J. Matousek, "Geometric range searching," ACM Computing Surveys, 26 (1994) 421--461.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of computational geometry, multidimensional range search, also known as orthogonal range search, was one of the fundamental areas of interest. This interest produced a wealth of results, of which only few can be mentioned here. A number of books (e.g. Sam89a, Sam89b] and surveys (e.g. Meh84, Mat94, AE97] cover the subject very thoroughly. In the context of geometric range search, the term used is orthogonal range search. 13 2.1.1 Quadtree and kd tree One of the earliest data structures was the quad tree, proposed by Finkel and Bentley [FB74] In its simplest form, the quadtree is a ....

....attracted considerable attention, and has motivated some of the most elegant results in the past two decades. Here, we shall mention selectively some basic techniques, that gave rise to recent results for external memory. For some excellent surveys, as well as more complete references, see [AE97, Mat94] The earliest technique was the partition tree, proposed by Willard [Wil82] It became the basis for most linear space data structures for this problem. The original data structure had query cost O(n log 4 3 t) for d = 2. This was improved by a series of papers, most notably by the seminal ....

J. Matousek. Geometric range searching. ACM Computing Surveys, 26(4):422--461, 1994.

....problem, depending on whether objects have zero extent (point objects) or not. Aggregation over point objects is a special case of orthogonal range searching which has received vast attention in the past 20 years in the field of computational geometry. For more details, we refer to the surveys [Meh84, PS85, Mat94, AE98]. Most of the solutions utilize some variation of the range tree ( Ben80] following the multi dimensional divide and conquer technique. In the database field, JL98] proposed the R a Gamma tree which stores aggregated results in the index. Aok99] proposed to selectively traverse a ....

J. Matousek, "Geometric Range Searching", Computing Surveys 26(4), pp. 422-461, 1994.

....in areas that have other simple geometric shapes; furthermore the assumption about even distribution of the sensors can be dropped as long as the union of the sensed regions cover the domain of interest. This falls under an area studied in computational geometry known as range searching [31, 3]. To our knowledge, little has been done on this in this direction in the sensor setting; the closest is work motivated by the data base view [10, 11] Cluster maintenance: In this scenario we have n friendly vehicles carrying sensors. These vehicles wish to organize themselves into clusters, ....

Jiri Matousek. Geometric range searching. Computing Surveys, 26:421--462, 1994.

....the 2D tag memory too. Thus, the associative search is organized hierarchically, on the two levels. This new design of associative architecture is shown to be effective in solving some exemplary problems, that involve complex search operations like: multiple search [28] geometric range search [19] and matrix problems [12] Six representative problems with various computational complexity characteristics are selected, and algorithms for them are derived with the multi comparand search as a basic operation. In the next section the machine model is described. Section 3 is devoted to ....

....bits of all elements of B. The following algorithm deals with the first case. Algorithm 1 1. DA A. 2. CA B. 3. DM, CM, SM1 1. 4. compute TM for the prescription function f = g. 5. C2 0. 6. SM2 1. 7. compute T2. 8. if w=0 then return YES else return NO. GEOMETRIC RANGE SEARCH [19]. There are many variants of geometric range search problems. We will consider two of them: Problem A: Given a system of ranges of the d dimentional Euclidean space R d and a n point set P in R d ; find for given range R, whether all points of P are lying in R. Remark: In [12] redundant ....

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Matousek J.: Geometric range searching, Computing Surveys, 26 (1994), No.4, pp. 421-461.

....are needed to report T output points. We refer to these bounds as linear and introduce the notation n = dN=Be and t = dT=Be. 1. 2 Previous results In recent years tremendous progress has been made on non isothetic range searching in the computational geometry community; see the recent surveys [3, 33] and the references therein. As mentioned, halfspace range searching is the simplest form of non isothetic range searching and thus the problem has been especially extensively studied. Unfortunately, all the results are obtained in main memory models of computation where I O efficiency is not ....

