D. Dobkin and H. Edelsbrunner. Space searching for intersecting objects. Journal of Algorithms, 8:348--361, 1987. 46  Home/Search   Document Not in Database   Summary   Related Articles   Check

....method [48] which is similar to a k d tree using arbitrary splitting planes has been proposed. Tokuyama [166] propose a theoretic method for orthogonal clipping with applications to nearest neighbour queries. A more general non orthogonal case has also been studied by Dobkin and Edelsbrunner [45]. Miller et al. 117] generalised the separator theorem of Lipton and Tarjan [107] to higher dimensions to obtain an algorithm using a neighbourhood system for point location and geometric divide and conquer in a fixed dimensional space. 2.4.1 Hyperplanes as Projections Despite the popularity of ....

D. Dobkin and H. Edeldbrunner. Space searching for intersecting objects. Algorithms, 8:348--361, 1987.

....segment intersection searching problem. Other colored intersection searching problems have been studied independently by Janardan and Lopez [16] Gupta et al. 14] and Nievergelt and Widmayer [19] In the last few years much work has been done on the segment intersection searching problem [2, 3, 5, 7, 10, 13, 21]. Agarwal and Sharir [3] showed that S can be preprocessed using O(n ) space and time, so that all k segments of S intersecting a query segment can be reported in time O(n k) The n factor in the size and query time can be reduced to log O(1) n factor using a more sophisticated ....

Dobkin, D. P. and H. Edelsbrunner, Space Searching for Intersecting Objects, J. of Algorithms 8 (1987), pp. 348--361.

....First, we consider ray shooting in axis parallel polyhedra. In this case it is possi ble to obtain O(log n) query time after O(n 2 ) preprocessing, by using the recently developed recursive partition trees of Chazelle et al. 9] instead of the conjugation trees of Dobkin and Edelsbrunner [14] in the second structure of [25] In fact, anything between near linear storage and roughly O(x ) query time and near quadratic storage and polylogarithmic query time is possible. We take a different approach and obtain a structure using the same amount of preprocessing time and space, namely ....

D.P. Dobkin and H. Edelsbrunner, Space Searching for Intersecting Objects, J. Algorithms 8 (1987), pp. 348-361.

....for the disjoint disks to which corresponds, as described above. This main tree is used to extract the disks with their center in the strip R. The other internal nodes have no associated structure. This technique of building hierarchical structures is based on ideas of Dobkin and Edelsbrunner [6]. A query with the line segment s = is performed as follows. Let I and 12 be the lines perpendicular to s, containing q and q2 respectively. Search with I and 12, by recursively continuing the search in the children of a node if the corresponding region is intersected by l or 12. If the region ....

Dobkin, D.P., and H. Edelsbrunner, Space searching for intersecting objects, J. of Algorithms $ (1987), pp. 348-361.

....to outside. 5 Discussion of the Technique The spatial index has been motivated in part by Vanecek s brep index [Van90b, Van90c] The brep index is a generalization of the binary space partition (BSP) trees [FKN80, Van90a] and motivated indirectly by the cut trees of Dobkin and Edelsbrunner [DE87]. It is useful to compare these two approaches. In the brep index, the boundary of an object, given by a boundary representation (brep) is recursively cut by oriented cut planes, thus fragmenting the boundary into pieces that do not cross the splitting planes. This partitions space into convex ....

D. P. Dobkin and H. Edelsbrunner, "Space Searching for Intersecting Objects." Journal of Algorithms, 8:348-361, 1987.

....can be cascaded together to answer more complex queries, at the increase of a logarithmic factor in their performance. This property has been implicitly used for a long time, see e.g. 108, 173, 176, 256, 225] The real power of the cascading property was first observed by Dobkin and Edelsbrunner [100], who used this property to answer several complex geometric queries. Since their result, several papers have exploited and extended this property to solve numerous geometric searching problems; see [10, 139, 244, 183, 215] In this subsection we briefly sketch the general cascading scheme, as ....

