearby) points together. ffl Whenever a transformation is used (e.g. a 2 dimensional rectangle corresponds to a 4 dimensional point [9, 12] a polyhedron is mapped to a high dimensionality point [14]) The problem we examine here is the following: Given a hyper rectangle of size s 1 Theta s 2 Theta : s n , Find the number of blocks that it will span on the average. Previous attempts have been restricted to 2 dimensional rectangles: Dyer in [6] presented an analysis for the best, worst ....

H. V. Jagadish. Spatial search with polyhedra. Proc. Sixth IEEE Int'l Conf. on Data Engineering, February 1990.

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H. V. Jagadish. Spatial search with polyhedra. In Proceedings of ICDE'90, pages 311--319, 1990.

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H.V. Jagadish: "Spatial Search with Polyhedra", Proceedings 6th IEEE ICDE Conference, pp.311-319, Orlando, FL, 1990.

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H. V. Jagadish. Spatial search with polyhedra. In Proceedings of the 6th IEEE Conference on Data Engineering, February 1990.

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