A. Efrat, M. J. Katz, F. Nielsen, and M. Sharir. Dynamic data structures for fat objects and their applications. Comput. Geom. Theory Appl., 15:215--227, 2000. A preliminary version appeared in WADS'97, LNCS 1272.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is the size of the piercing set found. This algorithm is based on the divide and conquer paradigm. Although not stated explicitly, the approximation on the radius of the k center problem in [1] implies a 4approximation algorithm for the minimum piercing set problem for and L 2 . Efrat et al. [10] introduced a dynamic data structure based on segment trees which can be used for the piercing set problem. They presented a sequential algorithm which gives a constant factor approximation for the minimum piercing set problem for fat objects with polynomial setup and update time. See [10] for ....

....et al. 10] introduced a dynamic data structure based on segment trees which can be used for the piercing set problem. They presented a sequential algorithm which gives a constant factor approximation for the minimum piercing set problem for fat objects with polynomial setup and update time. See [10] for the de nition of fatness and more details. All the results are for unit disks; Lp = L1 for any p in 1 dimension. A large number of clustering algorithms have been proposed and evaluated through simulations in the ad hoc network domain, as for example the work in [15, 27, 31, 28, 3, 4] ....

Alon Efrat, Matthew J. Katz, Franck Nielsen, and Micha Sharir. Dynamic data structures for fat objects and their applications. In Workshop on Algorithms and Data Structures, pages 297-306, 1997.

....the size of the piercing set found. This algorithm is based on the divide and conquer paradigm. Although not stated explicitly, the approximation on the radius of the k center problem in [1] implies a four approximation algorithm for the minimum piercing set problem for and L 2 . Efrat et al. [10] introduced a dynamic data structure based on segment trees which can be used for the piercing set problem. They presented a sequential algorithm which gives a constant factor approximation for the minimum piercing set problem for fat objects with polynomial setup and update time. See [10] for ....

....et al. 10] introduced a dynamic data structure based on segment trees which can be used for the piercing set problem. They presented a sequential algorithm which gives a constant factor approximation for the minimum piercing set problem for fat objects with polynomial setup and update time. See [10] for the de nition of fatness and more details. A large number of clustering algorithms have been proposed and evaluated through simulations in the ad hoc network domain, as for example in [3, 4, 15, 27, 28, 31] Gerla and Tsai in [15] considered two distributed clustering algorithms, the ....

Alon Efrat, Matthew J. Katz, Franck Nielsen, and Micha Sharir. Dynamic data structures for fat objects and their applications. In Proc. Workshop on Algorithms and Data Structures, pages 297-306, 1997.

....a PTAS for the piercing problem for fat objects of arbitrary size; it runs in n O(1= d ) time and requires linear space. To our knowledge, this is the rst PTAS for this case; previously, only a constant factor approximation result was known, with a simple algorithm described by Efrat et al. [11] (see Section 2) Our result thus extends Hochbaum and Maass by enlarging the class of objects with PTASs. 2 Preliminaries In the sequel, all boxes (including squares and hypercubes) are implicitly assumed to be axis aligned. 3 Given an object S, de ne the center and size of S to be the center ....

....and hypercubes) are implicitly assumed to be axis aligned. 3 Given an object S, de ne the center and size of S to be the center and side length of its smallest enclosing hypercube. There are a number of di erent de nitions of fatness in the geometry literature (e.g. see the many references in [6, 11]) The one that we nd the most appropriate for our purpose is the following: De nition. A collection C of objects is fat if for any r and size r box R, we can choose a constant number c of points such that every object that intersects R and has size at least r contains one of the chosen points. ....

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A. Efrat, M. J. Katz, F. Nielsen, and M. Sharir. Dynamic data structures for fat objects and their applications. Comput. Geom. Theory Appl., 15:215-227, 2000.

.... [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 for point intersection searching techniques ....

A. Efrat, M. Katz, F. Nielsen, and M. Sharir, Dynamic data structures for fat objects and their applications, Proc. 5th Workshop Algorithms Data Struct., 1997. To appear.

.... ) 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 intersection searching. Agarwal et al. 6] extended ....

A. Efrat, M. Katz, F. Nielsen, and M. Sharir, Dynamic data structures for fat objects and their applications, Proc. 5th Workshop Algorithms Data Struct., 1997. To appear.

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A. Efrat, M. J. Katz, F. Nielsen, and M. Sharir. Dynamic data structures for fat objects and their applications. In Proc. 5th Workshop Algorithms Data Struct., pages 297--306, 1997.

....ff fat objects in R 2 or in R 3 (or sets of (fi; ffi) covered objects in R 2 ) that find a piercing set whose size is larger than the optimal size by only a constant factor. The running time is close to O(n 4=3 ) in R 3 and close to linear in R 2 . A third application is described in [15]. Let C be a set of n convex ff fat objects in the plane. In the bounded length segment shooting problem, we wish to preprocess C, so that, for a given oriented query segment r = Gamma ab , whose length is at most some constant times the smallest diameter of an object in C, the first object ....

....efficiently. This is an object c for which there exists z 2 r such that z 2 c, and the relative interior of az does not meet any object of C) An efficient solution to this problem is presented in [17] for the special case where C consists of either (constant complexity) polygons or disks. In [15] we present a solution for the general (fat) case. The data structure we describe is based on the (static version of the) data structure for point enclosure (see remark just after Theorem 2.4) its size is nearly linear in n, and the query cost is polylogarithmic, as opposed to roughly O( p n) ....

A. Efrat, M.J. Katz, F. Nielsen, and M. Sharir, Dynamic data structures for fat objects and their applications, Tech. Report 99-06, Dept. Math & Computer Science, Ben-Gurion University, 1999.