D. P. Dobkin and H. Edelsbrunner, Space searching for intersecting objects, J. Algorithms, 8 (1987), 348--361.

....range queries, where h = O(1) This is unlike Section 2.1, where we used just one halfspace. Towards this end, we review a useful query composition result due to van Kreveld [vK92] which we will use often in Sections 2.3 2.5. This result is based on multi level range searching structures [DE87, Mat92a, CSW92]. Let S be a set of n geometric objects. Let D be a data structure for some query problem on S, with space and query time bounds O( f(n) and O(g(n) respectively. Suppose that we now wish to answer queries not on the entire set S but on a subset S 0 of S, where S 0 is specified by putting ....

D.P. Dobkin and H. Edelsbrunner. Space searching for intersecting objects. Journal of Algorithms, 8:348--361, 1987.

....report) the triangles intersected by any query ray or segment. One of the main building blocks for our collision free data structure is a solution for the dual problem: given a set of segments in 3 space count (or report) the segments intersected by any query triangle. Dobkin and Edelsbrunner in [DE84] solve the problem of counting the number of segments intersected by a query plane; but their approach does not seem to extend to query half planes. Our approach is based on reducing the problem to halfspace range searching and it attains the goal of a data structure for half plane queries. Once ....

D. Dobkin and H. Edelsbrunner. Space searching for intersecting objects. In 25th FOCS, pages 387--392, 1984.

....the family A of Cartesian products. Namely, for a finite P ae R d , we put C(P ) CANA k (CANA k Gamma1 ( CANA1 (P ) This uses the idea of multilevel partition trees, going back to Bentley [14] in the context of Cartesian products of intervals and to Dobkin and Edelsbrunner [17] for halfplanes. Given an A = A 1 Theta A 2 Theta Delta Delta Delta Theta A k 2 A, we express P A as a disjoint union of the sets of C(P ) as follows. First, we decompose 1 (P ) A 1 into sets of CANA 1 ( 1 (P ) as was described above (and interpret the resulting sets in the ....

D. P. Dobkin and H. Edelsbrunner. Space searching for intersecting objects. J. Algorithms, 8:348--361, 1987.

....the efficiency of the resulting data structure is roughly the same as for the worse one of the data structures we started with. A simple instance of multi level data structures is the range tree (see [29] An example more closely related to our application was given by Dobkin and Edelsbrunner [18]. Chazelle et al. 15] show how a multi level data structure can be used to extend a half space range searching data structure to answer simplex range queries. Other applications of multi level structures can be found in [25, 4] An abstract framework for multi level data structures has been ....

D. Dobkin and H. Edelsbrunner. Space searching for intersecting objects. Journal of Algorithms, 8 (1987), 348--361.

....together to answer more complex queries, at the increase of a logarithmic factor in their performance. This property has been implicitly used for a long time; see, for example, 112, 192, 196, 290, 252] The real power of the cascading property was first observed by Dobkin and Edelsbrunner [102], who used this property to answer several complex geometric queries. Since their result, several papers have exploited and extended this property to solve numerous geometric searching problems; see [10, 149, 274, 203, 238] In this subsection we briefly sketch the general cascading scheme, as ....

D. P. Dobkin and H. Edelsbrunner, Space searching for intersecting objects, J. Algorithms, 8 (1987), 348--361.

....last few years in all these areas. The problems of contact determination and interference detection have been extensively studied in different areas. The literature in computational geometry consists of a number of theoretically efficient algorithms for polyhedral objects in static environments [1, 2, 9, 10, 11, 12, 13, 17, 36]. There are a number of algorithms with good asymptotic bounds, however their practical utility is not clear since many of them are not implemented in a realistic environment. Many algorithms are known for intersection between curved objects represented as algebraic surfaces or piecewise spline ....

