Agarwal, P. K. (1997), Range searching, in J. E. Goodman & J. O'Rourke, eds, `Handbook of Discrete and Computational Geometry', CRC Press LLC, Boca Raton, FL, chapter 31, pp. 575--598.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... l i = 0, n 2 (5) 2 is a random 3 D vector uniformly distributed on the unit 3 D sphere and l is the fixed Euclidean distance between two consecutive points of the chain. 2 is sampled as follows: 2 = sin # cos # sin # sin # cos # (6) where # U[0, 2#] and cos # U [ 1, 1]. Computing the covariance matrix of 2 reveals that the off diagonal elements are identically 0, and as a result the three dimensions of each random step are uncorrelated. Since each step is independent of all other steps the distributions of the three dimensions of any point on the chain are ....

....approach is to evaluate the similarity measure (cRMS or dRMS) for all pairs and then report the k NNs for each sample. However, the quadratic complexity makes this approach scale badly. Spatial data structures such as the kd tree [4] can avoid this complexity under certain circumstances [1, 5, 17, 15, 23, 27]. Note that these data structures allow for exact search, i.e. they return the same NNs as would the brute force search. However, most of them require a Euclidean metric space of rather limited dimensionality. Unfortunately, cRMS is not a Euclidean metric. Although dRMS is a Euclidean metric, ....

P. K. Agarwal. Range searching. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, pages 575-- 598. CRC Press, 1997.

....technique was used. Surveys on kNN Research Finally, numerous surveys and books provide an overview of the Nearest Neighbour(s) family of problems and catalogues the numerous techniques and data structures developed to solve these problems. Notable ones includes surveys by Agarwal et al. [2,3], Arya and Mount [6] Smid [65] Gaede and Gunther [22] and Tsaparas [68] as well as books edited by Goodman et al. 25] and Sack et al. 59] 3 Block Hashing: Preliminaries We have developed a novel technique called block hashing to solve the approximate kNN (AkNN) problem in the context of, ....

P. K. Agarwal. Range Searching. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry. CRC Press, July 1997. 2.2

....hash value to each partition. Mathematically, let = t P be a monotonically increasing sequence of P 1 thresholds between 0 and 1. Assume t 0 = 0 and t P = 1, so there are P 1 degrees of freedom in this sequence. Define a one dimensional Locality Sensitive Hash Function h : [0, 1] # 0 . P 1 to be h (t) i, where t i t t i 1 . In other words, the hash value i can take on P different values, one for each bucket defined by the threshold pair [t i , t i 1 ) An example is shown in Figure 1. Figure 1: An example of h . The circles and boxes ....

P. K. Agarwal. Range Searching. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry. CRC Press, July 1997. 2

....is given in Sect. 2. emptiness problem (aka. existential range queries) We prove optimal bounds on maintaining points in the plane and check if a given rectanglecontains any points. Finding a lower bound for this problem is Open Problem 1 in a recent handbook chapter on range queries by Agarwal [1]. range searching and partial sums: Lower bounds for range searching problems in the plane are known for structured or algebraic models [13, 22, 39, 42] We extend these to the stronger cell probe model. priority search trees: We show that Willard s RAM improvement [40] of McCreight s classic ....

....at the heart of all range searching problems: maintain a set S [n] of points in the plane under insertions and deletions, and determine whether S R = for rectangle R. Finding a lower bound for this problem is Open Problem 1 in a recent handbook chapter on range queries by Agarwal [1]. Proposition 1. The planar emptiness problem requires time log log n) per operation. This is true even for dominance queries, where all query rectangles have their lower left corner in the origin. The bound holds for the incremental or decremental variants, and also for the amortised query ....

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Pankaj K. Agarwal. Range searching. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 31, pages 575-598. CRC Press LLC, 1997.

....the previous section we have focused our attention on standard elementary data structures. Now we shall consider some soft kinetic data structures from computational geometry. 3. 1 1D Range Trees We can use soft kinetic data structures for binary search trees to obtain the following result (see [1] for a formal definition of range trees) Theorem 7 There is a soft kinetic version of 1D range trees such that range queries are supported in O(log n k) time, where k is the number of reported points. The 1D range tree is O( log 2 n ) competitive. Further, if k denotes the number ....

