B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. Algorithmica, 8:407--429, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....general, the solutions to e.g. simplicial range searching only provide low query time at the cost of a relatively high storage requirement and preprocessing time or vice versa. At the one end, one finds solutions providing polylogarithmic query time and roughly ) storage and preprocessing (see [9, 25]) whereas, at the other end of the spectrum, The result is a generalization of an earlier reported result [29] which is restricted to d = 2; 3. All quoted rough bounds are adequate up to a factor n ffl n, for some arbitrarily small ffl and some constant c. In all quoted time bounds ....

B. Chazelle, M. Sharir, and E. Welzl, Quasi-optimal upper bounds for simplex range searching and new zone theorems, Algorithmica 8 (1992), pp. 407-429.

....we have also shown that for any query half space h , we can determine within the same bounds whether P h is a terrain in some direction. Furthermore, by choosing a different partition tree for the primary tree, we can trade query time for storage space, see e.g. Chazelle, Sharir and Welzl [6], Matousek [17] and Agarwal and Sharir [2] For any n N n , a data structure of size and preprocessing time O(N ) exists with query time O(n =N 1=3 ) The theorem above states the version we need for the casting problem. Corollary 3 For any constant ffl 0 and any simple polyhedron ....

....on the planes bounding Psi 0 (f ) Since these planes all pass through the origin, a 2 dimensional partition tree is indeed sufficient. It allows one to select all planes below and all planes above a given query ray (normal to the query plane) starting at the origin in O(n ) canonical subsets [2, 6, 17]. For any canonical subset of a node ffi, the vertices in the closures of the facets f for which the half space Psi 0 (f) appears in that canonical subset are further preprocessed into a secondary data structure with ffi for half space emptyness queries, see for example Clarkson and Shor [7] The ....

Chazelle, B., M. Sharir, and E. Welzl, Quasi-optimal upper bounds for simplex range searching and new zone theorems. Algorithmica 8 (1992), pp. 407--429.

....range searching, but it can be extended to triangular range searching, as explained later. It turns out to be possible to construct structures whose performance lies between these two extremes: for any m with n m n 2 there is a data structure that uses O(m) storage and has O(nl v ) storage [48]. It is also possible to generalize the results to simplex range searching in higher dimensional spaces: it is possible to report the points inside a query simplex in R d in O(n x 1 d k) time with a data structure that uses O(n) storage [93] or to report the points in O(logn k) time with a ....

.... results to simplex range searching in higher dimensional spaces: it is possible to report the points inside a query simplex in R d in O(n x 1 d k) time with a data structure that uses O(n) storage [93] or to report the points in O(logn k) time with a data structure that uses O(n ) storage [48]. Trade offs are possible as well. Notice that the performance of the data structure deteriorates quickly when the dimension gets higher. Unfortunately this cannot be unavoided, as follows from the lower bounds proved by Chazelle [35] 3.2 Multi level data structures Multi level data structures ....

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B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. Algorithmlea, 8:407-429, 1992.

....time and space into a data structure of size O(n ) so that all k connected components intersecting any query segment can be reported in O(n k) time, for any fixed ffl 0. Remark 5. 4: Using the space query time tradeoff for simplex range searching and line intersection data structures [3, 6], one can obtain a space query time tradeoff for Theorem 5.3. In particular, for any n N n , one can preprocess S into a data structure of size O(N ) so that all k connected components intersected by a query segment can be reported in time O( k) The preprocessing time is O(n ) ....

....n edges can be preprocessed in time O(n ) into data structure of size O(n ) so that all k polygons intersecting a query segment can be reported in time O(n k) Remark 6. 3: i) As in the previous section, one can obtain a space query time tradeoff by using the standard techniques; see [3, 6]. ii) The above algorithm returns only those polygons whose boundaries intersect a query segment. If one also wants to report the segments whose interiors contain fl, we triangulate each polygon and preprocess the set of resulting triangles in a data structure of size O(n log n) for point ....

Chazelle, B., M. Sharir, and E. Welzl, Quasi-Optimal Upper Bounds for Simplex Range Searching and New Zone Theorems, Algorithmica 8 (1992), 407--430.

....results, and we also obtain new results for other classes of objects. 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 ....

....of branching degree O(r2) The root of this tree stores the subdivision . W) preprocessed for point location queries using e.g. Kirkpatricks method [18] Furthermore, for each cell c in S(W) the set of points 8 IT e W(c) is stored, preprocessed for half planar range counting as described in [9]. This half planar range counting structure uses 0( IW, c)12 log IW, c)l) preprocessing time and O( IW (c)12) space and it allows us to count the number of points in query line in O(logn) time. Finally, the O(r 2) children of the root correspond to recursively defined structures on the set Next ....

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B. Chazelle, M. Sharir and E. Welzl, Quasi-Optimal Upper Bounds for Simplex Range Searching and New Zone Theorems, Proc. 6th A CM Syrup. on Compu- tational Geometry, 1990, pp. 23-33.

....will use the following result of Matousek [28] Lemma 9. 6 (Matousek [28] We can preprocess a set of n points in IR , in time and space O(n n) so that the k nearest neighbors to a query hyperplane can be found in time O(k log n) We make use of a technique developed by Chazelle et al. [9] for answering simplex range queries. Given a data structure used to answer some arbitrary geometric query, they build on top of it another structure, called a partition tree, that limits the query to the points within an arbitrary halfspace. The resulting data structure can be built in time O(n ....

B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. In 6th ACM Symp. Comput. Geom., pages 23--33, 1990.

....have also shown that for any query half space h , we can determine within the same bounds whether P h is a terrain in some direction. Furthermore, by 25 choosing a different partition tree for the primary tree, we can trade query time for storage space, see e.g. Chazelle, Sharir and Welzl [8], Matousek [21] and Agarwal and Sharir [2] For any n N n , a data structure of size and preprocessing time O(N ) exists with query =N 1=3 ) The theorem above states the version we need for the casting problem. Remark: de Berg [11] noted that the result for orthogonal cast removal ....

....on the planes bounding Psi 0 (F ) Since these planes all pass through the origin, a 2 dimensional partition tree is indeed sufficient. It allows one to select all planes below and all planes above a given query ray (normal to the query plane) starting at the origin in O(n ) canonical subsets [2, 8, 21]. For any canonical subset of a node ffi , the vertices in the closures of the facets f for which the half space Psi 0 (f) appears in that canonical subset are further preprocessed into a secondary data structure with ffi for half space emptyness queries, see for example Clarkson and Shor [9] ....

Chazelle, B., M. Sharir, and E. Welzl, Quasi-optimal upper bounds for simplex range searching and new zone theorems. Algorithmica 8 (1992), pp. 407--429.

....we have also shown that for any query half space h , we can determine within the same bounds whether P h is a terrain in some direction. Furthermore, by choosing a different partition tree for the primary tree, we can trade query time for storage space, see e.g. Chazelle, Sharir and Welzl [7], Matousek [15] and Agarwal and Sharir [2] For any n N n , a data structure of size and preprocessing time O(N ) exists with query time O(n =N 1=3 ) The theorem above states the version we need for the casting problem. Corollary 3 For any constant ffl 0 and any simple polyhedron ....

Chazelle, B., M. Sharir, and E. Welzl, Quasi-optimal upper bounds for simplex range searching and new zone theorems. Proc. 6th ACM Symp. on Comp. Geometry (1990), pp. 23--33.

....mapped into a point and we locate this point in a special cell complex, we have the so called primal approach. Rayshooting problems can be cast also in a dual approach where the queries are mapped into hyperplanes, and we ask for the points in a point set that are above the query hyperplane (see [CSW90, AS91] Using the dual approach Agarwal and Sharir [AS91] obtain an algorithm to solve decision line shooting queries that uses O(m) storage and answers the queries in time O(n 16=15 ffl =m 4=15 ) for any ffl 0 and n 1 ffl m n 4 ffl . As a lemma for their space query trade off ....