D. P. Dobkin and H. Edelsbrunner. Space searching for intersecting objects. J. Algorithms, 8:348--361, 1987.

....segment intersection searching problem. Other colored intersection searching problems have been studied independently by Janardan and Lopez [16] Gupta et al. 14] and Nievergelt and Widmayer [19] In the last few years much work has been done on the segment intersection searching problem [2, 3, 5, 7, 10, 13, 21]. Agarwal and Sharir [3] showed that S can be preprocessed using O(n 1 ffl ) space and time, so that all k segments of S intersecting a query segment can be reported in time O(n 1=2 ffl k) 1 (The n ffl factor in the size and query time can be reduced to log O(1) n factor using a more ....

Dobkin, D. P. and H. Edelsbrunner, Space Searching for Intersecting Objects, J. of Algorithms 8 (1987), pp. 348--361.

....other scene objects visible from A. A either is a scene object or is of the same type as the scene objects but does not intersecting them. In the solution of these problems we suggest, the visibility sets are represented as a union of stabber sets. The solution uses halfspace decomposition schemes [6, 1, 9]. A halfspace decomposition scheme is a data structure that consists of nested partitions. The nesting of partitions reflects a sequence of halfspace queries which yields the answer of the original problem. The partitions are built over finite sets of points. They are data structures into which ....

D. Dobkin, H. Edelsbrunner, Space searching for intersecting objects, J. Algorithms 8 (1987) 348--361

....range queries, where h = O(1) This is unlike Section 2.1, where we used just one halfspace. Towards this end, we review a useful query composition result due to van Kreveld [26] which we will use often in Sections 2.3 2.5. This result is based on multi level range searching structures [6, 7, 16]. 3 Let S be a set of n geometric objects. Let D be a data structure for some query problem on S, with space and query time bounds O(f(n) and O(g(n) respectively. Suppose that we now wish to answer queries not on the entire set S but on a subset S 0 of S, where S 0 is specified by ....

Dobkin, D.P. and Edelsbrunner, H. (1987). Space searching for intersecting objects. Journal of Algorithms, 8, 348--361.

....3.3 below, we will express intersection conditions as the conjunction of h 1 halfplane range queries, where h = O(1) Towards this end, we review a useful query composition result due to van Kreveld [vK92] which we will use. This result is based on multi level range searching structures [DE87, Mat92, CSW92]. Let S be a set of n geometric objects. Let D be a data structure for some query problem on S, with building time, space and query time bounds of p(n) f(n) and g(n) respectively. Suppose that we now wish to answer queries not on the entire set S but on a subset S 0 of S, where S 0 is ....

D.P. Dobkin and H. Edelsbrunner. Space searching for intersecting objects. Journal of Algorithms, 8:348--361, 1987.

....represent positions where a commodity is available. The region corresponding to a given site would then represent those points from which this is the site of choice for getting the commodity. Or, we might want to find for each site, the other site to which it is closest. The Delaunay triangulation [De] is also constructed from a set of sites. In the plane, we build a triangulation of a point set by connecting pairs of sites. This technique can be used to generate an exponential number of different triangulations. For many applications, some of these triangulations are preferable to others. For ....

....range searching extensions of k d trees. Recent results here apply deep results from probability theory (dealing with the Vapnik Chervonenkis dimension) to obtain theoretical improvements [EW] Other results give methods of extending range searching techniques to more sophisticated queries [DE]. 4. Conclusions and further results This paper has shown that the fields of computational geometry and computer graphics have had significant impact on one another. Indeed, a major frustration in writing this paper is that few of the exciting interactions can be explored. The two problems we ....

Dobkin, D. P. and Edelsbrunner, H., "Space searching for intersecting objects", J. Algorithms, vol. 8, 1987, pp. 348-361.

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D. Dobkin and H. Edelsbrunner. Space searching for intersecting objects. Journal of Algorithms, 8:348--361, 1987. 46

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D. P. Dobkin and H. Edelsbrunner. Space searching for intersecting objects. Journal of Algorithms, 8:348--361, 1987.

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