I. Kamel, C. Faloutos: "On Packing R-trees". Proc. Int. Conf. on Information and Knowledge Managemenet (CIKM'93), 1993, pp. 490499.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....for a given objective function. Thus, we have reduced the complexity of the optimal splitting algorithm from exponential to O(n ) where d 1 is the number of dimensions of the input data. In our experimental results we choose minimization of the expected number of node accesses as stated in [Kam93] (i.e. A q x Delta L y L x Delta q y q x Delta q y Delta N , where A; L x ; L y ; q x ; q y ; N are the sum of areas, sum of node extents in x, y, extents of query in x,y and number of nodes respectively) Note that for point queries it is just the sum of areas. The other major ....

....utilization. The Hilbert R tree [Kam94] achieves improved node utilization by using 2 3 splitting. In addition to the split algorithm (local optimization) overall tree structure (global optimization) has a significant impact on search performance. The superior performance of packing algorithms [Rou85, Kam93, Leu97] is attributable to both better node occupancy and better tree structure. The dynamic Hilbert R tree not only achieves good overall node utilization, but also improves overall tree structure by following the Hilbert order. We have devised a new dynamic algorithm incorporating improved node ....

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Kamel, I., Faloutsos, C., "On Packing R-trees", Proc. 2nd International Conference on Information and Knowledge Management, p. 490-499, Arlington, VA, November 1993 (CKIM-93).

....ID(S) for each data sequence S and then inserts it into a multi dimensional index. Here, ID(S) is the identifier of S. If there are a large number of data sequences at the stage of initial index construction, we can achieve high performance in construction by hiring bulk loading methods [6, 15, 16]. 4.3.2 Query Processing Algorithm 1 shows TW Sim Search, our query processing algorithm. Step 1 extracts a 4 tuple feature vector from the query sequence. Step 2 performs a square range query on a four dimensional index. The range query uses a 4 tuple feature vector obtained in Step 1 as a ....

I. Kamel, C. Faloutsos, "On Packing R-trees", Proc. ACM CIKM , pp. 490-499, 1993.

....the same situation also occurs. Obviously, this merging may lead to a great enlargement of not only the leaf node MBRs but also the overlap among the leaf nodes, which will cut down search performance of R tree. 3.2. Weakness of packed R trees Some packing algorithms on R trees are proposed in [8, 9, 10]. In general, the three packed algorithms are the same and summarized as follows: 1. Preprocess the data file so that all the objects (or rectangles) are ordered in consecutive groups of b objects, where b is the number of objects in each leaf node. Note that the last group may contain fewer than ....

I. Kamel, C. Faloutsos. "On Packing R-trees". In Proceedings of the 2nd International Conference on Information and Knowledge Management, pages 490--499, Arlington, VA, November 1993.

....of the R tree, and construct the R tree bottom up, rather than starting at the root and inserting data items one by one in a top down fashion. By doing so, we obtain an approachwhichiswell suited for parallel implementation. Unlike previous work on packing methods based on space filling curves [10, 15], in our approach clustering takes place in D dimensional space. It takes into account both the position and the spatial extent of the data in all D dimensions. We therefore obtain superior packing performance which becomes more evident with increasing dimensionalityandskewness of the data, as we ....

....and let r j be the extentofthei non leaf ( directory ) rectangle of the R tree on disk in the j dimension. For the case of a query distribution with fixed windows of above sizes and uniformly distributed centers, the expected response time of an R tree configuration is determined by [10, 14] E[T r ] X i=1 (r j q j ) S j (1) The above expression computes the fraction of the volume indexed in the data space, if every i (disk resident) directory rectangle is enlarged by q j in the j dimension. It is an approximation for the case q j S j , in which case ....

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I. Kamel and C. Faloutsos: "On packing R-trees", Proc. 2nd Int'l. Conf. on Information and Knowledge Management, pp. 490--499, 1993.

