D. E. Vengro# and J. S. Vitter. E#cient 3-d range searching in external memory. In Proc. ACM Symp. on Theory of Computation, pages 192--201, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... (designed as indexing schemes) to perform filtering search [Cha86] In various forms, this technique was employed by Ramaswamy and Subramanian in the P range tree [SR95] by Arge and Vitter in the External Interval Tree [AV96] and by Vengro# and Vitter in their 3 dimensional index structures [VV96a] Since our work, it has been employed by Agarwal et al. AAE 98] to indexing for half space queries, and moving points on the plane [AAE00] The main merit of bootstrapping as a general approach, is that it does not (potentially) su#er from the weaknesses of the Path Caching technique of ....

D. E. Vengro# and J. S. Vitter. E#cient 3-d range searching in external memory. In Proc. ACM Symp. on Theory of Computation, pages 192--201, 1996.

....Department of Computer Science, University of Aarhus, Denmark and I.N.R.I.A. Sophia Antipolis, France. Email: jsv cs.duke.edu. 1 Introduction There has recently been much effort toward developing worst case I O efficient external memory data structures for range searching in two dimensions [1, 2, 4, 8, 12, 13, 20, 26, 28, 29]. In their pioneering work, Kanellakis et al. 13] showed that the problem of indexing in new data models (such as constraint, temporal, and object models) can be reduced to special cases of twodimensional indexing. Refer to Figure 1) In particular they identified the 3 sided range searching ....

....the TPIE system [27] There are several interesting range searching problems in external memory that remain open, such as higher dimensional range searching and non orthogonal queries. Some recent work has been done on I Oefficient three dimensional range searching and halfspace range searching [1, 28, 29]. ....

D. E. Vengroff and J. S. Vitter. Efficient 3-d range searching in external memory. In Proc. ACM Symp. on Theory of Computation, pages 192--201, 1996.

....Note that the query bound consists of an O(log B N) search term corresponding to the familiar 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 ....

....a set of N rectangles in the plane, a query asks for the number of rectangles intersecting a query rectangle. Finally, they extended their structures to the d dimensional versions of the two problems. See also [157] and references therein. Higher dimensional range searching. Vengroff and Vitter [149] presented a data structure for 3 dimensional range searching with a logarithmic query bound. With recent modifications their structure answers queries in O(log B N T=B) I Os and uses O( N B log 3 N B = log log 3 B N) space [151] More generally, they presented structures for answering ....

D. E. Vengroff and J. S. Vitter. Efficient 3-D range searching in external memory. In Proc. ACM Symp. on Theory of Computation, pages 192--201, 1996.

.... either presented with a batch of queries (the batched scenario) or we want to build a data structure that can answer queries in o(scan(N) I Os per query, depending on the size of the output (the online scenario) I O ecient solutions for the online range searching problem have been presented in [3, 7, 20]. When solving the batched range searching problem, our goal is to minimize the total number of I Os spent on answering all queries. An I O ecient solution to this problem has been presented in [2] While range searching asks to report all points in P q, range counting asks to report a value N ....

....N takes scan(N) N DB I Os. EM algorithms for computing pairwise intersections of orthogonal line segments, answering range queries in the plane, nding all nearest neighbors for a set of N points in the plane, dominance problems, and other geometric problems in the plane are discussed in [2, 3, 7, 13, 20]. General line segment intersection problems have been studied in [6] For lower bounds on computational geometry problems in EM see [5] See [4] for bu er trees, priority queues, and their applications. Overview. In Sect. 2, we discuss our solution to the batched range counting problem. In Sect. ....

D. E. Vengro, J. S. Vitter. Ecient 3-D range searching in external memory. Proc. STOC, 1996.

....them balanced [35, 40] In contrast, our cross tree can be easily kept balanced. Many other powerful data structures for range queries were devised subsequently and we refer the reader to [11] for a comprehensive survey on this topic and a list of references. Recently, some elegant data structures [5, 24, 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 ....

D.E. Vengro and J.S. Vitter. Ecient 3-D range searching in external memory. Proc. 28th ACM Symp. on Theory of Computing (1996) pp.192{ 201.

