M. H. Overmars. Efficient data structures for range searching on a grid. J. Algorithms, 9:254--275, 1988.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....at least len, we can find the entries among SA0 [i] SA0 [j] whose mod value is t by means of the functions Psi 0 and Psi 0 0 and a certain two dimensional orthogonal range query. The range search can be done, using O(n) bits, either in O(log log n occ t ) time for len = log 2 n [43] or O(n fi occ t ) time for len = log n and any fixed fi 0 [6; 47] The total running time for all len range searches is thus the cost of the pattern search plus O( log 2 n) log log n occ) which is O(m= log n occ) when the pattern is Omega Gamma 473 3 n) log log n) bits long; for ....

M. H. Overmars. Efficient data structures for range searching on a grid. Journal of Algorithms, 9(2):254-- 275, June 1988.

....The graph labeled grid 14 corresponds to a grid with 14 3 cells; similarly for the other graphs. that the sweep plane could potentially intersect many bounding volumes although they are very far from each other. We have evaluated two space indexing approaches, namely the grid and octree [Ove88, GA93, MSH 92, HT92] The advantage of space indexing approaches is that they work locally (defined by the cell size) with the price that they do not work at object precision but at cell precision. Our implementations of grid and octree work incremental so that only those cells have to be ....

Overmars, M. H. Efficient data structures for range searching on a grid. J. Algorithms, 9:254--275, 1988.

....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 ....

....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 ....

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M. H. Overmars. Efficient data structures for range searching on a grid. Journal of Algorithms, 9(2):254--275, 1988.

....due to Chazelle [52, 55] a data structure that can answer a range reporting query in time O(log d Gamma2 n log n k) using O(n log d Gamma1 n) space, and Bozanis et al. 48] have proposed an a data structure with O(n log d n) size and O(log d Gamma2 n k) query time. Overmars [209] showed that if S is a subset of a u Theta u grid U in the plane and the vertices of query rectangles are also a subset of U , then a range reporting query can be answered in time O( p log u k) using O(n log n) storage and preprocessing, or in O(log log u k) time, using O(n log n) storage ....

M. H. Overmars, Efficient data structures for range searching on a grid, J. Algorithms, 9 (1988), 254--275.

....solved with the same time bounds. 5 Proof: Since the arrays are permutations, every number between 1 and n appears precisely once in each array. The coordinates of every number i are [x; y] where V [x] W [y] i. It is clear that the range search gives precisely the intersection. ut Overmars [22] shows an algorithm that preprocesses the points in time and space O(n log n) and the query time is O(k p log U) where k is the number of points in the range [a; b] Theta [c; d] Therefore we have the following. Theorem 5.3 Let T = t 1 Delta Delta Delta t n and P = p 1 Delta Delta ....

....can be solved with O(n log 2 n) preprocessing time and O(tocc m log n log log n) query time, where tocc is the number of occurences of the pattern in the text with at most one error. Proof: The time of the preprocessing stage is affected only by the range query preprocessing. By Theorem 7. 2 of [22], half infinite range queries on the three dimensional range can be preprocessed in time and space O(n log 2 n) Therefore O(n log 2 n) is the total preprocessing time. If we use the method suggested in [22] and give the highest level to one of the index coordinates and use the alphabet ....

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M. H. Overmars. Efficient data structures for range searching on a grid. J. of Algorithms, 9:254--275, 1988.

....structure for orthogonal range searching in d dimension. For d 3 we give a data structure that occupies O(n log d Gamma1 n) space and answers queries in O(IL (n) log d Gamma2 n t) time. The time space product of this data structure is better than that of previously known data structures [5,25]. We also give a lower bound for range searching in secondary memory that shows that 2 dimensional range searching is inherently harder in secondary storage than in main memory. Our lower bound shows that to implement an efficient secondary memory data structure for 2 dimensional range ....

....query q in O(IL (n) t) time where t denotes the number of points of P that belong to q. The data structure of Theorem 4.2 can be used to build a multidimensional range searching data structure for dimension three or more. By using a normalization technique due to Karlsson and Overmars [14, 25] we can reduce the general d dimensional range searching problem (with arbitrary coordinates) to one in which all the coordinates have value in the range [1: n] This reduction only adds a factor of d log n to the query time. Therefore, using the p range tree and the d dimensional range searching ....