....expected number of references per page Table 1. Notations used throughout the analysis. Assume that the query rectangle centroids obey a uniform distribution and that the dataspace dimensions are normalized to the unit square. The expected number of page references to satisfy the query 6 is [7]: 798 ; 6 = 6 A CBEDGFIHI J6 =1K 7 HI J6 ,K 7 =GHLDG MK J6 =9K 6 (1) where 6 = and 6 are the and extends of the window query 6 . Equation (1) is independent of the R tree construction method as well as independent of the data object distribution. Also, ....

I. Kamel and C. Faloutsos: "On packing R-trees", Proceedings of the 2nd Conference on Information and Knowledge Management (CIKM), Washington DC, 1993.

....mentation uses Guttman s quadratic node splitting algorithm. 4 . Linearized insertion load. As above, but the data set is sorted in Hilbert curve order as a data clustering heuristic [JAGA90] before insertion. Linearized bulk load. We used Hilbert curve order as the page packing heuristic [KAME93]. Non linearized bulk load. We used STR [LEUT97] because of its simplicity. Estimators. The traversal and aggregation interfaces of [AOKI98a] allow us to implement estimation using prioritized traversal, breadth first or level at a time traversal (la[ANDE88,WHAN94] and acceptance rejection ....

I. Kamel and C. Faloutsos, "On Packing R-trees," Proc. 2nd Int'l Conf. on Inf. & Knowledge Mgmt., Arlington, VA, Nov. 1993, 490-499.

....table can strongly affect the build time and final structure of the indices built over it. The base tables used in the B tree studies were loaded in both numerically sorted and random key order, whereas the base tables used in the R tree studies were loaded in both least Hilbert value order [KAME93] and the alphabetic order in which the USGS distributes the data. The following conventions apply to all I O measurements described hereafter. Page access counts include both reads and writes. Counts are measured in the file system routines below the buffer manager (i.e. they measure the number ....

....of the movement operation to the time when both the target table and its index are available. Retrieval Performance: Since we are concerned with the performance of actual index instances, we assess the goodness of an R tree using the bounding box coverage metric of Kamel and Faloutsos [KAME93]: P( q) N n=1 S D i=1 P (x i,n q i ) where x n = x 1,n , x D,n ) and q = q 1 , q D ) are D dimensional node bounding boxes and 17 query boxes with side length x i,n and q i , respectively, and P( q) is the expected mean number of R tree nodes visited while searching for ....

I. Kamel and C. Faloutsos, "On Packing R-trees," Proc. 2nd Int. Conf. on Information and Knowledge Management, Arlington, VA, Nov. 1993, 490-499.

....e.g. when query processing requires answering a set of sub queries with multi attribute restrictions or when further processing the stream in different sort orders than the original one. So far there has been already some work on bulk loading for multidimensional index structures, such as RTrees [8], Gridfiles [13] and quad trees [7] These algorithms have an I O complexity of O(Plog P) for an input size of P pages, which is usually due to the fact that these approaches do not utilize a pre sorted input and thus require external sorting of the input data. For B Trees and ....

I. Kamel and C. Faloutsos, "On Packing R-trees," presented at CIKM, 1993.

....(MBRs) R trees can be updated incrementally [1, 5] however, incremental construction often produces trees with poor space utilization and structure. For static data, a preprocessing step may be used to build more query efficient R trees. Examples of such pre processing methods include [4, 6, 8]. In this paper we propose a new pre processing algorithm, Top down Greedy Split (TGS) that builds the tree top down using a data adaptive step which rearranges the database objects under a node in construction. Trees built by TGS achieve up to three times improvement in query performance. This ....

....Hilbert vs TGS 3 Experimental Results We now discuss the methodology used to provide a sound comparison of the packing algorithms and the results obtained from this comparison. 3. 1 Methodology For each data set, R trees are built with each of the three packing algorithms Hilbert Sort (HS) [4], Sort Tile Recursive (STR) 6] and Top down Greedy Split (TGS) Resulting trees are searched with the same sets of random queries. A query set has 10,000 equal size squares. Lower left corner of queries are generated as uniformly distributed points within the unit square. Queries were truncated ....

