B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17:3:427--462, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... to represent the data structure for range searching, Range, where we store n block identifiers k (representing the pair (k; k 1) Among the plethora of data structures offering different space time tradeoffs for range searching [1, 10] we prefer one of minimal space requirement by Chazelle [4]. This structure is a perfect binary tree dividing the points along one coordinate plus a bucketed bitmap for every tree node indicating which points (ranked by the other coordinate) belong to the left child. There are in total n log 2 n bits in the bucketed bitmaps plus an array of the point ....

....disregard in this case blocks totally containing P , since these occurrences extend others of the other two types) Finally, we can uncompress and show the text of length L surrounding any occurrence reported in O(L log oe) time, and uncompress the whole text T 1: u in O(u log oe) time. Chazelle [4] permits several space time tradeoffs in his data structure. In particular, by paying O space, reporting time can be reduced to O(log n) If we pay for this space complexity, then our search time becomes O(m log(moe) m log n R log n) 6 Conclusions We have presented an index ....

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17(3):427--462, 1988.

....n) nodes of T that v j updates. We further process each node in T so that it contains a list of cover two life ranges. This is a list of intervals consisting of the pairwise intersections of the life spans in the node. For example, assume that a node s contains the life spans [1, 7] 3, 4] and [5, 6]. The list of cover two at s is [3, 4] and [5, 6] If a vertical segment is to be deleted from s at instances x 1 , x 2 or x 7 , then s will be exposed after the deletion. But if the deletion occurs at instance x 3 , x 4 , x 5 or x 6 then, since the cover of s is 2 at this instance, s will not be ....

....each node in T so that it contains a list of cover two life ranges. This is a list of intervals consisting of the pairwise intersections of the life spans in the node. For example, assume that a node s contains the life spans [1, 7] 3, 4] and [5, 6] The list of cover two at s is [3, 4] and [5, 6]. If a vertical segment is to be deleted from s at instances x 1 , x 2 or x 7 , then s will be exposed after the deletion. But if the deletion occurs at instance x 3 , x 4 , x 5 or x 6 then, since the cover of s is 2 at this instance, s will not be exposed by deleting v j . Our parallel algorithm ....

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B. Chazelle, "A functional approach to data structures and its use in multidimensional searching ", SIAM J. Comput., 17(1988), pp. 427--462.

....the highlighted nodes. With leftrank and rightrank we nd that their ranges are [1,9] and [13,14] respectively. Figure 9 shows the second part. We search for the reverse pre xes of ala, namely la and a, in RevT rie. The nodes reached are highlighted. Their ranges are, respectively, 10,10] and [2,5]. Finally, Figure 10 shows the last part of the search. We join pre x a with sux la, obtaining a 2 dimensional rank range (2,13) 5,14) and pre x al with sux a, obtaining a 2 dimensional range (10,1) 10,9) Both ranges are searched for in Range, and all the block identi ers found are ....

.... need to represent the data structure for range searching, Range, where we store n block identi ers k (representing the pair (k; k 1) Among the plethora of data structures o ering di erent space time tradeo s for range searching [2, 13] we prefer one of minimal space requirement by Chazelle [5]. This structure is a perfect binary tree dividing the points along one coordinate plus a bucketed bitmap for every tree node indicating which points (ranked by the other coordinate) belong to the left child. There are in total n log 2 n bits in the bucketed bitmaps plus an array of the point ....

[Article contains additional citation context not shown here]

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17(3):427-462, 1988.

....prefix sum cube P d1 d2 measure value d1 d2 0 0 3 0 1 7 0 2 2 0 3 3 . 4 2 3 4 3 3 Data set D Figure 1: The original array and prefix sum array A number of highly sophisticated aggregation techniques for sparse data have been proposed for computational geometry applications [8, 7, 23]. However, typically the storage overhead is super linear, e.g. O(N log d Gamma1 N) for a data set of size N , which is infeasible for large multidimensional data sets in data warehousing applications. Also, since the data structures are fairly involved, they are rarely used in practice. ....

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17(3):427--462, 1988.

....hierarchy. 1.2 Previous Results Range searching has been studied extensively in the RAM model. In the planar case, for example, some of the best known structures answer queries in O(log N T log (2N T ) time using linear space and in O(log N T ) time using N) space, respectively [18, 19]. Refer to a recent survey for further results [3] In the I O model, the B tree [21, 9] supports one dimensional range queries in O(log B N T B) memory transfers using linear space. In two dimensions, one has to use #(N log B log B N ) space to obtain an O(log B N T B) query bound [8, ....