J. Matousek. Geometric range searching. ACM Comput. Surv., 26:421--461, 1994.

....As mentioned in the introduction, simplex range searching has received considerable attention during the last few years. Besides its direct applications, simplex range searching data structures have provided fast algorithms for numerous other geometric problems. See the survey paper by Matousek [184] for an excellent review of techniques developed for simplex range searching. Unlike orthogonal range searching, no simplex range searching data structure is known that can answer a query in polylogarithmic time using near linear storage. In fact, the lower bounds stated below indicate that there ....

....the best known lower bounds for offline range searching. Lower bounds for emptiness problems apply to counting and reporting problems as well. No nontrivial lower bound was known for any offline range searching problem under the group model until Chazelle s result [61] See the survey papers [60, 184] for a more detailed discussion on lower bounds. 28 Pankaj Agarwal and Jeff Erickson Range Problem Model Query Time Source Simplex Semigroup Semigroup (d = 2) n p m [56] Semigroup Semigroup (d 2) n m 1=d log n [56] Reporting Pointer machine n 1 Gamma m 1=d k [77] Hyperplane ....

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J. Matousek, Geometric range searching, ACM Comput. Surv., 26 (1994), 421--461.

....S that are contained in the ball with center q and radius r. This ball is called a query ball. In the two dimensional case a query ball is also called a query disc. 4.1.1. 1 Circular range searching For an overview of different kinds of range searching problems we refer to the survey of Matousek [58] and Agarwal and Erickson [1] Time optimal solutions for the 69 two dimensional circular range searching problem use higher order Voronoi diagrams. Informally, the ith order Voronoi diagram V D i (S) of a set S of n sites in the d space, 1 i n, partitions the space into regions such that each ....

J. Matousek. Geometric range searching. ACM Computing Surveys, 26:421--461, 1991.

....space IR d . We consider the problem of preprocessing P so that given any query halfspace fl, one can quickly report all points in P fl. A vast literature in computational geometry has been devoted to this fundamental problem called halfspace range reporting , a special case of range searching [4, 37, 49, 53, 55]. Here are some of the major known results. First, halfspace range reporting in the planar case (d = 2) was solved optimally by Chazelle, Guibas, and Lee [28] Their data structure takes linear space and answers a query in O(log n k) time, where k is the number of reported points. The ....

J. Matousek. Geometric range searching. ACM Comput. Surv., 26:421--461, 1994. 17

....up to some factor f 1. f spanners enabled Rao and Smith to construct a FPTAS (Fully Polynomial Time Approximation Scheme) for the Euclidean Travelling Salesperson Problem [21] Further applications are closest point queries [25] motion planning [7] as well as many range searching problems [1, 18]. For example, the objective of a circular range query is reporting all those points p of P lying within a circle of given radius r and center c. Having constructed an f spanner G for P ae IR D of outdegree k, queries with centers c 2P can be answered in nearly output sensitive running time, ....

Jir Matousek: "Geometric Range Searching", ACM Computing Surveys 26(4), 1994, 422-461.

....up to some factor f 1. f spanners enabled Rao and Smith to construct a FPTAS (Fully Polynomial Time Approximation Scheme) for the Euclidean Travelling Salesperson Problem [12] Further applications are closest point queries [15] motion planning [4] as well as many range searching problems [1, 11]. For example, the objective of a circular range query is reporting all those points p of P lying within a circle of given radius r and center c. Having constructed an f spanner G for P ae IR D of outdegree k, queries with centers c 2 P can be answered in nearly output sensitive running time, ....

Jir'i Matousek: "Geometric Range Searching", ACM Computing Surveys 26(4), 1994, 422-461.

.... enabled Rao and Smith to construct a FPTAS (Fully Polynomial Time Approximation Scheme) for the Euclidean Travelling Salesperson Problem [21] Among other applications are closest point queries [25] motion planning [7] min cost perfect matching [27] as well as many range searching problems [1, 18]. For example, the objective of a circular range query is to report all those points p of P lying within a circle of given radius r and center c. Having constructed an f spanner G for P ae IR D of outdegree k, queries with centers c 2 P can be answered in nearly output sensitive running time, ....