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Kamel, I., Faloutsos, C., "On Packing R-trees", Proc. 2nd International Conference on Information and Knowledge Management, 490-499, 1993.

....condition that their nearest neighbor is q. The points that satisfy this condition are returned as the answer to the RNN(q) query. 3. 2 Algorithm Development In the past years several papers developed index structures based on R trees, and discussed possible optimizations ( SR87, Gut84, BKSS90, KF93, KF94, BKK96, PF99, EKK00] RKV95] proposed a branchand bound method to find the nearest neighbor to a J J K L A B C D E F G H I S1 S2 S3 S4 S5 S6 y x B A K C H I L G F E D q Figure 3: A section of an R tree and the corresponding regions S i and minimum bounding rectangles ....

I. Kamel and C. Faloutsos. On packing rtrees. In Proceedings of the 2nd International Conference on Information and Knowledge Management (CIKM), pages 490--499, 1993.

....only the MBR (Minimum Bounding Rectangle) of a real object. The notations to be used in this paper are summarized in Table 1. 2 Under the assumption that the placement distribution is uniform, the formula for estimating the result size of the window query on r i is shown in the following formula [3, 4]: Size(oe q (r i ) Nr i X j=1 (s r i ;j;x q x ) s r i ;j;y q y ) 1) In the above formula, a query optimizer does not know object information such as s r i ;j;x and s r i ;j;y in optimization time. For the estimation of the query result size, a query optimizer generally keeps ....

....and s r i ;j;y in optimization time. For the estimation of the query result size, a query optimizer generally keeps statistics such as N r i ; L r i x; L r i y and L r i xy in the system catalog. Therefore, we must transform the above formula into the expression in terms of statistics as follows [3]: Size(oe q (r i ) L r i xy q x L r i y q y L r i x N r i q x q y (2) The following shows the formula for estimating the result size of the spatial join between r i and r j under the uniform assumption of the placement distribution [2, 6] Size(r i 1 intersect r j ) Nr i X k=1 ....

I. Kamel and C. Faloutsos, "On Packing R-tree," Proc. of CIKM, 490-499, 1993.

....version of R tree that employs a sophisticated insertion algorithm, achieving best quality of intermediate nodes. The R tree and R tree are dynamic SAMs that build and maintain their structure incrementally, thus serving as efficient index methods for spatial data. Packing algorithms [RL85, KF93, vdBSW97] build optimal R tree structures from a static set of objects in space. The resulting packed R trees have full leaf nodes, and thus minimum number of nodes and height, leading to minimization of search time. Among the most important spatial queries is the spatial join, which retrieves ....

....(i) is a viable choice, only when the size of input B is small enough for the expected number of accesses in R A not to exceed the total number of pages in the index; in the general case it is too expensive. Patel and DeWitt [PD96] use a bulk loading technique that builds a Hilbert packed R tree [KF93] for set B, under the assumption that the size of B is smaller than the available buffer. For typical situations (i.e. the size of B is greater than the buffer) however, method (ii) is expensive because of the large overhead of external sorting prior to building R B . LR94] shows that method ....

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Kamel, I., Faloutsos, C. "On Packing R-trees". ACM CIKM, 1993.

....ID(S) i for each data sequence S and then inserts it into a multi dimensional index. Here, ID(S) is the identifier of S. If there are a large number of data sequences at the stage of initial index construction, we can achieve high performance gains in construction by hiring bulk loading methods [6, 14, 15]. 4.3.2 Query Processing Algorithm 1 shows TW Sim Search, our query processing algorithm. Step 1 extracts a 4 tuple feature vector from the query sequence. Step 2 performs a square range query on a four dimensional 12 index. The range query uses a 4 tuple feature vector obtained in Step 1 as a ....