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM J. Comput., 17(3):427--462, June 1988.

....data structures in the internal memory model have been developed in computational geometry; see [1] for a survey. The best known data structure in R is the range tree, which uses O(N log 2 N) space and can answer a rangeaggregate query such as range SUM in O(log 2 N) time [1] Chazelle [7] developed the compressed range tree data structure that uses O(N) space under the so called bit model (in which a word can store log 2 N bits and each bit can be manipulated individually) This structure can be used to answer a range COUNT query in O(log 2 N) time. Both of these structures use ....

....2 N) time has to use super linear storage. 3 Our results. Our main result, described in Section 2, is a new indexing scheme, called the Compressed Range B tree (or CRB tree) for answering two dimensional range COUNT queries. This structure is an external version of the compressed range tree [7]. It uses O(n) disk blocks, answers a query in O(log B n) I Os, and can be bulk loaded using O(n log B n) I Os. This is the first optimal indexing scheme for the 2D range COUNT problem in the I O model. Using a partial rebuilding scheme [5] a point can be inserted deleted in O(log B n) I Os. ....

[Article contains additional citation context not shown here]

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM J. Comput., 17(3):427--462, June 1988.

....is not effective if requests are not accessing sectors in order or if there is a large gap between groups of target sectors. There is an inherent dimensionality curse in computing range aggregates. The best known algorithms whose runtime is provably sub linear in the size of the data set (e.g. [7, 38]) have polylogarithmic query cost and storage overhead. However, for dimensionalities d 9 a polylogarithmic cost is practically worse than a linear cost. Let n denote the number of data points, i.e. tuples in the data warehouse s table. Then for d = 9 the polylogarithmic value log n is only ....

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17(3):427--462, 1988.

.... we need to represent the data structure for range searching, Range, where we store n block identi ers k (representing the pair (k; k 1) Among the plethora of data structures o ering di erent spacetime tradeo s for range searching [1, 11] we prefer one of minimal space requirement by Chazelle [4]. This structure is a perfect binary tree dividing the points along one coordinate plus a bucketed bitmap for every tree node indicating which points (ranked by the other coordinate) belong to the left child. There are in total n log 2 n bits in the bucketed bitmaps plus an array of the point ....

....disregard in this case blocks totally containing P , since these occurrences extend others of the other two types) Finally, we can uncompress and show the text of length L surrounding any occurrence reported in O(L log ) time, and uncompress the whole text T 1: u in O(u log ) time. Chazelle [4] permits several space time tradeo s in his data structure. In particular, by paying O space, reporting time can be reduced to O(log n) If we pay for this space complexity, then our search time becomes O(m log(m ) m log n R log n) 6 Conclusions We have presented an index for ....

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17(3):427-462, 1988.

....Computational Geometry. A range reporting query reports all objects in the query box [22, 6, 2] 5] proposed the range tree for the range reporting query. The range tree is very similar to the ECDF tree. The best internal memory solution for the 2 dimensional range counting problem is given in [9]. For the static case (i.e. all n points are known in advance) the solution uses O(n) space and has O(log 2 n) query time. In particular, a range tree is compressed using the functional approach technique. To extend the solu tion to the d dimensional case, the multi dimensional divideand ....

....a range tree is compressed using the functional approach technique. To extend the solu tion to the d dimensional case, the multi dimensional divideand conquer technique of [5] can be used which leads to O(n log 2 n) space and O(log 2 n) query time. However, note that the solution of [9] applies only to range counting and not to range sum. Furthermore, the data structure is rather complex to implement in practice. Note that the ECDF tree and the range tree are both static and internal memory structures. To dynamize a static data structure some standard techniques can be used ....

B. Chazelle, \A Functional Approach to Data Structures and Its Use in Multidimensional Searching", SIAM J. Comput. 17, 1988.

....in high dimensional space. In range searching problems, there is a need to find points in a k dimensional space which intersect a given query window. Orthogonal query problems have been well investigated, and a number of good data structures, such as the range search tree have been proposed [185, 180, 21, 22]. In particular, the bucketing method [48] which is similar to a k d tree using arbitrary splitting planes has been proposed. Tokuyama [166] propose a theoretic method for orthogonal clipping with applications to nearest neighbour queries. A more general non orthogonal case has also been studied ....

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17:427--462, 1988.