Jir Matousek: "Geometric Range Searching", ACM Computing Surveys 26(4) 1994, 422-461

....for set systems with sets defined by a constant number of bounded degree inequalities can be computed using the techniques for simplices. Bibliography and remarks. Literature on geometric range searching is extensive; recent survey papers are Agarwal and Erickson [AE97] and Matousek [Mat95b] The original idea for producing geometric partitions is due to Willard [Wil82] Theorem 4.5 was proved by Matousek [Mat92a] by generalizing the ideas of Chazelle and Welzl [CW89] Interestingly, the s = 2 case has a version for general set systems, while for s = n=O(1) a similar result cannot ....

J. Matousek. Geometric range searching. ACM Comput. Surveys, 26:421--461, 1995.

....proportional to n. Questions of this type, the so called range searching problems, have been studied quite intensively and in a much more general form in higher dimensions, with different query shapes, with more space allowed, etc. there is a survey by Agarwal in [20] and another survey is [25]) But many interesting aspects can be demonstrated on the particular problem formulated above. In this case, it is possible to answer the query in O( p n) time, and with some restriction on the type of algorithm used, this is asymptotically optimal. Ironically, while the known data structures ....

....One almost wouldn t believe that after thousands of years of geometry, it is still possible to discover such pretty theorems about points in the plane. This was later generalized to a partition of an n point set into r parts of size roughly n r , with any line crossing O( p r) parts only (see [25]) Both these results are asymptotically optimal. The research in range searching also initiated a fruitful theory related to the so called Vapnik Chervonenkis dimension of set systems, with applications, e.g. in discrepancy theory; this is surveyed in [27] Lower bounds for range searching were ....

J. Matousek. Geometric range searching. ACM Comput. Surv., 26:421--461, 1994.

....in R d , the improvement to d i = d i follows from known results; more details on this will be given in the proof. The proof is based on techniques developed in computational geometry for the rangesearching problem; detailed information about range searching can be found in the surveys [1, 24]. We will use results of the following type: Given a finite set P ae R d and a family A of Tarski cells in R d , one can define a small system C of subsets of P , such that each intersection A P , where A 2 A, can be written as a disjoint union of few of the sets of C, most of which, ....

J. Matousek. Geometric range searching. ACM Comput. Surveys, 26:421--461, 1995.

....Finally, Section 4 mentions connection of more general distance selection problems to halfspace range counting : how to preprocess a given n point set so that one can quickly count the number k of points inside a given query halfspace. The problem has been studied by a number of researchers; see [2, 28] for surveys. In the planar case, O(n 1=2 polylog n) query time is attainable after O(n log n) preprocessing time [26] On the other hand, results on halfspace range reporting imply an output sensitive query time of O(log n k) with the same preprocessing time [9] We point out that combining ....

J. Matousek. Geometric range searching. ACM Comput. Surveys, 26:421--461, 1994.

....IR d . We consider the problem of preprocessing P so that given any query halfspace fl, one can quickly report all points in P fl. A vast literature in computational geometry has been devoted to this fundamental problem called halfspace range reporting , a special case of range searching [4, 34, 44, 48, 49]. Here are some of the major known results. First, halfspace range reporting in the planar case (d = 2) was solved optimally by Chazelle, Guibas, and Lee [25] Their data structure takes linear space and answers a query in O(log n k) time, where k is the number of reported points. The ....

J. Matousek. Geometric range searching. ACM Comput. Surv., 26:421--461, 1994.

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J. Matousek, "Geometric range searching," ACM Computing Surveys, 26 (1994) 421--461.

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J. Matousek, "Geometric Range Searching", Computing Surveys 26(4), 1994.

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J. Matousek, "Geometric Range Searching", Computing Surveys 26(4), 1994.

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Jir Matousek. Geometric range searching. ACM Comput. Surv., 26(4):421--461, 1994.

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Matousek, J. Geometric range searching, ACM Comput. Surv. 26 (1994), 421--461.

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J. Matousek, Geometric Range Searching, Computing Surveys 26,4 (Dec. 1994), 421461.

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J. Matousek, "Geometric Range Searching", ACM Computing Surveys 26(4), 1994.

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J. Matousek, "Geometric Range Searching", Computing Surveys 26(4), pp. 422-461, 1994.

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J. Matousek. Geometric range searching. ACM computing surveys, Vol.26, No.4, December 1994.

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