....do not asymptotically modify the space or the time needed to construct the 2 level tree. Using the above data structure and using the batching technique it is possible to solve the batched ray shooting problem of n rays and n axisoriented boxes in time O(n 4=3 ffl ) With techniques in [CSW90] and [AS91] it is possible to trade off space and query time. Using O(m) storage for n 1 ffl m n 2 ffl , we obtain query time O(n 1 ffl = p m) 8 Querying isotopy classes In this section we adapt the techniques used in the Section 3 in order to solve the following problem: Given n ....

B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. In Proceedings of the 6th ACM Symposium on Computational Geometry, pages 23--33, 1990. 26

....bilinear forms depending on P and v. The same result is obtained by dualizing point and surfaces of Lemma 1, in this case the problem is transformed in a series of halfplane range searching on sets of points in 2 spaces. Such queries can be solved using techniques for half space range searching [CSW90, Mat92] in n 1 ffl m n 2 ffl storage and T = O(n 1 ffl =m 1=2 ) query time. When we apply the parametric search to superlogarithmic algorithms (as it is the case for the query time after the trade off) we use the more sophisticated form of parametric search that needs a parallel version ....

.... we apply the parametric search to superlogarithmic algorithms (as it is the case for the query time after the trade off) we use the more sophisticated form of parametric search that needs a parallel version of the halfplane range searching algorithm [Meg83] The data structures of Chazelle et al. CSW90] and Matousek [Mat92] are based on a partition tree approach and the query time depends on the number of nodes in the partition tree visited during the query. We can allocate dynamically processors to the nodes visited during the search, thus we need p = O(n 1 ffl =m 1=2 ) processors. The ....

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B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. In Proceedings of the 6th ACM Symposium on Computational Geometry, pages 23--33, 1990. 17

.... The technique by Matousek and Welzl has also been applied to solve some other geometric searching problems, including ray shooting and intersection searching [34] The first data structure with roughly n 1 Gamma1=d query time and near linear space, for d 3, was obtained by Chazelle et al. [79]. Given a set S of n points in R d , they construct a family F = f Xi 1 ; Xi k g of triangulations of R d , each of size O(r d ) For any hyperplane h, there is at least one Xi i so that only O(n=r) points lie in the simplices of Xi i that h intersects. Using this observation, ....

....the space is due to the fact that they maintain a family of partitions instead of a single partition. Another consequence of maintaining a family of partitions is that, unlike partition trees, this data structure does not directly extend to answering simplex range queries. Instead, Chazelle et al. [79] construct a multi level data structure (see Section 5.1) to answer simplex range queries. Matousek [183] developed a simpler, slightly faster data structure for simplex range queries, by returning to the theme of constructing a single partition, as in the earlier partition tree papers. His ....

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B. Chazelle, M. Sharir, and E. Welzl, Quasi-optimal upper bounds for simplex range searching and new zone theorems, Algorithmica, 8 (1992), 407--429.

.... preprocessing time and a query time which is of the form O(polylog(n) k polylog(n) except, of course, for axis parallel rectangles (using standard orthogonal range searching techniques [23] When removing the fatness assumption, the best known solutions using nearly linear storage (see e.g. [8, 20, 21]) have roughly O( p n k) query time. The generalized problem was studied in a series of papers by Janardan and Lopez [17] and by Gupta et al. 12, 13, 14] They present a solution for the case of disks [14] which requires O(n log 2 n) preprocessing time and O(n log n) storage, and the query ....

.... problem, it does not seem to help to assume that the underlying objects belong to some class of objects for which the combinatorial complexity of the union is small (e.g. fat triangles or disks) When removing the fatness assumption, the best known solutions using nearly linear storage (see e.g. [8, 20, 21, 33]) require roughly O( p n k) time. We present efficient solutions to the bounded size range searching problem when the objects of C are either (convex ff fat) polygons or disks, and the query ranges are either not necessarily convex and fat polygons or disks. Our solutions require nearly ....

B. Chazelle, M. Sharir and E. Welzl, Quasi-optimal upper bounds for simplex range searching and new zone theorems, Algorithmica 8 (1992), 407--429.