I. Kamel, C. Faloutsos, "On Packing R-trees", Proc. ACM CIKM , pp. 490-499, 1993.

....QUERIES An R tree is an extension of B trees for multidimensional objects. An object in an R tree is represented by its minimum bounding rectangle (MBR) Details of R trees are discussed in [3] There are various R tree variants, such as the R tree [17] R tree [1] and the packed R tree [10][16] An R tree is based on heuristic optimization to minimize the area of each enclosing rectangle in the inner nodes. Most of the research on the R trees focuses on minimizing overlapping MBRs and optimizing storage utilization. However, for a GIS system, more specific research on data ....

Ibrahim Kamel, Christos Faloutsos, "On Packing Rtrees, " CIKM, pp. 490-499, 1993.

....have been presented in the past. Faloutsos et al. Falo87] presented a model that estimates the performance of R trees [Gutt84] and R trees [Sell87] assuming uniform distribution of data and packed trees (i.e. all the nodes of the tree are full of data) Recently, Kamel and Faloutsos [Kame93] and Pagel et al. Page93] independently presented a formula that gives the average number of disk accesses in an R tree index accessed by a query, provided that the sizes of the Rtree nodes are known. The proposed fomula is qualitative, i.e. it does not predict the average number of disk ....

....to be answered. A formula that estimates the average number DA of disk accesses with only the knowledge of the data properties is the goal to be reached. Related work has reached only parts of this goal, as will be described in detail, but, obviously, can be used as a basis for our study. In [Kame93, Page93] the following formula is presented (the summation extends over all the nodes of the tree) DA q s q j i i n j ( 1 where qi and sj,i denote the sizes of the query window sides and the node MBR sides, respectively. The above formula allows us to estimate the number of ....

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I. Kamel, C. Faloutsos, "On Packing R-trees", Proceedings of the 2nd CIKM Conference, 1993.

....the number of expected I Os incurred during the filter step without considering the buffer effect (we assume a zero buffer size) We call it ZEIO for Zero buffer Expected I Os. Our estimation of ZEIO for an R tree join is based on the spatial density concept of the R tree cost model presented in [13, 19] for window queries. However, we extend the model beyond window queries to also capture the I O cost for the more complex spatial join processing. Unlike for window query processing where each tree page is accessed at most once, each R tree page is typically accessed multiple times during an ....

....encloses MBRs of all entries in that child node. A leaf node contains en tries of the form oid; mbr where oid refers to a spatial object stored in the database and mbr is the MBR of that spatial object. In most R tree variants, MBRs stored in the tree nodes are allowed to overlap one another [2, 8, 13]. This means that, unlike B trees, there may be more than one search path to identify a desired object. To improve this weakness, recently proposed R tree variants try to minimize the overlap between the MBRs [2, 13] in their tree nodes. Among them, R tree [2] introduces more sophisticated ....

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Kamel, I. and Faloutsos, C., "On Packing R-tree," Proc. of the CIKM, 1993, pp. 490 -- 499.

....situation are highly desirable because this setting is rather typical for geographical applications. Surprisingly little work, however, has been devoted to the static case so far. To our knowledge, only two approaches addressing the static situation exist in the literature. The packed R tree [KF93] arranges the objects according to a space filling curve, e.g. the Hilbert curve, in a preprocessing step and then an R tree is built bottomup, similar to the bottom up construction of an optimal B tree. As a result, space utilization of the R tree is high, its height is small and also because ....

....packed Rtree outperforms conventional R trees mainly for large window queries. In [BPS94] it is shown that the static clustering problem turns out to be a classical optimization problem, if the expected number of data bucket accesses needed to perform a window quer is used as performance measure [PSTW93, KF93]. The solution of the optimization is a set of data buckets with given bucket capacity inducing an optimal data clustering w.r.t. the performance measure. The bucket optimization problem can be solved for bucket capacity c b = 2 by mapping the optimization problem onto a well known graph matching ....

