B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing kth- order Voronoi diagrams. IEEE Trans. Comput., C-36:1349--1354, 1987. 79  Home/Search   Document Not in Database   Summary   Related Articles   Check

....arises when the planes are the images of a set of points in the plane under the transformation that maps the order k Voronoi diagram of these points to the k level of the planes. Most known algorithms for computing the k level in three dimensional space actually compute the k level [Mul91b, CE87, BDT93] Since the complexity of the k level is Theta(nk ) in the situation sketched above, the running time of these algorithms is at least Omega Gamma n log n) The randomized incremental algorithm by Aurenhammer and Schwarzkopf [AS92] maintains only the k level, but it can be shown ....

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing kth- order Voronoi diagrams. IEEE Trans. Comput., C-36:1349--1354, 1987.

....n) For smaller values of k,theO(n log n k n) time algorithm of Aggarwal et al. is faster. For larger values of k, Agarwal and Matousek s Voronoi diagram algorithm is faster. Finally,fork= Omega 1; the fastest algorithm is based on another Voronoi algorithm of Chazelle and Edelsbrunner [6] and runs in time O(n n) Mulmuley describes an algorithm that constructs the kth order Voronoi diagram 2 e b 2 c log n k ) 30] To find minimum variance sets in higher dimensions, we use Mulmuley s algorithm as a subroutine within eachneighbor set. We improve the previous ....

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing k th -order Voronoi diagrams. IEEE Trans. Comput., C-36:1349--1354, 1987.

.... a set S of n point sites has O(k(n Gamma k) vertices, edges, and faces, and can be obtained from the order k Gamma 1 implicit Voronoi diagram V k Gamma1 (S) by intersecting each face of V k Gamma1 (S) with the (order 1) Voronoi diagram of a suitable subset of the vertices of S [42] As shown in [42, 15], V k (S) can be computed in O(k(n Gamma k) p n log n) time. Since the construction is based on iteratively computing Voronoi diagrams by using the incircle test, which is the most expensive operation in terms of degree, the overall degree of the preprocessing is 4 (Lemma 5) Hence, we obtain. ....

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing kth-order Voronoi diagrams. IEEE Trans. Comput., C-36:1349--1354, 1987.

.... IR 2 can be constructed in expected time O(n log n nk 1=3 log 2=3 k) 10 year time bound references 1982 O(nk 2 log n) Lee [46] 1986 O(n 3 ) Edelsbrunner, O Rourke, and Seidel [39] 1986 O(nk p n log n) Edelsbrunner [36] 1987 O(n 2 nk log 2 n) Chazelle and Edelsbrunner [26] 1987 O(n 1 k) rand. Clarkson [30] 1989 O(n log n nk 2 ) Aggarwal, Guibas, Saxe, and Shor [9] 1991 O(n log n nk 2 ) rand. Mulmuley [52] 1992 O(nk log 2 n nk 2 ) rand. online Aurenhammer and Schwarzkopf [14] 1993 O(n log n nk 3 ) rand. online Boissonnat, Devillers, and ....

....in time O(n bd=2c k dd=2e (log n= log k) O(1) 13 A similar approach works for k levels and order k Voronoi diagrams. We mention that for the latter problem in two dimensions, the best deterministic result can be achieved with Chazelle and Edelsbrunner s O(n 2 log 2 n) bound [26] for the base cases: Theorem 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 ....

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing kth-order Voronoi diagrams. IEEE Trans. Comput., C-36:1349--1354, 1987.

....case. We give an algorithm that guarantees O(n log n nk 1=3 ) expected time. 1 Introduction The notion of k levels [2, 20, 26, 34] has proved to be an important one in computational geometry, exploited directly or indirectly in various algorithms for computing higher order Voronoi diagrams [1, 11, 14] (used, for instance, in finding clusters of points [3, 23] designing data structures for halfspace range searching [16, 17] and solving hyperplane partitioning problems (such as hamsandwich cuts [32] and weak line separators [25] The concept is also fundamental in the combinatorics of ....

....vertices in the intersection of two consecutive levels; the algorithms in Sections 2 and 3 are actually sensitive to this latter parameter. Higher order Voronoi diagrams. For an application of our deterministic output sensitive algorithm, we improve an early algorithm of Chazelle and Edelsbrunner [14] for higher order Voronoi diagrams [2, 20, 34] which originally ran in O(n 2 log 2 n) worst case time. These diagrams are used to solve a number of problems on a planar point set, such as finding a k point subset of minimal variance [3, 23] and finding the smallest circle enclosing all but k ....

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing kth-order Voronoi diagrams. IEEE Trans. Comput., C-36:1349--1354, 1987.