....as can be seen from the table, we obtain polylogarithmic query time at one extreme and linear or almost linear space at the other extreme. We remark that other intermediate trade offs are also easily derived from our results. Techniques for obtaining such intermediate trade offs can be found in [CSW92]. Next, we consider a generalization of the standard intersection searching problems involving curved objects: Here S consists of n linear or curved objects and the objects come aggregated in disjoint groups. If we assign each object a color, according to the group it belongs to, then our goal ....

....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 ....

B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. Algorithmica, 8:407--429, 1992.

....A problem intimately connected to the point location problem is the half space range searching problem. Given a set S of n points in R d , build a data structure such that, for every query half space h , the number of points in S h is computed efficiently. This problem is solved in [10, 28] using partition trees. In a partition tree, each node is associated with a region in R d such that only a fraction of the children intersect the hyperplane h supporting the query half space. During the query we retrieve the number of points of S within the regions completely contained in h ....

....structures are a basic paradigm in computational geometry [30] They are used to search for elements satisfying a complex property. Usually the complex property is split into elementary properties and each elementary property is tested at a specific level of the data structure. For example, in [10, 28] sets of points are organized in multilevel partition trees to answer simplex range queries, where each level of the data structure tests the position of the data points with respect to the hyperplane spanning a facet of the simplex. We have this fundamental theorem in [10] Theorem 1 (Theorem ....

[Article contains additional citation context not shown here]

B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. In Proceedings of the 6th ACM Symposium on Computational Geometry, pages 23--33, 1990.

.... these twoalgorithms, achieving a running time of O(n 1:412 ) Edelsbrunner et al. 15]developed a randomized algorithm with expected running time O(n 4=3 ) 1 Further research replaced the n term in this upper bound with a succession of smaller and smaller polylogarithmic factors [10, 14, 1, 8]. The fastest known algorithm, due to Matousek [22] runs in time n 4=3 2 O(log n) 2 Matousek s algorithm can be tuned to detect incidences among n points and m lines in the plane in time O(n log m n 2=3 m 2=3 2 O(log (n m) m log n) 5] or more generally among n points and m ....

B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. Algorithmica, 8:407--429, 1992.

....of constant complexity) of the robot are part of the query. We consider first a solution which, at the expense of large storage, answers the queries 2 All the bounds presented hold for every ffl 0 and the multiplicative constants depend on ffl. 2 in logarithmic time. Applying the methods in [CSW90, Mat91, AM92a] see also [AS91a] we obtain a space query time trade off. In [Pel93, AS91a] algorithms for the following problem are presented: given a set of triangles, count (or report) the triangles intersected by any query ray or segment. One of the main building blocks for our ....

....A problem intimately connected to the point location problem is the half space range searching problem. Given a set S of n points in R d , build a data structure such that, for every query half space h , the number of points in S h is computed efficiently. This problem is solved in [CSW90, Mat91] using partition trees. In a partition tree, each node is associated with a region in R d such that only a fraction of the children intersect the hyperplane h supporting the query half space. During the query we retrieve the number of points of S within the regions associated with ....

[Article contains additional citation context not shown here]

B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. In Proceedings of the 6th ACM Symposium on Computational Geometry, pages 23--33, 1990.

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B. Chazelle, M. Sharir and E. Welzl, Quasi-optimal upper bounds for simplex range searching and new zone theorems, Proc. 6th ACM Symp. on Computational Geometry, 1990, pp. 23-33. (Also to appear in Algorithmica.)

....can be determined in O(log jS v j) time. Next, we construct a two level data structure on S v . For each segment e 2 S v , we mark one of its endpoints; let A v be the set of these points. We preprocess A v into a halfplane range searching data structure, using the algorithm by Chazelle et al. [16]. Their algorithm chooses a parameter r (greater than a constant speci ed by their algorithm) and constructs a family of canonical subsets of A v so that there are O( jS v j=r ) canonical subsets of size between r j 1 , for any integer 1 j log r n; here 0 is an arbitrarily small ....