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I. Kamel and C. Faloutsos. On packing Rtrees. In Proc. 2nd Int. Conf. on Information and Knowledge Management, pages 490--499, Washington D.C., 1993.

....oid refers to a spatial object stored in the database and mbr is the MBR of that spatial object. R trees are dynamically balanced by queries such as insert or delete. Therefore, no periodic reorganization is necessary. In most R tree variants, however, entry MBRs are allowed to overlap one another [2, 7, 5]. This means that there may not be only one search path as in the case of B trees. To improve this weakness, recently proposed R tree variants tried to minimize the overlap between the entry MBRs. Among them, R tree [2] introduces heuristics that yields a better query performance. In [5] R trees ....

....[2, 7, 5] This means that there may not be only one search path as in the case of B trees. To improve this weakness, recently proposed R tree variants tried to minimize the overlap between the entry MBRs. Among them, R tree [2] introduces heuristics that yields a better query performance. In [5], R trees are constructed in a bottom up approach called the packed R tree based on the Hilbert curve transformation. As a result, the node occupancy rate is maximized whereas the overlap between entry MBRs is minimized. We exploit these previous results in this paper by basing our performance ....

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Faloutsos, C. and Kamel, I. "On Packing R-tree," Proc. of the CIKM, 1993, pp. 490 -- 499.

.... a priori and, when done properly, results in R trees with nearly 100 space utilization and improved query times (due to the fact that fewer nodes need to be accessed while performing a query) Such packing algorithms were first proposed by Roussopoulos [12] and later by Kamel and Faloutsos [6]. Kamel and Faloutsos [6] propose a packing algorithm based on the Hilbert Curve ordering, and compare it with the Nearest X algorithm proposed by Roussopoulos [12] The latter is simpler to implement and in some cases results in better trees for point queries. However, due to the smaller ....

.... done properly, results in R trees with nearly 100 space utilization and improved query times (due to the fact that fewer nodes need to be accessed while performing a query) Such packing algorithms were first proposed by Roussopoulos [12] and later by Kamel and Faloutsos [6] Kamel and Faloutsos [6] propose a packing algorithm based on the Hilbert Curve ordering, and compare it with the Nearest X algorithm proposed by Roussopoulos [12] The latter is simpler to implement and in some cases results in better trees for point queries. However, due to the smaller perimeter of the bounding ....

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Kamel, I., Faloutsos, C., "On Packing R-trees," Proc. 2nd International Conference on Information and Knowledge Management (CKIM-93), p. 490-499, Arlington, VA, November 1993.

....of q joins. Implementation algorithms for general q joins were presented and evaluated for various probability distributions. Later, Aref and Samet [AS94] proposed analytical formulas for the execution cost and the selectivity of spatial joins, based on Kamel and Faloutsos R tree analysis [KF93]. The basic idea of that work was the consideration of the one dataset as the underlying database and the other dataset as a source for query windows in order to estimate the cost of a spatial join query based on the cost of range queries. Experimental results were also presented to show the ....

....of R trees Several proposals about the analytical estimation of the search performance of the R trees have been presented in the past. The earliest one [FSR87] assumed uniform distribution of data and packed trees (i.e. all nodes of the tree are full of entries) Later, Kamel and Faloutsos [KF93] and Pagel et al. PSTW93] independently presented an analytical formula that estimates the average number of node accesses as a function of the sizes of the R tree nodes. The proposed # formula is qualitative, i.e. it does not really predict the average number of node accesses but, ....

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I. Kamel, C. Faloutsos, "On Packing R-trees", Proceedings of the 2nd Conference on Information and Knowledge Management (CIKM), 1993.

....to analytically predict the performance of spatial data structures and, particularly R trees [Gut84] have concentrated on the range query performance. Some efforts towards the analytical estimation of the range query performance of R tree based data structures have been presented in the past [FSR87, KF93, PSTW93, FK94]. As a further step, in this paper we propose an analytical model that supports datasets of any type (point or region data) and distribution (uniform or non uniform) In particular, we propose a model that predicts the R tree performance using only data properties and, especially, the amount N and ....