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M. H. Overmars, "Efficient Data Structures for Range Searching on a Grid.," Department of Computer Science, University of Utrecht, Technical Report RUU-CS-87-2, 1987.

.... computational geometry has so far only been studied by a few researchers (overviews can be found in [KK81, Ove88a, Ove88b] More efficient solutions in comparison with their Euclidean counterparts have been found for the nearest neighbor searching problem [KM85] range searching on a grid [Ove88b, Ove88c], the point location problem [M l85] the computation of rectangle intersections and maximal elements by divide and conquer [KO88b] computing the convex hull of a set of points, reporting all intersections of a set of arbitrarily oriented line segments, and the calculation of rectangle ....

M.H. Overmars. Efficient Data Structures for Range Searching on a Grid. Journal of Algorithms, vol. 9, 254-275, 1988.

....n= log log n) based on fusion trees, that can answer an orthogonal range reporting query in time O(log d Gamma1 n= log log n k) Fusion trees were introduced by Fredman and Willard [129] for an O(n p log n) sorting algorithm in a RAM model that allows bitwise logical operations. Overmars [231] showed that if S is a subset of a u Theta u grid U in the plane and the vertices of query rectangles are also a subset of U , then a range reporting query can be answered in time O( p log u k) using O(n log n) storage and preprocessing, or in O(log log u k) time, using O(n log n) storage ....

M. H. Overmars, Efficient data structures for range searching on a grid, J. Algorithms, 9 (1988), 254--275.

....has coordinates of the form (x; y) with x; y 2 f1; 2; kg. We keep the k corners of the loop in an array of size O(k) with a start and an end pointer. The corners of the loop are also organized in the balanced binary tree structure for two dimensional range searching as in [5] see also[16]) where instead of storing the coordinates of the corners, we store a pointer to the appropriate entry in the array. This structure takes O(k log k) time to build. If the loop has more than three corners, then we do the following. ffl We find the point a with minimum y coordinate in O(log k) ....

M. H. Overmars. Efficient Data Structures for Range Searching on a Grid. Journal of Algorithms, 9 (1988), pp. 254-175.

....EPM n log 4 n All the results mentioned in Table 1 can be extended to higher dimensions at a cost of log d Gamma2 n factor in the preprocessing time, storage, and query search time. Table 2 summarizes a few additional results on higher dimensional orthogonal range searching results. Overmars [96] showed that if S is a subset of a u Theta u grid U in the plane and the Range Searching 5 TABLE 2 Higher dimensional orthogonal range reporting S(n) Q(n) Source Notes n log d Gamma1 n log n log log n d Gamma1 k [88] Pointer machine m log n log 2m=n d Gamma1 [30] ....

M. H. Overmars, Efficient data structures for range searching on a grid, J. Algorithms, 9 (1988), 254--275.

....better lower bound must take the form of a space time tradeoff. 1 Sublogarithmic or even constant query times can be obtained for axis aligned rectangular queries in models of computation that allow bit manipulation and require integer inputs within a known bounded universe; see, for example, [4, 5, 13, 44, 45, 53]. No such result is known for non orthogonal ranges, however. We will take the traditional computational geometric view that geometric objects are represented by arbitrary real coordinates, for which bit manipulation is impossible. 2 Jeff Erickson Space Preprocessing Query Time Source O(n d = ....

M. H. Overmars. Efficient data structures for range searching on a grid. J. Algorithms 9:254--275, 1988.

.... by Greene and Yao [GrY86] as well as Yao [Ya92] Finite precision geometry has so far only been studied by a few researchers (overviews can be found in [KeK81, Ov88b, Ov88c] Problems considered are, for example, the nearest neighbour searching problem [KaM85] range searching on a grid [Ov88a, Ov88b], the point location problem [M 85] the computation of rectangle intersections and maximal elements by divide and conquer [KaO88b] computing the convex hull of a set of points, reporting all intersections of a set of arbitrarily oriented line segments, and the calculation of rectangle ....

Overmars, M.H., Efficient Data Structures for Range Searching on a Grid. Journal of Algorithms 9 (1988), 254-275.

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M. H. Overmars. Efficient data structures for range searching on a grid. J. Algorithms, 9:254--275, 1988.

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M. H. Overmars. Efficient data structures for range searching on a grid. J. Algorithms, 9:254--275, 1988.

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M. H. Overmars. Efficient data structures for range searching on a grid. Journal of Algorithms 9, pp. 254--275, 1988.

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