....key reason for this discrepancy is the important practical restriction that the structures must use near linear space. Recently, some progress has been made on the construction of structures with provably good performance for (special cases of) two dimensional [5, 29, 38, 43] and three dimensional [44] isothetic range searching. Even though the practical data structures mentioned above are often presented as structures for performing isothetic range searching, most of them can be easily modified to answer non isothetic queries and thus also halfspace range queries. However, the query ....

....layers; see Figure 3 (ii) The basic idea underlying our approach is to find the layer containing the query point and report all lines of L lying below the query point. Similar ideas have been used previously to build range searching data structures in internal memory [4, 12] and external memory [44]. How exactly we do this efficiently is described next. For each layer 0 i m, as stated in Lemma 3.1, we compute a 3 i clustering Gamma i of A i (L) of size at most N= i . By Lemma 2.2, the expected number of vertices in A i (L) is O(N ) so Gamma i can be computed using O(N(log 2 n) log B ....

D. E. Vengroff and J. S. Vitter. Efficient 3-d range searching in external memory. In Proc. of the 28th Annual ACM Symposium on Theory of Computing (STOC '96), pages 192--201, Philadelphia, PA, May 1996.

....papers also deal with fundamental problems such as permutation, sorting and matrix transposition. The problem of implementing various classes of permutations has been addressed in [47, 48, 50] More recently researchers have moved on to more specialized problems in the computational geometry [11, 15, 34, 40, 67, 74, 79, 110, 121, 130, 137], graph [12, 40, 42, 97] and string areas [44, 56, 57] As already mentioned the number of I O operations needed to read the entire input is N=B and for convenience we call this quotient n. We use the term scanning to describe the fundamental primitive of reading (or writing) all elements in a ....

....problems. A number of researchers have considered the design of worst case efficient external memory on line data structures, mainly for the range searching problem. While B trees [21, 51, 82] efficiently support range searching in one dimension they are inefficient in higher dimensions. In [27, 74, 79, 110, 121, 130] data structures for (special cases of) two and three dimensional range searching are developed. In [Interval] we develop an optimal on line data structure for the equally important problem of dynamic interval management. This problem is a special case of two dimensional range searching with ....

[Article contains additional citation context not shown here]

D. E. Vengroff and J. S. Vitter. Efficient 3-d range searching in external memory. In Proc. ACM Symp. on Theory of Computation, pages 192--201, 1996.

....time under the RAM model since that model, among other things, does not take the memory management costs into account. In this sense the RAM model is too powerful to model modern computers well. There are several results dealing with the I Ocomplexity of various algorithms and data structures (cf. [4, 5, 96, 102]) but for our problem this will not be a good model of computing either. More relevant to us is the work on hierarchical memory management by Aggarwal et al. 2] On the other hand, several papers have lately shown that the RAM model is not powerful enough. A number of interesting techniques ....

D.E. Vengroff and J.S. Vitter. Efficient 3-d range searching in external memory. In 28 th ACM Symposium on Theory of Computing, 1996.

.... Time Source d = 1 Interval N Log n k=B [35, 92] Quadrant N log log B Log n k=B [220] d = 2 3 sided rectangle N Log n k=B log B [240] 3 sided rectangle N log B log log B Log n k=B [220] Rectangle N log N= log Log n Log n k=B log B [240] d = 3 Octant N log N fi(n) Log n k=B [245] Rectangle N log 4 N fi(n) Log n k=B [245] Table 2. Asymptotic upper bounds for secondary memory structures; here N = n=B, Log = log B , and fi(n) log log Log n. 3.4 Practical data structures None of the data structures described in Section 3.1 are used in practice, even in two ....

.... 92] Quadrant N log log B Log n k=B [220] d = 2 3 sided rectangle N Log n k=B log B [240] 3 sided rectangle N log B log log B Log n k=B [220] Rectangle N log N= log Log n Log n k=B log B [240] d = 3 Octant N log N fi(n) Log n k=B [245] Rectangle N log 4 N fi(n) Log n k=B [245] Table 2. Asymptotic upper bounds for secondary memory structures; here N = n=B, Log = log B , and fi(n) log log Log n. 3.4 Practical data structures None of the data structures described in Section 3.1 are used in practice, even in two dimensions, because of the polylogarithmic overhead in ....