....than the proposal of one more effective variant. Faloutsos et al. FSR87] presented a model that estimates the performance of R trees and R trees [SRF87] assuming uniform distribution of data and packed trees (i.e. all the nodes of the tree are full of data) Later, Kamel and Faloutsos [KF93] and Pagel et al. PSTW93] independently presented the following formula that gives the average number DA of disk accesses in a n dimensional R tree index accessed by a query window q = q 1 , q n ) provided that the sides (s j,1 , s j,n ) of each R tree node s j are known (the ....

[Article contains additional citation context not shown here]

I. Kamel, C. Faloutsos, "On Packing R-trees", In Proceedings of the 2nd International Conference on Information and Knowledge Management (CIKM), November 1993.

....data structures have been presented in the past. Faloutsos et al. Falo87] presented a model that estimates the performance of R trees and R trees assuming uniform distribution of data and packed trees (i.e. all the nodes of the tree are full of data) Recently, Kamel and Faloutsos [Kame93] and Pagel et al. Page93] independently presented the following formula that gives the average number DA of disk accesses in a n dimensional R tree index accessed by a query window q = q 1 , q n ) provided that the sides (s j,1 , s j,n ) of each R tree node s j (j = 1, N) are ....

....i j n , 1 (7) 1 The proofs of all lemmas are omitted because of space limitations. 4 In order to reach Eq. 7 we assumed that the nodes are square like (i.e. s j,1 = s j,2 = s j,n , j = 1, h 1) This simplification is a reasonable property for a good R tree [Kame93]. What remains is an estimation of D using data properties. Suppose that, at level j, N nodes with average size (s j,i ) n are organized in N j 1 parent nodes with average size (s j 1,i ) n . Each parent node groups, on the average, f nodes. The average size s j 1,i of a parent node at ....

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I. Kamel, C. Faloutsos, "On Packing R-trees", Proceedings of the 2nd CIKM Conference, 1993.

....model that predicts the number of disk accesses for an LRU buffer manager given the minimum bounding rectangles (MBRs) of the R tree nodes and a buffer size. Our model can be used to evaluate the quality of any R tree update operation, such as node splitting policies [3] or packing algorithms [4, 6, 8], as measured by query performance of the resulting tree. The model is very accurate and simple to understand, making it easy for researchers to integrate it into their studies. It can easily be modified to accommodate different buffer management policies for example, keeping the top few levels ....

.... R ij B ij = Number of accesses to R ij N = Number of queries performed so far N = Expected number of queries to fill the buffer B = Buffer size D(N) Number of distinct nodes (at all levels) accessed in N queries For random queries and objects within the unit square, Kamel and Faloutsos [4] argue that A ij is the area of R ij for point queries, and the area of R ij Phi Q for region queries of size Q 1 . Bhide et al. [2] analyze the LRU buffer replacement policy for databases consisting of a number of partitions with uniform page access within each partition. While modeling ....

I. Kamel, C. Faloutsos, "On Packing R-trees", Proc. 2nd International Conf. on Information and Knowled Management (CKIM-93), p. 490-499, Arlington, VA, November 1993.

....support of q joins. Implementation algorithms for general q joins were presented and evaluated for various probability distributions. Later, Aref and Samet [AS94] proposed analytical formulae for the execution cost and the selectivity of spatial joins, based on Kamel and Faloutsos Rtree analysis [KF93]. The basic idea of that work was the consideration of the one data set as the underlying database and the other data set as a source for query windows in order to estimate the cost of a spatial join query based on the cost of range queries. Experimental results showing the accuracy of the ....

