Pankaj K. Agarwal, Alon Efrat, and Micha Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM Journal on Computing, 29(3):912--953, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....distance calculations among nodes. Mobile Base Stations occupy the centers of gn avity of the clusters and are moved according to a fast practical bipartite matching algorithm which tries to minimize both total and maximum distance. We show that the best known computational geometry algorithms [1] become unfeasible for our application when an high number of mobile base stations is required. On the other hand our proposed 8 average error solution requires approximatively O(klogk) running time instead of the approximatively O(k 2) exact algorithm [1] Other efficient clustering ....

....computational geometry algorithms [1] become unfeasible for our application when an high number of mobile base stations is required. On the other hand our proposed 8 average error solution requires approximatively O(klogk) running time instead of the approximatively O(k 2) exact algorithm [1]. Other efficient clustering algorithms [10, 15] may be used instead of the Antipole Tree. However the nice hierarchical structure of the Antipole Tree and the fact that it can be constructed in a distributed fashion makes it applicable to other types of mobile wireless (Ad Hoc) and wired ....

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P. K. Agarwal, A. Efrat, and M. Sharir. Vertical Decomposition of Shallow Levels in 3- Dimensional Arrangements and its Applications. SIAM Journal on Computing, 1999.

....distance calculations among nodes. Mobile Base Stations occupy an approximate centtold of the clusters and are moved according to a fast practical bipartite matching algorithm which tries to minimize both total and maximum distance. We show that the best known computational geometry algorithms [1] become infeasible for our application when a high number of mobile base stations is required. On the other hand our proposed 8 average en or solution requires O(k log k) running time instead of the approximatively O(k 2) exact algorithm [1] Communication among nodes is realized by a ....

....the best known computational geometry algorithms [1] become infeasible for our application when a high number of mobile base stations is required. On the other hand our proposed 8 average en or solution requires O(k log k) running time instead of the approximatively O(k 2) exact algorithm [1]. Communication among nodes is realized by a Clusterhead Gateway Switching Routing (CGSR) protocol [15] where the mobile base stations are organized in a suitable network. Other efficient clustering algorithms [11, 17] may be used instead of the Antipole Tree. However the nice hierarchical ....

[Article contains additional citation context not shown here]

P. K. Agarwal, A. Efrat, and M. Sharir. Vertical Decomposition of Shallow Levels in 3- Dimensional Arrangements and its Applications. SIAM Journal on Computing, 1999.

....A.3 The order k Voronoi diagram of n point sites in IR 2 can be constructed deterministically in time O(nk log 2 k (log n= log k) O(1) Remarks : 1. The use of the shallow cutting lemma to construct levels deterministically has been noted before in a paper by Agarwal, Efrat, and Sharir [3]; however, our deterministic bounds appear new. 2. Theorem A.2 is worst case optimal if k = Omega Gamma n ) for some constant 0. For small k, optimal derandomization for arbitrary dimensions appears difficult, as can be seen from Chazelle s work on convex hulls [24] Update. The author ....

P. K. Agarwal, A. Efrat, and M. Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. In Proc. 11th ACM Sympos. Comput. Geom., pages 39--50, 1995.

....a query pattern A in a larger target pattern B and have to deal with noise points. 4. 2 Minimum weight matching The minimum total distance (weight) is the minimum over all 1 1 correspondences f between A and B of the sum of the distances d(a; f(a) It can be computed in O(n 2 ) time [AES95] Here, the constant stands for a positive constant which can be chosen arbitrarily small with an appropriate choice of other constants of the algorithm. For the L1 distance, it can be computed in time O(n 2 log 3 n) Vai89] 4.3 Uniform matching The most uniform distance is the minimum ....

Pankaj K. Agarwal, Alon Efrat, and Micha Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. In Proceedings of the 11th Annual ACM Symposium on Computational Geometry, pages 39-50, 1995.

....Voronoi diagram of n point sites in IR 2 can be constructed deterministically in time O( n log n nk log 2 k) log n= log k) O(1) Remarks: 1. The use of the shallow cutting lemma to construct levels deterministically has been noted before in a paper by Agarwal, Efrat, and Sharir [3]; however, our deterministic bounds appear new. 2. Theorem A.2 is worst case optimal if k = Omega Gamma n ) for some constant 0. For small k, optimal derandomization for arbitrary dimensions appears difficult, as can be seen from Chazelle s work on convex hulls [21] A.3 A deterministic ....