....counting queries. The data structure is O( log n log 2 n) competitive and answers interval counting queries in time O(log n) with absolute error with high probability. 3. 2 2D Range Trees In this section we describe and analyze 2 dimensional soft kinetic range trees (see, e.g. [1] for a formal definition) We consider a standard implementation of 2D range trees which can be regarded as a double level 1D range tree: there is one 1D range tree for the first level structure and n 1D range trees (one for each node in the first level structure) for the second level structure ....

[Article contains additional citation context not shown here]

P. K. Agarwal. Range searching. In Handbook of Discrete and Computational Geometry, J. E. Goodman and J. O'Rourke, eds., CRC Press, Boca Raton, FL, 1997.

....Count in constant time and uses O(n) space. The preprocessing time for the mentioned data structures is expected time O(n p log u) 1. 1 Related work Efficient static data structures for range searching have been studied intensively over the past 30 years, for surveys and books see e.g. [1, 18, 20]. In one dimension there has been much focus on the following two fundamental problems: the membership problem and the predecessor problem. These problems address the following queries respectively: Member(a) a 2 U : Return yes iff a 2 S. Pred(a) a 2 U : Return the predecessor of a, i.e. ....

....the m and M values of v:left and v:right is in [a; b] If one of the four values belongs to [a; b] we return such a value. Otherwise is returned. As an exampled consider the query FindAny(8; 13) for the set in Figure 1. Here d = 2, u = 8 4) # 3 = 3, z = 3 # ( 3 2) 1) 3 # 1 = 1. Since B[1] = 1, we have D[u] 1, and v = V [u # D[u] V [3 # 1] V [1] The four values tested are the m and M values of V [2] and V [7] i.e. f3; 7; 12; 14g, and we return 12. Theorem 4 The data structure supports FindAny in constant time and Report in time O(k) where k is the number of elements ....

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P. K. Agarwal. Range searching. In Handbook of Discrete and Computational Geometry, CRC Press. 1997. 10

....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 ....

....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 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 ....

P. Agarwal. Range searching. In Handbook of Discrete and Computational Geometry, CRC Press. 1997.

....of fixed radius centered at m. The current implementation uses a naive method that checks every new milestone m 0 against all the milestones currently in T . Thus, for every new milestone, updating w takes linear time in the number of milestones in T . More efficient range search techniques [Aga97] would certainly improve the planner s running time for problems requiring very large roadmaps. Implementing PROPAGATE Given a milestone m and a control function u, PROPAGATE uses the Euler method with a fixed step size to integrate (6) from m and computes a trajectory of the system under the ....

P. K. Agarwal. Range searching. In J. E. Goodman and J. O'Rourke, editors, Handbook of discrete and computational geometry, pages 575--598. CRC Press, 1997.

....Count# in constant time, for any constant # 0. The preprocessing time for the mentioned data structures is expected time O(n # log u) 1. 1 Related work E#cient static data structures for range searching have been studied intensively over the past 30 years, for surveys and books see e.g. [1, 18, 20]. In one dimension there has been much focus on the following two fundamental problems: the membership problem and the predecessor problem. These problems address the following queries respectively: Member(a) a # U : Return yes if and only if a # S. Pred(a) a # U : Return the predecessor ....

....values of v.left and v.right is in [a, b] If one of the four values belongs to [a, b] we return such a value. Otherwise # is returned. As an example consider the query FindAny(8, 13) for the set in Figure 1. Here d = 2, u = 8 4) # 3 = 3, z = 3 # ( 3 2) # 1) 3 # 1 = 1. Since B[1] = 1, we have D[u] 1, and v = V [u # D[u] V [3 # 1] V [1] The four values tested are the m and M values of V [2] and V [7] i.e. 3, 7, 12, 14 , and we return 12. Theorem 4. The data structure supports FindAny in constant time and Report in time O(k) where k is the number of ....

[Article contains additional citation context not shown here]

P. K. Agarwal. Range searching. In Handbook of Discrete and Computational Geometry, CRC Press. 1997.

....searching is greatly facilitated by k d trees. For orthogonal range searching, a host of particular data structures have been developed, such as the range tree and variations or improvements of it (for surveys, see Bentley and Friedman, 1979; Bentley, 1979; Yao, 1990; Samet, 1990a, 1990b; and Agarwal, 1997). However, the k d tree o#ers several advantages it takes O(kn) space for n data points, it is easily updated and maintained, it is simple to implement and comprehend, and it is useful for other operations besides orthogonal range search. 0 1 2 3 4 5 6 7 8 9 11 14 16 17 18 0 1 2 3 4 5 6 7 8 9 11 ....

P. K. Agarwal, "Range searching," in: Handbook of Discrete and Computational Geometry, ed. J. E. Goodman and J. O'Rourke, pp. 575--598, CRC Press, Boca Raton, FL, 1997.

....author was at BRICS. 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 ....

....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 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 ....

P. Agarwal. Range searching. In Handbook of Discrete and Computational Geometry, CRC Press. 1997.

....the heart of all range searching problems: maintain a set S # [n] 2 of points in the plane under insertions and deletions, and given rectangle R determine whether S # R is empty. Finding a lower bound for this problem is Open Problem 1 in a recent handbook chapter on range queries by Agarwal [1]. Proposition 1 The planar emptiness problem requires time #(logn log log n) per operation. This is true even for dominance queries, where all query rectangles have their lower left corner in the origin. The bound holds for the incremental and decremental variants, and also for the amortised ....

P. K. Agarwal. Range searching. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 31, pages 575--598. CRC Press LLC, 1997.

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Agarwal, P. K. (1997), Range searching, in J. E. Goodman & J. O'Rourke, eds, `Handbook of Discrete and Computational Geometry', CRC Press LLC, Boca Raton, FL, chapter 31, pp. 575--598.

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Agarwal, P. K. (1997), Range searching, in J. E. Goodman & J. O'Rourke, eds, `Handbook of Discrete and Computational Geometry', CRC Press LLC, Boca Raton, FL, chapter 31, pp. 575--598.

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P.K. Agarwal. Range Searching. In Handbook of Discrete and Computational Geometry, J.E. Goodman and J. O'Rourke (eds.), CRC Press, pp. 575-598, 1997.

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P. K. Agarwal. Range searching. In Handbook of Discrete and Computational Geometry, pp. 575-598. CRC Press, Boca Raton, FL, 1997.

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P. K. Agarwal. Range searching. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 31, pages 575-598. CRC Press LLC, Boca Raton, FL, 1997.

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P.K. Agarwal. Range Searching. In: J. Goodman and J. O'Rourke (Eds.), CRC Handbook of Computational Geometry, CRC Press, pages 575--598, 1997.

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P.K. Agarwal. Range searching. In: J.E. Goodman and R. Pollack. Handbook of Discrete and Computational Geometry. CRC Press, 1997, pages 575--598.

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Pankaj K. Agarwal. Range searching. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 31, pages 575--598. CRC Press, 1997.

No context found.

P.K. Agarwal. Range searching. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 31, pages 575-598. CRC Press LLC, Boca Raton, FL, 1997.

No context found.

Pankaj K. Agarwal. Range searching. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 31, pages 575--598. CRC Press LLC, Boca Raton, FL, 1997.

No context found.

P. K. Agarwal. Range searching. In Handbook of Discrete and Computational Geometry, pp. 575--598. CRC Press, Boca Raton, FL, 1997.

No context found.

P. K. Agarwal. Range searching. In Handbook of Discrete and Computational Geometry, pp. 575--598. CRC Press, Boca Raton, FL, 1997.

No context found.

P. K. Agarwal. Range searching. In Handbook of Discrete and Computational Geometry, pp. 575--598. CRC Press, Boca Raton, FL, 1997.

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