.... of the tree (the root is assumed to be at level j=h, and the leafnodes at level j=1) the expected retrieval cost, in terms of node accesses, for an n dimensional query window q with extents (q 1 , q 2 , q n ) at each direction is given by the following formula, originally proposed in [KF93, PSTW93]: 1 1 1 , h j n k k k j j q s N q NA (1) i.e. the expected number NA of node accesses is equal to the sum of the total coverage (at each level j) of R tree nodes s, assuming that their size has been extended by the size of the query window q at each ....

I. Kamel, C. Faloutsos, "On Packing R-trees", Proc. 2nd CIKM Conf., 1993.

....is the R tree [Gut84] The R tree [SRF87] the R tree [BKSS90] and the X tree [BKK96] are improved versions of the R tree. These dynamic SAMs maintain their structure on each insertion deletion. In the case of static collections which are not often updated, packing algorithms [RL85, KF93, LEL96] build optimal R trees, called packed R trees. Spatial joins algorithms can be classified into three categories depending on whether each relation is indexed or not. i. no index: for the case where no index exists on any relation, several partitioning techniques have been proposed which ....

....be read and written during the growing phase, thus rendering the method ineffective. STR The second variant of Build And Match algorithm implemented, called Sort Tile Recursive (STR) and proposed in [LEL96] constructs on the fly a packed R tree. We also experimented the Hilbert packed Rtree [KF93] but found that the comparison function (based on the Hilbert values) was more expensive than the centroid comparison of STR. The algorithm is as follows. First the rectangles from the source are sorted 4 by x coordinate of their centroid. At the end of this step, the size N of the dataset ....

I. Kamel and C. Faloutsos. On Packing Rtrees. In Proc. Intl. Conf. on Information and Knowledge Management (CIKM), 1993.

....is an improved version of R tree that employs a sophisticated insertion algorithm, achieving optimal tree quality. The R tree and R tree are dynamic SAMs that build and maintain their structure incrementally, thus serving as efficient index methods for spatial data. Packing algorithms [RL85] KF93] vdBSW97] build optimal R tree structures for a static set of objects in space. The resulting packed R trees have full leaf nodes, and thus minimum number of nodes and height, leading to minimization of search time. Among the most important spatial queries is the spatial join, which retrieves ....

....small, and the expected number of accesses in R A will not exceed the total number of pages in the index; in the general case it is too expensive. Method (ii) too, is applicable only for small sizes of B. Patel and DeWitt [PD96] use a bulk loading technique that builds a Hilbert packed R tree [KF93] for set B, under the assumption that the size of B is smaller than the available buffer. For typical situations (the size of B is greater than the buffer) however, method (ii) is too expensive because of the large overhead of external sorting prior to building R B ; for such cases, LR94] shows ....

[Article contains additional citation context not shown here]

Kamel, I., Faloutsos, C. "On Packing R-trees". In Proc. 2 nd International Conference on Information and Knowledge Management (CIKM '93), 1993.

....query and its execution procedure. Spatial queries addressed by users of SDBMS usually involve selection (point or range) and join operations. In the literature, most efforts towards the analytical prediction of the performance of spatial data structures have focused on point and range queries [FSR87, KF93, PSTW93, FK94, TS96] and, recently, on spatial join queries [Gun93, HJR97, TSS98] Some proposals support both uniform like and non uniform data distributions, which is an important advantage keeping in mind that modern database applications handle large amounts of real (usually non uniform) multidimensional data. In ....

....and cost models. Although research on multi dimensional histograms has recently appeared in the literature [PI97] in the spatial database literature, cost models for selectivity and cost estimation seem to be the most promising solutions. Proposals in this area include models for selection [KF93, PSTW93] and join [Gun93] queries. However, most proposals require knowledge of index properties and make a uniformity assumption, thus rendering them incomplete tools for the purposes of real query optimization. Appropriate extensions to solve those problems are presented in the rest of the section. ....

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I. Kamel, C. Faloutsos, "On Packing R-trees", In Proceedings of the 2nd International Conference on Information and Knowledge Management (CIKM), 1993.