P. K. Agarwal, A. Efrat, and M. Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. In Proc. 11th ACM Sympos. Comput. Geom., pages 39--50, 1995.

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P. K. Agarwal, A. Efrat and M. Sharir, Vertical decomposition of shallow levels in 3dimensional arrangements and its applications, SIAM J. Comput. 29 (2000), 912--953.

....Linear optimization queries can be used to answer many other queries. For example, using Matousek s technique and a dynamic data structure for halfspace range searching, the 1 center of a set S of points in R can be maintained dynamically, as points are inserted into or deleted from S. See [7, 11, 201] for additional applications of multidimensional parametric searching for query type problems. 11.3 Extremal placement queries Let S be a set of n points in R . We wish to preprocess S into a data structure so that queries of the following form can be answered efficiently: Let Delta(t) for ....

P. K. Agarwal, A. Efrat, and M. Sharir, Vertical decomposition of shallow levels in 3dimensional arrangements and its applications, Proc. 11th Annu. ACM Sympos. Comput. Geom., 1995, pp. 39--50.

....simply connected cells of dimensions ## ## ## #, each having constant description complexity, so that the size of the conflict list of each cell with respect to # # is at most ###. Since each # # is a two dimensional algebraic set of constant description complexity, it follows from the results in [2, 3] that there exists a ##### cutting # of size ### # # ####,where# is 2 plus the maximum number # ## # ## # ## # ## # #, over all quadruples of curves # # ## # ## # ## # in #,of vertical lines # that pass through both intersection curves # ## # ## in .More precisely, # ## # ## ....

P. K. Agarwal, A. Efrat and M. Sharir, Vertical decomposition of shallow levels in 3dimensional arrangements and its applications, SIAM J. Comput. 29 (2000), 912--953.

....we mention a few concrete examples) For the first two of these problems, there exist algorithms for the geometric versions that are more efficient than the corresponding more general algorithms. The best algorithm for computing a minimum weight Euclidean matching was given by Agarwal et al. [3]; it runs in time O(n 2 ) as opposed to the O(n 3 ) time algorithm that is obtained from the standard Hungarian method [17, 18] The algorithm of Agarwal et al. is based on a previous O(n 2:5 log n) time algorithm of Vaidya [23] Recently Efrat and Itai [7] have proposed an O(n 1:5 log ....

....graph was Fair and Bottleneck Matchings May 15, 1996 Computing a Most Uniform Matching 2 considered by Gupta and Punnen [11] who gave an O(n 4 ) time solution. For this problem we present an O(n 10=3 ) time solution 1 in the geometric setting. This solution is based on the algorithm of [3] for computing the minimum weight Euclidean matching. We also study the (non bipartite version of the) Euclidean bottleneck matching problem in higher dimensions, that is, given a set A of 2n points in d space, compute a bottleneck matching in G, the (complete) Euclidean graph over A. The weight ....

[Article contains additional citation context not shown here]

P.K. Agarwal, A. Efrat and M. Sharir, Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications, Proceedings 11 Annual Symposium on Computational Geometry, 1995, 39--50.

....important cases of these problems that admit more efficient solutions that exploit Geometry. In the Euclidean versions, the set of vertices V is a set of points in R d , and G is the complete graph over V . The weight associated with an edge (a; b) is the Euclidean distance between a and b. See [2, 5, 6, 14, 16] for a sample of results concerning Euclidean matching. In this paper we consider the Euclidean version of the bottleneck matching problem. Let A be a set of 2n points in R d , and let G be the complete graph over A. The weight of an edge (a; b) is simply the Euclidean distance between a and b, ....

P.K. Agarwal, A. Efrat and M. Sharir, Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications, Proc. 11th ACM Symp. Comput. Geom., 1995, 39--50.

....E of these functions. Indeed, a query point q lies in the union of C if and only if r q E ( q ; OE q ) where (r q ; q ; OE q ) are the spherical coordinates of q about p . The maintenance of this envelope can be accomplished using the shallow levels data structure of Agarwal et al. [2]. This structure has size O( n ) 2 ) and can be constructed in time O( n ) 2 ) where n = jC j. Using this structure, we can determine whether r q E ( q ; OE q ) in O(log n ) O(log n) time, or report all k objects of C that contain q in time O(log n k) An insertion or ....