....n) time whether P (Z 0 ) intersects any segment of v2V 2 S v . If it does not, we also obtain Z 2 = Z 0 ; v2V 2 S v ) Now (Z 0 ; E) is the lowest of Z 1 and Z 2 . The query time can be improved to O(log mn) by constructing the segment tree with a larger fan out, e.g. as described in [16], without increasing the asymptotic size and preprocessing time. Omitting the technical details of this improvement, we summarize the analysis in the following theorem: Theorem 3.3 Given a parameter 0, a convex polygon P with m edges, and a polygonal environment Q with a total of n edges, we ....

B. Chazelle, M. Sharir, and E. Welzl, Quasi-optimal upper bounds for simplex range searching and new zone theorems, Algorithmica 8 (1992), 407-429.

....fi j Gamma1 ) and the center line of this strip lies to the right (resp. to the left) of p 1 p 2 . Let M denote the set of these n(n Gamma 1) strips. Perform n(n Gamma 1) batched range searching queries on the set P with these strips. Using standard techniques, such as those of [31] 32] or [13], we can perform these queries so that the output consists of a collection of pairs (M t ; P t ) where, for each t, P t is a subset of P , M t is a subset of M , for each p 2 P t and each A 2 M t we have p 2 A, and each such containment is obtained in exactly one pair M t Theta P t . Moreover, ....

B. Chazelle, M. Sharir and E. Welzl, Quasi-optimal upper bounds for simplex range searching and new zone theorems, Algorithmica 8 (1992), 407--429.

....Space query time tradeoff. If the bound (n) is rather large, say, O(n 2 ) we can obtain a space query time tradeoff by combining Theorem 6. 1 with the linear size data structure of [5] for nearest neighbor searching, mentioned at the beginning of this section, in a rather standard manner [22]. For example, if (n) O(n 2 ) then, for any parameter n m n 2 , we can store S into a data structure of size O(m 1 ) so that a nearest neighbor query can be answered in time O(n 1 = p m) Insertions and deletions can be performed in O(m 1 =n) time, per update. Hence, we ....

B. Chazelle, M. Sharir, and E. Welzl, Quasi-optimal upper bounds for simplex range searching and new zone theorems, Algorithmica 8 (1992), 407--430.

.... corresponding to the marked segments of Gamma , each oriented so that its xy projection is rightward directed; note that jGj X i t i s 1 X i t i s 2 ( X i t i ) 2 3 : We will construct on G a data structure based on a partitioning scheme due to Chazelle et al. [18]. For a segment fl 2 R 3 , this structure decomposes G further into canonical subsets, so that, for each canonical subset of G, either all the corresponding original polyhedra edges lie above fl or all of them lie below fl. Next, for each canonical subset Q in the output, we determine whether ....

....the second data structure Psi 2 (P) which, given a query segment fl, determines whether there is a face f of some polyhedron P i such that fl intersects f and that f contains one of the endpoints of fl . The data structure is again based on the partitioning scheme due to Chazelle et al. [18]. As in the preceding subsection, we consider here only 8 Segment Intersection Detection 9 the top portions of the given polyhedra. Choosing a sufficiently large constant parameter r, we partition the plane, in O(n) time, into a collection Delta of O(r 2 ) triangles, so that each triangle ....

B. Chazelle, M. Sharir and E. Welzl, Quasi-optimal upper bounds for simplex range searching and new zone theorems, Algorithmica 8 (1992), 407--430.

....Space query time tradeoff. If the bound (n) is rather large, say, O(n 2 ) we can obtain a space query time tradeoff by combining Theorem 6. 1 with the linear size data structure of [5] for nearest neighbor searching, mentioned at the beginning of this section, in a rather standard manner [22]. For example, if (n) O(n 2 ) then, for any parameter n m n 2 , we can store S into a data structure of size O(m 1 ) so that a nearest neighbor query can be answered in time O(n 1 = p m) Insertions and deletions can be performed in O(m 1 =n) time, per update. Hence, we ....

B. Chazelle, M. Sharir, and E. Welzl, Quasi-optimal upper bounds for simplex range searching and new zone theorems, Algorithmica 8 (1992), 407--430.

No context found.

B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. Algorithmica, 8:407--429, 1992.

No context found.

B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. Algorithmica, 8:407-429, 1992.

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