K. V. R. Kanth and A. K. Singh. Optimal dynamic range searching in non-replicating index structures. In Proc. International Conference on Database Theory, LNCS 1540, pages 257--276, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....attempts [SR01, TPS02, BJKS02] addressing variations of the problem, incur the same complexity. On the other hand, there is a significant amount of theoretical results on conventional spatial queries for static and moving objects. For orthogonal range search on static points, Kanth and Singh [KS99] prove that the best possible query time using any structure consuming linear O(N B) space is K B) I Os, where K is the number of objects retrieved. This bound is tight and has been realized by the O tree [KS99] and the cross tree [GI99] Applying the theory of indexability [HKP97] Arge et al. ....

....and moving objects. For orthogonal range search on static points, Kanth and Singh [KS99] prove that the best possible query time using any structure consuming linear O(N B) space is K B) I Os, where K is the number of objects retrieved. This bound is tight and has been realized by the O tree [KS99] and the cross tree [GI99] Applying the theory of indexability [HKP97] Arge et al. ASV99] show that a structure achieving optimal query cost O(log B (N B) K B) must occupy #( N B)log B (N B) loglog B (N B) space. They propose the external range tree that achieves these bounds. A special RS ....

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Kanth, K., Singh, A. Optimal Dynamic Range Searching in Non-Replicating Index Structures. ICDT, 1999.

.... They include K D B trees [22] hB trees [18, 10] and R trees [13, 6] If is the total number of points and is the number of points that fit in a disk block, 50 455 is the theoretical lower bound on the number of I Os needed by a linear space index to answer a window query [15]. Here is the number of points in the query rectangle. In practice, the above indexing structures often answer queries in much fewer I Os. However, their query performance can seriously deteriorate after a large number of updates. Recently, a number of linear space structures with guaranteed ....

....indexing structures often answer queries in much fewer I Os. However, their query performance can seriously deteriorate after a large number of updates. Recently, a number of linear space structures with guaranteed worst case efficient query and update performance have been developed (see e.g. [5, 15, 12]) The so called cross trees [12] and O trees [15] answer window queries in the optimal number of I Os and can be updated, theoretically, in I Os, but they are of limited practical interest because a theoretical analysis shows that their average query performance is close to the ....

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K. V. R. Kanth and A. K. Singh. Optimal dynamic range searching in non-replicating index structures. In Proc. Intl. Conf. on Database Theory, volume 1540 of Lecture Notes Comput. Sci., pages 257--276, 1999.

....tree Subramanian and Ramaswamy [SR95] can answer two dimensional three sided range queries with slightly suboptimal O(log B n t B IL # (B) I Os per query, using linear space. For two dimensional range search, a structure of O( log log B n ) blocks can be used. Ravi Kanth and Singh [KS89] study the case of non replicating structures, and provide a structure with O( n B) d 1) d t B) access cost and optimal update cost O(log B n) They also show that this IL # (B) is the iterated log # function, i.e. the number of times we must apply log # to B before the result becomes ....

K.V. Ravi Kanth and A.K. Singh. Optimal dynamic range searching in non-replicating index structures. In Proc. Intl. Conf. Database Theory, pages 257--276, 1989.

....updates (insertions) Unfortunately, a similar claim cannot be made about the underlying kd tree, and thus good query efficiency cannot be maintained. Recently, a number of theoretical worst case efficient dynamic external data structures have been developed. The cross tree [15] and the O tree [17], for example, both use linear space, answer range queries in the optimal number of I Os, and they can be updated I O efficiently. However, their practical efficiency has not been investigated, probably because a theoretical analysis shows that their average query performance is close to the ....

K. V. R. Kanth and A. K. Singh. Optimal dynamic range searching in nonreplicating index structures. In Proc. International Conference on Database Theory, LNCS 1540, pages 257--276, 1999.

....O(log N) internal memory search term, and an O(T=B) reporting term accounting for the O(T=B) I Os needed to report T elements. Recently, the above bounds have been obtained for a number of problems (e. g [30, 26, 149, 5, 47, 87] but higher lower bounds have also been established for some problems [141, 26, 93, 101, 106, 135, 102]. We discuss these results in later sections. B trees come in several variants, like B and B trees (see e.g. 35, 63, 95, 30, 104, 3] and their references) A basic B tree is a Theta(B) ary tree (with the root possibly having smaller degree) built on top of Theta(N=B) leaves. The degree of ....

....use linear space. We discuss this further in Section 7. Grossi and Italiano developed the elegant linear space cross tree data structure which answers planar range queries in O( p N=B T=B) I Os [89, 90] This is optimal for linear space data structures as e.g. proven by Kanth and Singh [102]. The O tree of Kanth and Singh [102] obtains the same bounds using ideas similar to the ones used by van Kreveld and Overmars in divided k d trees [146] In Section 5.2 below we discuss the cross tree further. 5.1 Logarithmic query structure The O(log B N T=B) query data structure is based on ....

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K. V. R. Kanth and A. K. Singh. Optimal dynamic range searching in nonreplicating index structures. In Proc. International Conference on Database Theory, LNCS 1540, pages 257--276, 1999.

....36, 38, 42, 43] were devised to support fast range queries in external memory, and Arge et al. 4] have dealt with some decomposable problems in external memory. However, none of these data structures seems to be able to support eciently split and concatenate along any coordinate. Ravi and Singh [39], following up on a previous lower bound of Hellerstein et al. 22] considered the complexity of range searching in a simple index tree model. In this model, they proved a lower bound of N=B) 1 1=d r=B) for a range query retrieving the pointers to r out of N items, where B is the capacity ....

....crosstrees use some auxiliary pointers for supporting eciently split and concatenate operations, they are not strictly considered non replicating in the terminology of Ravi and Singh: however, we do not believe this to be a substantial infringement of the non replicating assumption. Ravi and Singh [39] introduced also the Otree, a pointer based strictly non replicating data structure which also matches their lower bound. Once again, O trees support only queries, insertions and deletions but are not able to support eciently split and concatenate operations. The remainder of this paper consists ....

K.V. Ravi Kanth and A.K. Singh. \Optimal dynamic range searching in non-replicating index structures". Technical Report TRCS97-13, University of California at Santa Barbara, Dpetartment of Computer Science, July 1997.

....during updates (insertions) Unfortunately, a similar claim cannot be made about the underlying kd tree and thus good query efficiency cannot be maintained. Recently, a number of theoretical worst case efficient dynamic data structures have been developed. The cross tree [21] and the O tree [27] for example, both use linearspace, answer range queries in the optimal number of I Os, and they can be updated I O efficiently. However, their practical efficiency has not been investigated, probably because a careful theoretical analysis shows that their average query performance is close to the ....

K. V. R. Kanth and A. K. Singh. Optimal dynamic range searching in non-replicating index structures. In Proc. International Conference on Database Theory, pages 257--276, 1999.

....many enter and or leave bundle B j between t l and t r . of the slice S i on the two planes. As previously, we store a set of critical levels j of A(S x ) within the projected window S x i in a persistent B tree T . However, inspired by previous approaches for two dimensional range searching [19, 23], instead of storing every B i th level, we now only store every ( p NB i )th level of A(S x ) Refer to Figure 6 (i) We have p N=B i critical levels, and hence we define bundle B j to be the p NB i levels between critical level j Gamma1 and j . Using Lemma 3.4, we can prove ....

K. V. R. Kanth and A. K. Singh. Optimal dynamic range searching in non-replicating index structures. In Proc. International Conference on Database Theory, pages 257--276, 1999.

....searching results of Chazelle to external memory structures with an arbitrary node capacity b. All these structures use non linear storage space which may be too high for large database systems. When only linear space is used, Bentley s multi level range structures [4] and its dynamic variants [29, 21] obtain O(n ffl ) query times for any ffl 0. However, these structures replicate each data item at least (1=ffl) d times. This storage cost may be still quite high for most database systems. In this paper, we concentrate on external memory structures that do not replicate data items. We ....

....time for each layer B tree is amortized over the number of updates that occur in it. This adds a logarithmic amortized cost to each update due to reconstruction of each layer B tree. Since the number of such layers is d, the total accrued cost for an update is O(log n) More details are given in [21]. Combining the above three costs, a C tree for n points supports range queries in optimal time and insertions deletions in O(log n (log 2 n) log(log 2 n) time. Theorem 3 There exists a non replicating data structure that supports range queries on n d dimensional points in O(n ....

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K. V. Ravi Kanth and Ambuj K. Singh. Optimal dynamic range searching in non-replicating index structures. Technical Report TRCS97-13, UCSB, June 1997.

....have been proposed in the database literature. These structures achieve low storage costs by storing each data point exactly once. However, such simple structures have high query time complexity, which is bounded below by Omega Gamma n (d Gamma1) d ) for n d dimensional data [Mehlhorn 1984, Ravi and Singh 1997a] This complexity can, however, be improved by allowing for replication of data. Using such replication (but still using linear storage) Bentley s nonoverlapping k level d range structure [Bentley 1980] achieves a query time complexity of O(n (d Gamma1) k log n) Bentley 1980] By choosing k ....

Ravi Kanth, K. V., and Ambuj K. Singh. "Optimal dynamic range searching in non-replicating index structures", Technical Report TRCS97-13, Department of Computer Science, University of California, Santa Barbara, July 1997.

.... a user specified query box (range intersection queries) and retrieve data items that have specific attributes (exact match query) Image objects are characterized by high dimensional feature vectors and queries retrieve images similar to user specified patterns (similarity queries) In [RS97] we show that the lower bound on time complexity of range queries on n d dimensional objects (d 1) using conventional tree structures such as the R trees [BKSS90] is Omega Gamma n 1 Gamma1=d ) In contrast, for 1 dimensional data, range queries can be answered in logarithmic time. Given ....

K. V. Ravi Kanth and A. Singh. Optimal dynamic range searching in non-replicating index structures. Technical Report TRCS97-13, Comp. Sci. Dept., University of California, Santa Barbara, 1997. http://www.cs.ucsb.edu/TRs/TRCS97-13.html.

....[Freeston 1987] have been proposed in the database literature. These structures achieve low storage costs by storing each data point exactly once. However, such simple structures have high query time complexity, which is bounded below by Omega Gamma n (d Gamma1) d ) for n d dimensional data [Ravi and Ambuj 1997]. This complexity can, however, be improved by allowing for replication of data. Using Work supported in part by a research grant from NSF ARPA NASA IRI 9411330 and NSF instrumentation grant CDA 9421978. such replication (but still using linear storage) Bentley s nonoverlapping k level ....

....worst case bounds. 2 5 Conclusions In this paper we considered the problem of dynamic range searching using linear space structures. Since current structures such as the R trees [Beckman et al. 1990, Guttman 1984] and Bang files [Freeston 1987] have high query time complexity (O(n) for n points [Ravi and Ambuj 1997]) we focussed on alternate means to improve query performance in dynamic environments. We observed that Bentley s non overlapping range structure [Bentley 1980] achieves O(n ffl ) query time for any ffl 0. However, this structure is static in nature. We demonstrated that the same query ....

Ravi Kanth, K. V. and Ambuj K. Singh "Optimal dynamic range searching in non-replicating

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K. V. R. Kanth and A. K. Singh. Optimal dynamic range searching in non-replicating index structures. In Proc. International Conference on Database Theory, LNCS 1540, pages 257--276, 1999.

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