.... u,1 q l,1 d) P l,2 q u,2 d) P u,2 q l,2 d) P l,n q u,n d) P u,n q l,n d) Table II Constraints for the intermediate nodes Several analytical models that estimate R tree performance on overlap queries have been proposed in the past (Pagel et al. 1993; Kamel and Faloutsos, 1993; Faloutsos and Kamel, 1994) In this paper we use a recent one (Theodoridis and Sellis, 1996) which predicts R tree performance using knowledge of the dataset only, and is applicable to point and non point datasets. According to this model, equation 1 describes the expected retrieval cost ....

.... N j is the expected number of nodes, and s j is the node extent on each dimension, at level j (the root is assumed at level j = h, and the leaf nodes at level j = 1) We assume that the sides of the nodes are equal on each dimension, a simplification that is reasonable for well structured R trees (Kamel and Faloutsos, 1993). The expression for computing the height h of the R tree is: h N c M c M = 1 log (2) where N is the number of distinct objects (leaf MBRs) in the database, M is the maximum number of entries in a R tree node, and c is the average node capacity (typically c=67 ; c M denotes ....

Kamel, I., Faloutsos, C., "On Packing R-trees", In Proceedings of the 2nd International Conference on Information and Knowledge Management (CIKM), 1993.

....effect on the effectiveness of an index. We used a variety of loading algorithms, each of which represented a class of related algorithms: insertion load using randomly ordered records, 9 insertion load using (Hilbert )clustered records [JAGA90] bulk load using (Hilbert )clustered records [KAME93], bulk load using (STR )tiled records [LEUT97] Estimators. The traversal and aggregation interfaces of [AOKI98a] allow us to implement estimation using prioritized traversal, breadth first or level at a time traversal, and acceptance rejection sampling in about 500 lines of C . These extensions ....

I. Kamel and C. Faloutsos, "On Packing R-trees," Proc. 2nd Int'l Conf. on Inf. & Knowledge Mgmt., Arlington, VA, Nov. 1993, 490-499.

....mbr is the MBR that encloses MBRs of all entries in that child node. A leaf node contains entries of the form oid; mbr where oid refers to a spatial object stored in the database and mbr is the MBR of that spatial object. In most R tree variants, entry MBRs are allowed to overlap one another [2, 7, 5]. This means that there may be more than one search path. Recently proposed R tree variants tried to minimize the overlap between the entry MBRs. Among them, R tree [2] introduces heuristics that yield a better query performance. In [5] R trees are constructed in a bottom up approach called ....

....entry MBRs are allowed to overlap one another [2, 7, 5] This means that there may be more than one search path. Recently proposed R tree variants tried to minimize the overlap between the entry MBRs. Among them, R tree [2] introduces heuristics that yield a better query performance. In [5], R trees are constructed in a bottom up approach called the packed R tree based on the Hilbert curve transformation. As a result, the node occupancy rate is maximized whereas the overlap between entry MBRs is minimized. The experimental results presented in this paper are based on packed ....

[Article contains additional citation context not shown here]

Faloutsos, C. and Kamel, I. "On Packing R-tree," Proc. of the CIKM, 1993, pp. 490 -- 499.

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I. Kamel, C. Faloutos: "On Packing R-trees". Proc. Int. Conf. on Information and Knowledge Managemenet (CIKM'93), 1993, pp. 490499.

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I. Kamel and C. Faloutsos: "On packing R-trees", Proceedings of the 2nd Conference on Information and Knowledge Management (CIKM), Washington DC, 1993.

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I. Kamel and C. Faloutsos: "On Packing R-trees", Proceedings 2nd CIKM Conference, pp.490499, Washington, DC, 1993.

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I. Kamel and C. Faloutsos. On packing Rtrees. In Proc. 2nd Intl. Conf. on Information and Knowledge Management,pages 490--499, Washington D.C., 1993.

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I. Kamel and C. Faloutsos: "On Packing R-trees", Proceedings 2nd CIKM Conference, 1993.

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