....treatment for decades. Thus, we shall only survey a few of the major, or more recent results. More extensive surveys can be found in [Meh84, PS85, AE97] The best data structures known today are due to Alstrup et al. ABR00] They improved on the previous best bounds due to Chazelle [Cha86, Cha88] The work of [ABR00] introduces two data structures for orthogonal range 18 search in R , one requiring O(log log n t) time and O(n log n) space, and the other requiring O( log log n) t log log n) time and O(n log log n) space. They also show that any data structure for R , with ....

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal of Computing, 17(3):427--462, 1988. 170

....sex, weight, salary etc. A typical orthogonal range query is of the form find all males of age between 30 and 40 years with an income between 20,000 and 40,000 . The orthogonal range searching problem has numerous applications and has been studied extensively for the last decades, see e.g. [1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 20, 22, 24, 25, 26, 27, 30, 31, 40, 41, 42, 43, 45, 46, 47]. Willard [43] gives a comprehensive list of references on the subject and gives applications to the theory of databases. For surveys see, e.g. the survey by Agarwal [1] and the books by Mehlhorn [27] and Preparate and Shamos [31] In this paper we consider various orthogonal range searching ....

....see, e.g. the survey by Agarwal [1] and the books by Mehlhorn [27] and Preparate and Shamos [31] In this paper we consider various orthogonal range searching problems on static point sets. We give new techniques for static orthogonal range searching problems improving the previous best results [11, 14, 18, 30, 32, 41, 42] for various models, problems and dimensions: general range reporting in R d , for fixed d # 3, two dimensional range reporting in rank space, and for the two dimensional semi group range sum problem. In the following we let n denote the number of stored points and k the number of points to be ....

[Article contains additional citation context not shown here]

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17(3):427--462, 1988.

....uses linear space and supports updates and queries in O(log 2 B N) I Os. Range counting. Given a set of N points in the plane, a range counting query asks for the number of points within a query rectangle. Based on ideas utilized in an internal memory counting structure due to Chazelle [52], Agarwal et al. 6] designed an external data structure for the range counting problem. Their structure use linear space and answers a query in O(log B N) I Os. Based on a reduction due to Edelsbrunner and Overmars [74] they also designed a linear space and O(log B N) query structure for the ....

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM J. Comput., 17(3):427--462, June 1988.

....Geometry. A range reporting query reports all objects in the query box [Mat94, BKO 97, AE98] Ben80] proposed the range tree for the range reporting query. The range tree is very similar to the ECDF tree. The best internal memory solution for the 2 dimensional rangecounting problem is given in [Cha88]. For the static case (i.e. all n points are known in advance) the solution uses O(n) space and has O(log 2 n) query time. In particular, a range tree is compressed using the functional approach technique. To extend the solution to the d dimensional case, the multi dimensional divide and conquer ....

....is compressed using the functional approach technique. To extend the solution to the d dimensional case, the multi dimensional divide and conquer technique of [Ben80] can be used which leads to O(n log d Gamma1 2 n) space and O(log d Gamma1 2 n) query time. However, note that the solution of [Cha88] applies only to rangecounting and not to range sum. Furthermore, the data structure is rather complex to implement in practice. Note that the ECDF tree and the range tree are both static and internal memory structures. To dynamize a static data structure some standard techniques can be used ....

B. Chazelle, "A Functional Approach to Data Structures and Its Use in Multidimensional Searching", SIAM J. Comput. 17, pp. 427-462, 1988.

....constructing bounding box hierarchies, both in main and external memory, that have low stabbing number, and consequently, low query complexity. Previous results. As noted above, several efficient data structures have been proposed for answering a rectangle intersection query. For example, Chazelle [5] showed that a compressed range tree can be used to answer a # dimensional rectangle intersection query in time ##### # # # # ## using ### ### # # ## ### ### ## space (where # is the number of rectangles reported) This data structure is too complex to be practical even in # # . As for ....

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal of Computing, 17:427--462, 1988.

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B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17:3:427--462, 1988.

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B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17(3):427-462, 1988.

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B. Chazelle. A functional approach to data structures and its use in multi-dimensional searching. SIAM J. Comput. 17: 427--462 (1988).

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B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM J. Comput., 17:427--462, 1988.

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B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17:3:427--462, 1988.

No context found.

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM J. Comput., 17(3):427--462, June 1988.

No context found.

B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal of Computing, 17(3):427-462, June 1988.

No context found.

B. Chazelle, \A Functional Approach to Data Structures and Its Use in Multidimensional Searching", SIAM J. Comput. 17, 1988.

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B. Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17:427--462, 1988.

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B. Chazelle, "A Functional Approach to Data Structures and Its Use in Multidimensional Searching", SIAM J. Comput. 17, 1988.

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