....at is at least ae. The complexity of the upper envelope of F is O( s 1 (jC fl v j) where s 1 is an appropriate constant, and s (n) is the maximum length of (n; s) Davenport Schinzel sequences [24] We construct Psi (3) v;fl; which is the shallow level data structure of Agarwal et al. [2] to maintain the functions of F . Let Psi (2) v;fl be the list of roots of the structures Psi (3) v;fl; 2 P fl . We construct a balanced search tree Psi (1) v over the cells in Gamma oe v that are Dynamic Data Structures October 5, 1999 Dynamic Data Structures for Fat Objects 10 c ....

P.K. Agarwal, A. Efrat and M. Sharir, Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications, Proc. 11th ACM Symp. Comput. Geom., 1995, 39--50.

.... which there exists a copy e 0 of e and a subset S 0 S of cardinality k, such that S 0 H(e 0 ; d) This problem is a natural extension of the well studied problem of computing the smallest disc enclosing k points of S; see [10, 12, 14, 21] A recent attack on this problem is given in [2], See also [13] ffl The segment center problem is actually a special case of the more general problem of computing the one directional Hausdorff distance, under euclidean motion, between two sets of objects, which can be stated as follows (see [16] Given two sets, S 1 ; S 2 , of objects in ....

P. Agarwal, A. Efrat and M. Sharir, Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications, Proc. 11th ACM Symp. on Computational Geometry (1995), 39--50.

....It presents an O(n log n) time algorithm in the special case where l(p, q) pq 2 r( pq ) for some monotone increasing function r. Regarding the general case, Section 2 presents an O(n 4 3 # ) time 1 algorithm that uses a (relatively complicated) dynamic data structure of Agarwal et al. [AES95]. A more practical O(n 3 2 # ) time algorithm is also described (assuming monotonicity) that uses much simpler data structures. It turns out that the running time of both algorithms is related to the combinatorial complexity of a geometric construct that we will call the cheapest path map: ....

....of the bivariate functions g i (p) l(p, q i ) q i # T , and data structure (ii) can be obtained by considering the lower envelope of the bivariate functions f i (q) d[p i ] l(p i , q) p i # S . Both dynamic data structures can be devised from the work of Agarwal et al. [AES95], which, with the appropriate query update time tradeo#, achieves T (n) O(n 1 3 # ) Thus, each iteration of Dijkstra s algorithm is doable in amortized time O(n 1 3 # ) and the overall time for the algorithm is O(n 4 3 # ) Theorem 2.3 For any cost function l of constant descriptive ....

[Article contains additional citation context not shown here]

P. K. Agarwal, A. Efrat, and M. Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 39--50, 1995.

....Linear optimization queries can be used to answer many other queries. For example, using Matousek s technique and a dynamic data structure for halfspace range searching, the 1 center of a set S of points in R d can be maintained dynamically, as points are inserted into or deleted from S. See [7, 11, 172] for additional applications of multidimensional parametric searching for query type problems. 1 2 t Delta(t) Figure 7: An extremal placement query 11.3 Extremal placement queries Let S be a set of n points in R d . We wish to preprocess S into a data structure so that queries of the ....

P. K. Agarwal, A. Efrat, and M. Sharir, Vertical decomposition of shallow levels in 3dimensional arrangements and its applications, Proc. 11th Annu. ACM Sympos. Comput. Geom., 1995, pp. 39--50.

....By constructing a multi level data structure, point intersection queries for S can be answered in time O(log n) using O(n ) space, or in time O(n 1 Gamma1=fl ) using O(n) space. Table 7 gives some of the specific bounds that can be attained using this general scheme. Agarwal et al. [6] extended the approach for dynamic halfspace range searching to answer point intersection queries amid the graphs of bivariate algebraic functions, each of bounded degree. Let F be a collection of bivariate polynomials, each of bounded degree, and let (m) denote the maximum size of the lower ....

.... Source Disks Counting m (n 4=3 =m 2=3 ) log(m=n) 8] Disks Reporting n log n log n k [17] d = 2 Triangles Counting m n p m log 3 n [10] Fat triangles Reporting n log 2 n log 3 n k [163] Tarski cells Counting n 2 log n [68] d = 3 Functions Reporting n 1 log n k [6] Fat tetrahedra Reporting m n 1 p m k [111] Simplices Counting m n m 1=d log d 1 n d 3 Balls Counting n d log n [8] Balls Reporting m n m 1=dd=2e polylog n k [179] Tarski cells Counting n 2d Gamma3 log n [68] n fl log n [8] Table 7. Asymptotic upper bounds ....