J. S. Vitter and D. E. Vengroff, Efficient 3-d range searching in external memory, Proc. 28th Annu. ACM Sympos. Theory Comput., 1996, p. to appear.

....red blue line segment intersection, trapezoidal decomposition, triangulation of simple polygons, planar point location. General techniques presented include an external memory version of fractional cascading using the buffer tree, and an extended external segment tree. Vengroff and Vitter [85] describe external memory algorithms for three dimensional range searching and query processing. In [24] Chiang studied the performance of algorithms to report the intersections of orthogonal line segments. The study compared the optimal EM algorithm reported in [55] with various versions of a ....

D. E. Vengroff and J. S. Vitter. Efficient 3-d range searching in external memory. In Proc. ACM Symp. on Theory of Computation, pages 192--201, 1996. BIBLIOGRAPHY 165

....sweeping can also be applied to finding pairwise rectangle intersections and some other problems. 2.2 Red Blue Line Segment Intersection Given a set of non intersecting red segments and a set of non intersecting blue segments, report the intersections between red and blue segments. Arge et al. AVV95] were the first to solve this problem I O optimally in O(n log m n k) They used an I O optimal algorithm to sort segments according to the aboveness relation via an external memory segment tree and employed a variant of distribution sweeping with a branching factor of p m. 2.3 General Line ....

....to the aboveness relation via an external memory segment tree and employed a variant of distribution sweeping with a branching factor of p m. 2. 3 General Line Segment Intersection A first step to compute the intersections of arbitrary line segments I O efficiently was taken by Arge et al. AVV95] They presented a very involved algorithm using an extended external segment tree and a variant of distribution sweeping to achieve an I O bound of O( n k) log m n) Crauser et al. CFM 98] used the important techinque of randomized incremental construction with gradations to obtain an ....

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D.E. Vengroff and J.S. Vitter. Efficient 3-d range searching in external memory. In Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, 1995. 8

.... k=B [35, 96] Quadrant N log log B Log n k=B [244] 3 sided rectangle N Log n k=B log B [270] d = 2 3 sided rectangle N log B log log B Log n k=B [244] Rectangle N log N= log Log n Log n k=B log B [270] Rectangle cN k=B 1 Gamma1=2c [270] d = 3 Octant N log N fi(n) Log n k=B [277] Rectangle N log 4 N fi(n) Log n k=B [277] Table 2. Asymptotic upper bounds for secondary memory structures; here N = n=B, Log n = log B n, and fi(n) log log Log n. bounds proved in Section 3.2 imply that we cannot hope for small worst case bounds and it should support insertions and ....

.... [244] 3 sided rectangle N Log n k=B log B [270] d = 2 3 sided rectangle N log B log log B Log n k=B [244] Rectangle N log N= log Log n Log n k=B log B [270] Rectangle cN k=B 1 Gamma1=2c [270] d = 3 Octant N log N fi(n) Log n k=B [277] Rectangle N log 4 N fi(n) Log n k=B [277] Table 2. Asymptotic upper bounds for secondary memory structures; here N = n=B, Log n = log B n, and fi(n) log log Log n. bounds proved in Section 3.2 imply that we cannot hope for small worst case bounds and it should support insertions and deletions of points. Keeping these goals in ....

J. S. Vitter and D. E. Vengroff, Efficient 3-d range searching in external memory, Proc. 28th Annu. ACM Sympos. Theory Comput., 1996, pp. 192--201.

....one disk block. Previous Related Work We first briefly review the work on I O techniques. In addition to early work on sorting and scientific computing [2, 28, 44] recently various researchers have been investigating external memory algorithms for graphs [1, 12] and for computational geometry [1, 3, 5, 6, 7, 10, 18, 21, 29, 38, 43]. As mentioned before, most of the results are theoretical, and yet the experiments of Chiang [11] Vengroff and Vitter [42] and Arge et al. 5] on some of these techniques show that they result in significant improvements over traditional algorithms in practice. As for isosurface extraction, ....

D. E. Vengroff and J. S. Vitter. Efficient 3-D range searching in external memory. In Proc. Annu. ACM Sympos. Theory. Comput., pages 192--201, 1996.

....log B n k=B n=B Quadrant log B n k=B (n=B) log log B [100] d = 2 3 sided rect. log B n k=B log B n=B [106] 3 sided rect. log B n k=B (n=B) log B log log B [100] Rectangle log B n k=B log B (n=B) log(n=B) log log B n [106] d = 3 Octant fi(n; B) log B n k=B (n=B) log(n=B) [110] Rectangle fi(n; B) log B n k=B (n=B) log 4 (n=B) 110] 16 Pankaj K. Agarwal OPEN PROBLEMS 1. Can a ball range counting query be answered in O(log n) time using O(n 2 ) space 2. Can a Gamma f range counting query be answered in time O(n 1 Gamma1=d ) using nearlinear space 3. A ....

....[100] d = 2 3 sided rect. log B n k=B log B n=B [106] 3 sided rect. log B n k=B (n=B) log B log log B [100] Rectangle log B n k=B log B (n=B) log(n=B) log log B n [106] d = 3 Octant fi(n; B) log B n k=B (n=B) log(n=B) 110] Rectangle fi(n; B) log B n k=B (n=B) log 4 (n=B) [110] 16 Pankaj K. Agarwal OPEN PROBLEMS 1. Can a ball range counting query be answered in O(log n) time using O(n 2 ) space 2. Can a Gamma f range counting query be answered in time O(n 1 Gamma1=d ) using nearlinear space 3. A solution to the following problem, which is interesting in its ....

J. Vitter and D. Vengroff, Efficient 3-d range searching in external memory, Proc. 28th ACM Symp. Theory of Computing, 1996, p. to appear.

....and slower external memory (disk) becomes a major bottleneck. Algorithms specifically designed to reduce the I O bottleneck are called external memory algorithms. In recent years various researchers have been investigating external memory algorithms for graphs [10] and for computational geometry [2, 4, 5, 8, 13, 16, 23, 31, 34], in addition to early work on sorting and scientific computing [1, 21, 35] Although most of the results are theoretical, the experiments of Chiang [9] and of Vengroff and Vitter [33] on some of these techniques show that they result in significant improvements over traditional algorithms in ....

D. E. Vengroff and J. S. Vitter. Efficient 3-d range searching in external memory. In Proc. Annu. ACM Sympos. Theory. Comput., pages 192--201, 1996.

....to the research papers for a discussion of this. For completeness it should be mentioned that recently a number of researchers have considered the design of worst case efficient external memory on line data structures, mainly for (special cases of) two and three dimensional range searching [20, 25, 59, 61, 78, 84, 93]. While B trees [22, 37, 65] efficiently support range searching in one dimension they are inefficient in higher dimensions. Similarly the many sophisticated internal memory data structures for range searching are not efficient when mapped to external memory. This has lead to the development of a ....

D. E. Vengroff and J. S. Vitter. Efficient 3-d range searching in external memory. In Proc. of the 28th Annual ACM Symposium on Theory of Computing (STOC '96), Philadelphia, PA, May 1996.

....denotes the iterated log function, that is, the number of times one must apply log to get below 2. It should be mentioned that the p range tree can be extended to answer general 2 dimensional queries, and that very recently a static structure for 3 dimensional queries has been developed in [43]. The segment tree [6] can also be used to solve the stabbing query problem, but even in internal memory it uses more than linear space. Some attempts have been made to externalizing this structure [8, 35] and they all use O( N=B) log 2 N) blocks of external memory. The best of them [35] is ....

D. E. Vengroff and J. S. Vitter. Efficient 3-d range searching in external memory. In Proc. ACM Symp. on Theory of Computation, pages 192--201, 1996.

