P. Gupta, R. Janardan, and M. Smid. Algorithms for generalized halfspace range searching and other intersection searching problems. Computational Geometry Theory and Applications, 5:321-340, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....examples and provide pointers to related work. We conclude with a discussion of possible directions for further research. 1. 2 Summary of known results Generalized intersection searching problems were introduced by Janardan and Lopez in [23] Subsequent work in this area may be found in [2, 3, 6, 7, 8, 16, 18, 19, 20, 21, 22, 29]. Some of these results are also reported in two Ph.D. theses [17, 30] In this section, we give a broad overview of the work on these problems to date; details may be found in the cited references. 1.2.1 Axes parallel objects In [23] ecient solutions were given for several generalized ....

....were given in [23] for generalized reporting on non intersecting line segments using a query line segment. Special, but interesting, cases of intersecting line segments, such as when each color class forms a polygon or a connected component, were considered in [3] Ecient solutions were given in [19] for input query pairs consisting of points halfspace in R , points fat triangle, and fat triangles point in R . A fat triangle is a triangle where each internal angle is at least a user speci ed constant, hence wellshaped . Some of these results were improved subsequently in [6] In ....

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P. Gupta, R. Janardan, and M. Smid. Algorithms for generalized halfspace range searching and other intersection searching problems. Computational Geometry Theory and Applications, 5:321-340, 1996.

.... to the group it belongs to, then our goal is to report the distinct colors of the objects intersected by a query object (rather than reporting all the intersected objects as in the case of the standard problem) Such generalized intersection searching problems have been considered recently in [3, 11, 12, 13] in the context of linear input and query objects. The challenge in these problems is to obtain solutions whose query times are sensitive to the output size, namely the number, i, of distinct colors intersected (not the number, k, of intersected objects, which can be much larger) For the ....

....the search is O(i log n) At each such node, v, we spend time f(jS(v)j) which is O(f(n) since jS(v)j n and f is non decreasing. Thus the total time spent in doing queries at the visited nodes is O(i Delta f(n) log n) The claimed query time follows. 2 Theorem 3. 1 generalizes a method used in [12] for the generalized halfspace range searching problem in d 4 dimensions. 3.2 Some applications of the general technique First we consider the problem where the input is a set of colored disks and the query is a point. We need to solve the following (standard) problem: Preprocess a set of n ....

Gupta, P., Janardan, R. and Smid, M. (1996). Algorithms for generalized halfspace range searching and other intersection searching problems. Computational Geometry: Theory and Applications, 5, 321--340.

....colors intersected, rather than K. Typically, our goal is to obtain query times of the form O(f(n) C) or O(f(n) C Delta g(n) where f and g are polylogarithmic. Generalized searching problems first appeared in [6] Subsequently, several papers were published on these type of problems. See [1, 3, 4, 5, 7, 8]. In this paper, we consider the problem of transforming generalized searching problems into other problems by adding a range restriction. This transformation was introduced for standard searching problems by Bentley [2] Let PR(q; S) denote the answer to a generalized searching problem PR with ....

....structure for solving the standard problem PR into a data structure for the standard problem TPR. Later, other general techniques were developed by Willard and Lueker [11] Generalized Problem Space Query Time Reference Querying Points with Fat Triangles n(log n) 3 (log n) 4 C(log n) 2 [5] n 1 ffl (log n) 2 C this paper Querying Fat Triangles with Points n(log n) 2 (log n) 3 C log n [5] n 1 ffl (log n) 2 C [5] n 1 ffl (log n) 2 C this paper Table 1: Overview of results for generalized problems on fat triangles. All bounds given are big oh . C denotes ....

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P. Gupta, R. Janardan, and M. Smid. Algorithms for generalized halfspace range searching and other intersection searching problems. Computational Geometry: Theory and Applications 5 (1996), pp. 321--340.

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