P. K. Agarwal, A. Efrat, and M. Sharir, Vertical decomposition of shallow levels in 3dimensional arrangements and its applications, Proc. 11th Annu. ACM Sympos. Comput. Geom., 1995, pp. 39--50.

....Matousek presented in [81] efficient algorithms for linear optimization queries, where we wish to preprocess a set H of halfspaces in R d into a linear size data structure, so that, given a query linear objective function c, we can efficiently compute the vertex of T H that minimizes c. See [3, 7, 81] for additional applications of multi dimensional parametric searching for query type problems. 6 Abstract Linear Programming In this section we present an abstract framework that captures both linear programming and many other geometric optimization problems, including computing smallest ....

P. Agarwal, A. Efrat, and M. Sharir, Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications, Proc. 11th ACM Symp. Comput. Geom., 1995, 39--50.

.... Time Source Disks Counting m (n 4=3 =m 2=3 ) log(m=n) 8] Disks Reporting n log n log n k [16] d = 2 Triangles Counting m n p m log 3 n [10] Fat triangles Reporting n log 2 n log 3 n k [182] Tarski cells Counting n 2 log n [71] d = 3 Functions Reporting n 1 log n k [6] Fat tetrahedra Reporting m n 1 p m k [115] Simplices Counting m n m 1=d log d 1 n d 3 Balls Counting n d log n [8] Balls Reporting m n m 1=dd=2e polylog n k [199] Tarski cells Counting n 2d Gamma3 log n [71] n fl log n [8] Table 7. Asymptotic upper bounds for point ....

....m k [115] Simplices Counting m n m 1=d log d 1 n d 3 Balls Counting n d log n [8] Balls Reporting m n m 1=dd=2e polylog n k [199] Tarski cells Counting n 2d Gamma3 log n [71] n fl log n [8] Table 7. Asymptotic upper bounds for point intersection searching. Agarwal et al. [6] extended the approach for dynamic halfspace range searching to answer point intersection queries amid the graphs of bivariate algebraic functions, each of bounded degree. Let F be an infinite family of bivariate polynomials, each of bounded degree, and let (m) denote the maximum size of the lower ....

P. K. Agarwal, A. Efrat, and M. Sharir, Vertical decomposition of shallow levels in 3dimensional arrangements and its applications, Proc. 11th Annu. ACM Sympos. Comput. Geom., 1995, pp. 39--50.

.... Objects S(n) Q(n) Source Notes Disks m (n 4=3 =m 2=3 ) log(m=n) Counting Disks n log n log n k [14] Reporting d = 2 Triangles m n p m log 3 n [8] Counting Fat triangles n log 2 n log 3 n k [73] Reporting Tarski cells n 2 log n [37] Counting d = 3 Functions n 1 log n k [4] Reporting Simplices m n m 1=d log d 1 n Counting d 3 Balls n d log n [6] Counting Balls m n m 1=dd=2e log c n k [79] Reporting Tarski cells n 2d Gamma3 log n [37] Counting Point location in arrangement of surfaces, especially determining whether a query point lies above a given ....

P. K. Agarwal, A. Efrat, and M. Sharir, Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications, Proc. 11th Annu. ACM Sympos. Comput. Geom., 1995, pp. 39--50.

No context found.

Pankaj K. Agarwal, Alon Efrat, and Micha Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM Journal on Computing, 29(3):912--953, 1995.

No context found.

P. K. Agarwal, A. Efrat, and M. Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput., 29:912-- 953, 1999.

No context found.

Pankaj K. Agarwal, Alon Efrat, and Micha Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM Journal on Computing, 29(3):912--953, 1995.

No context found.

P. K. Agarwal, A. Efrat, and M. Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput., 29:912-953, 1999.

No context found.

P. K. Agarwal, A. Efrat, and M. Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM Journal on Computing, 29(3):912--953, 1995.

No context found.

P. K. Agarwal, A. Efrat, and M. Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput., 29:912-953, 1999.

No context found.

P. K. Agarwal, A. Efrat, and M. Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput., 29:912--953, 1999.

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