.... Gamma is at most ff, where s is the smallest ball containing c and s Gamma is a largest ball that is contained in c. Often the input set in practical instances of geometric problems consists of fat objects. Fat objects have several desirable A preliminary version of this paper appeared as [14]. Work by M. Katz has been supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. Work by M. Sharir has been supported by NSF Grant CCR 97 32101, by a grant from the U.S. Israeli Binational Science Foundation, by the ESPRIT IV LTR project No. 21957 ....

A. Efrat, M.J. Katz, F. Nielsen, and M. Sharir, Dynamic data structures for fat objects and their applications, Proc. 5th Workshop on Algorithms and Data Structures, 1997, 297--306, LNCS 1272.

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A. Efrat, M.J. Katz, F. Nielsen and M. Sharir, Dynamic data structures for fat objects and their applications. Proc. 5rd Workshop on Algorithms and Data Structures, (1997), 297--396.

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A. Efrat, M.J. Katz, F. Nielsen, and M. Sharir. Dynamic data structures for fat objects and their applications. Comput. Geom. Theory Appls. 15 (2000), pp. 215-227.

....##(#C) is larger than # (otherwise parent(v) would have been a leaf) Let R be a square whose center coincides with the center of R v , and whose edge length is larger by some factor # # 1, than the edge length of R, where # # 1 depends on #. Using arguments similar to the ones used in [8], we can show that at least one of the following cases must occur: There is an object c # C such that R contains z, a rightmost, a leftmost, a highest or a lowest point of #c. R contains a vertex z of #(#C) There is an object c # C such that the area of R # c is at least a ....

Efrat, A., Katz, M. J., Nielsen, F., and Sharir, M. Dynamic data structures for fat objects and their applications. In Proceedings of the 5th Workshop on Algorithms and Data Structures (WADS), 1997, 297--396.

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Efrat, A., Katz, M. J., Nielsen, F., and Sharir, M. Dynamic data structures for fat objects and their applications. In Proceedings of the 5th Workshop on Algorithms and Data Structures (1997), pp. 297--396.

.... is only O(n log log n) Since then many authors considered various definitions of fatness (which are all more or less equivalent at least for convex objects) and obtained either interesting combinatorial results or efficient geometric algorithms for various classes of fat objects (see e.g. [2, 3, 5, 7, 8, 9, 11, 13, 14]) However, the question which properties suffice so that the number of vertices on the boundary of the union of a set of planar objects having these properties is always subquadratic remained open for quite a few years. Recently, Efrat and Sharir [6] showed that if the objects are convex, fat, ....

....n curved objects is presented. In a point location query, one must determine whether a query point q lies in the union of the input objects, and, if so, report a witness object containing q, or, alternatively, report all k objects containing q. The data structure is similar to those described in [5]; its size is O(n 1 ) the cost of an insertion or deletion of an object is O(n ) and the cost of a query is O(log 3 n) alternatively, O(log 3 n k) ....

A. Efrat, M.J. Katz, F. Nielsen and M. Sharir, Dynamic Data Structures for Fat Objects and their Applications, Proc. 5th Workshop Algorithms and Data Struct., 1997, 297--306.

....Remark 4.8: A convex object c in R d is ff fat, for some parameter ff 1, if there exists an axis parallel cube s containing c and an axis parallel cube s Gamma that is contained in c, such that the ratio between the edge lengths of s and s Gamma is at most ff. In a recent paper [22], we obtained results concerning matching points into fat shapes that contain them in two and three dimensions. These algorithms use the matching procedure of Section 3.1, but use different data structures D r (S) than those used in this paper. Some of the results obtained in [22] are listed ....

....In a recent paper [22] we obtained results concerning matching points into fat shapes that contain them in two and three dimensions. These algorithms use the matching procedure of Section 3.1, but use different data structures D r (S) than those used in this paper. Some of the results obtained in [22] are listed below, for the sake of completeness. Theorem 4.9 Let B be a set of n convex ff fat objects in R d (for d = 2; 3) and let A be a set of n points in R d . Then we can either find a one to one matching between A and B, such that each point p 2 A is contained in the object of B ....

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A. Efrat, M. Katz, F. Nielsen and M. Sharir, Dynamic data structures for fat objects and their applications, manuscript.

....are only 14 phases. This discussion and Remark 5.5 yield the following theorem: Theorem 5.10 Let A and B be sets of points in R 2 and L p the underlying norm for any 1 p 1. Then Match(A; B) can be found in time O(n 1:5 log n) 6 Additional Settings 6. 1 Points in 3 space In a recent paper [22], Efrat et al. obtained results concerning matching points into fat shapes that contain them in two and three dimensions. These algorithms use the matching procedure of Section 3.1, but use different data structures D r (S) than those used here. The following result will be useful: Theorem 6.1 ....

....al. obtained results concerning matching points into fat shapes that contain them in two and three dimensions. These algorithms use the matching procedure of Section 3.1, but use different data structures D r (S) than those used here. The following result will be useful: Theorem 6. 1 (Efrat et al. [22]) Let A be a set of n points in R 3 , and B a set of n balls in R 3 . Then in time O(n 11=6 ) we can either find a matching between A and B, such that each point a 2 A is contained in the object of B matched to a, or determine that no such matching exists. An immediate consequence of ....

A. Efrat, M.J. Katz, F. Nielsen, and M. Sharir, Dynamic Data Structures for Fat Objects and their Applications, Proceedings 5 Workshop on Algorithms and Data Structures, 1997 LNCS Vol. 1272, 297--306.

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A. Efrat, M. J. Katz, F. Nielsen, and M. Sharir. Dynamic data structures for fat objects and their applications. Comput. Geom. Theory Appl., 15:215--227, 2000. A preliminary version appeared in WADS'97, LNCS 1272.

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