....algorithms that minimize the input output communication (I O) performed when solving a given problem. The area was effectively started in the late eighties by Aggarwal and Vitter [6] and subsequently I O algorithms have been developed for several problem domains, including computational geometry [29, 7, 13, 14, 4, 15, 31, 38, 39, 41, 3, 44, 2, 12, 13, 16, 28, 30, 44], graph algorithms [17, 7, 33, 1, 21, 8, 27, 35, 40] and string processing [25, 26, 11, 20] Also I O performance can often be improved if many disks can efficiently be used in parallel and the use of parallel disks has received a lot of theoretical attention. Recent surveys of theoretical ....

D. E. Vengroff and J. S. Vitter. Efficient 3-d range searching in external memory. In Proc. ACM Symp. on Theory of Computation, pages 192--201, 1996.

....reason for this discrepancy is the important practical restriction that the structures must use near linear space. Recently, some progress has been made on the construction of structures with provably good performance for (special cases of) two dimensional [5, 34, 44, 49, 6] and three dimensional [50] isothetic range searching. Even though the practical data structures mentioned above are often presented as structures for performing isothetic range searching, most of them can be easily modi ed to answer nonisothetic queries and thus also halfspace range queries. However, the query performance ....

....increasing size; see Figure 5. Our approach is rst to nd the layer containing the query point and then to report all lines of L lying below the query point. Similar ideas have been used previously to build range searching data structures in internal memory [4, 15] and external memory [50]. How exactly we do this eciently is described next. # 3 # 2 # 1 4 # ###### ## Layers of levels For each i, we compute a 3 # clustering # of # # # (L) of size at most N= # , as described by Lemma 3.1. By Lemma 2.2, the expected number of vertices in # # # (L) is O(N ) so # can be ....

D. E. Vengro and J. S. Vitter, Ecient 3-d range searching in external memory, Proc. ACM Symp. on Theory of Computation, 1996, pp. 192-201.

....reason for this discrepancy is the important practical restriction that the structures must use near linear space. Recently, some progress has been made on the construction of structures with provably good performance for (special cases of) two dimensional [5, 34, 44, 49, 6] and three dimensional [50] isothetic range searching. Even though the practical data structures mentioned above are often presented as structures for performing isothetic range searching, most of them can be easily modi ed to answer nonisothetic queries and thus also halfspace range queries. However, the query performance ....

....increasing size; see Figure 5. Our approach is rst to nd the layer containing the query point and then to report all lines of L lying below the query point. Similar ideas have been used previously to build range searching data structures in internal memory [4, 15] and external memory [50]. How exactly we do this eciently is described next. l 3 l 2 l 1 4 l Figure 5. Layers of levels For each i, we compute a 3 i clustering i of A i (L) of size at most N= i , as described by Lemma 3.1. By Lemma 2.2, the expected number of vertices in A i (L) is O(N ) so i can be ....

D. E. Vengro and J. S. Vitter, Ecient 3-d range searching in external memory, Proc. ACM Symp. on Theory of Computation, 1996, pp. 192-201.

....denotes the iterated log function, that is, the number of times one must apply log to get below 2. It should be mentioned that the p range tree can be extended to answer general 2 dimensional queries, and that very recently a static structure for 3 dimensional queries has been developed in [10]. In this talk we present an optimal external memory data structure for the stabbing query problem [2] This result leads to the first known optimal solution to the interval management problem. In Figure 1 we Supported in part by the U.S. Army Research Office MURI grant DAAH04 96 1 0013. ....

D. E. Vengroff and J. S. Vitter. Efficient 3-d range searching in external memory. In Proc. ACM Symp. on Theory of Computation, pages 192--201, 1996.

....bound for sorting in internal memory. Work has also been done on matrix algebra and related problems arising in scientific computation [3, 51, 52] More recently, researchers have designed external memory algorithms for a number of problems in different areas, such as in computational geometry [32, 5, 53, 31, 2, 11, 34, 44, 47, 12, 50, 17, 1], string processing [28, 29, 9] and graph theoretic computation [6, 24, 38, 35] Some encouraging experimental results regarding the practical merits of the developed algorithms have also been obtained [23, 51, 11, 33] Recent surveys can be found in [7, 8] 1.3 Our Results In this paper, we ....

D. E. Vengroff and J. S. Vitter. Efficient 3-d range searching in external memory. In Proc. ACM Symp. on Theory of Computation, pages 192--201, 1996.

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