.... n) Thus the order k Voronoi diagram (in fact, the family of all order j Voronoi diagrams for 1 j k the order k Voronoi diagrams for short) can be constructed in time O(k 2 n log n) This bound can be tightened to O(n log n k 2 n) using the result of [5] Chazelle and Edelsbrunner [6] developed two versions of an algorithm that is better for large values of k. The first one takes O(n 2 log n k(n Gammak) log 2 n) time and O(k(n Gammak) storage while the other takes O(n 2 k(n Gamma k) log 2 n) time and O(n 2 ) storage. A radically different approach, pioneered by ....

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing k th - order Voronoi diagrams. In First ACM Symposium on Computational Geometry in Baltimore, pages 228--234, June 1985.

.... in IR 2 can be constructed in expected time O(n log n nk 1=3 log 2=3 k) year time bound references 1982 O(nk 2 log n) Lee [41] 1986 O(n 3 ) Edelsbrunner, O Rourke, and Seidel [35] 1986 O(nk p n log n) Edelsbrunner [33] 1987 O(n 2 nk log 2 n) Chazelle and Edelsbrunner [23] 1987 O(n 1 k) rand. Clarkson [27] 1989 O(n log n nk 2 ) Aggarwal, Guibas, Saxe, and Shor [9] 1991 O(n log n nk 2 ) rand. Mulmuley [47] 1992 O(nk log 2 n nk 2 ) rand. online Aurenhammer and Schwarzkopf [14] 1993 O(n log n nk 3 ) rand. online Boissonnat, Devillers, ....

....in time O( n log n n bd=2c k dd=2e ) log n= log k) O(1) A similar approach works for k levels and order k Voronoi diagrams. We mention that for the latter problem in two dimensions, the best deterministic result can be achieved with Chazelle and Edelsbrunner s O(n 2 log 2 n) bound [23] for the base cases: Theorem A.3 The order k 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 ....

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing kth-order Voronoi diagrams. IEEE Trans. Comput., C-36:1349--1354, 1987.

.... unsolved and has been regarded by some as one of the foremost open problems in the area [13, 25] Besides its natural appeal, such a dynamic data structure has a wide range of applications, as it is often used as subroutines for tackling more difficult geometric problems (see the references [5, 11, 12, 15, 16, 19, 20, 21, 23, 24, 26, 27, 34] for a mere sampling) Among the earliest proposed methods for dynamic hull maintenance in the plane was one by Overmars and van Leeuwen [31] and dated back to 1981. The worst case update time is O(log 2 n) where n is the maximum number of points. Since this data structure actually stores the ....

....it is unclear whether the space complexity of either randomized algorithm can be made O(n) ffl The order k Voronoi diagram of n points in the plane is another well known structure in computational geometry, with various applications of its own. An early algorithm of Chazelle and Edelsbrunner [12] used also dynamic convex hulls as subroutines. Our result improves their worst case time bound from O(n 2 log 2 n) to O(n 2 log 1 n) For smaller values of k, faster methods are known, as the size of the diagram is O(nk) Indeed, by using some complicated machinery (namely, shallow ....

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing kth-order Voronoi diagrams. IEEE Trans. Comput., C-36:1349--1354, 1987.

....arises when the planes are the images of a set of points in the plane under the transformation that maps the order k Voronoi diagram of these points to the k level of the planes. Most known algorithms for computing the k level in three dimensional space actually compute the k level [Mul91b, CE87, BDT93] Since the complexity of the k level is Theta(nk 2 ) in the situation sketched above, the running time of these algorithms is at least Omega Gamma nk 2 n log n) The randomized incremental algorithm by Aurenhammer and Schwarzkopf [AS92] maintains only the k level, but it can be ....

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing kth- order Voronoi diagrams. IEEE Trans. Comput., C-36:1349--1354, 1987.

....of k, the O(n log n k 2 n) time algorithm of Aggarwal et al. is faster. For larger values of k, Agarwal and Matousek s Voronoi diagram algorithm is faster. Finally, for k = Omega Gamma n 1 Gamma ) the fastest algorithm is based on another Voronoi algorithm of Chazelle and Edelsbrunner [6] and runs in time O(n 2 log 2 n) Mulmuley describes an algorithm that constructs the kth order Voronoi diagram of a set of n points in IR d , in time O(k d d 1 2 e n b d 1 2 c log n k d n 2 ) 30] To find minimum variance sets in higher dimensions, we use Mulmuley s ....

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing k th -order Voronoi diagrams. IEEE Trans. Comput., C-36:1349--1354, 1987.

No context found.

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing kth- order Voronoi diagrams. IEEE Trans. Comput., C-36:1349--1354, 1987. 79

No context found.

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing kth-order Voronoi diagrams. IEEE Trans. Comput., C-36:1349-1354, 1987.

No context found.

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing k th -order Voronoi diagrams. In Proc. First Symp. on Comp. Geometry, pages 228--234, 1985.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC