B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT-31:509-517, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....until S becomes empty. The iteration index in which a point is a maximum is de ned to be its layer. More formally, for p 2 S, layer(p) 1 if p is a maximum point; otherwise, layer(p) 1 maxflayer(q) q dominates pg. The layers of maxima problem, which is related to the convex layers problem [4], is to determine layer(p) for each p 2 S, given S. With some e ort [1] the three dimensional layers of maxima problem can be solved in time O(n log n log log n) using techniques from dynamic fractional cascading [13] We sketch this result in Section 2. In addition, Atallah, Goodrich, and ....

B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inf. Thy., IT31: 509-17, 1985.

....1: The dual polygons A and B. 2. 2 The data structure Since the vertices edges of the dual polygons A and B correspond to the edges vertices of the convex hull of P , we can maintain A and B by a known dynamic convex hull structure; in the decremental case, the required time bound is O(n log n) [8, 13]. In addition, we need structures for the vertices of A and for the edges of B: 1. We maintain a set V A of O(n) points that contains all vertices of polygon A. We permit extra non vertex points in the set, as long as they are inside A. 4 Lemma 2.1, when translated to the dual, allows us ....

B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT-31:509-517, 1985.

....n) However, when computing (r; 1) triangles, we do not need to re insert points into the convex hull maintenance structure in order to compute l c . Since, we do not need to insert points into the convex hull but simply delete them; we opt for a deletion only convex hull maintenance structure [Cha85, HS92], which provides better amortized time complexities for point deletions than Overmars and van Leeuwen s method. In [HS92] the point deletion operation removes a point from the convex hull maintenance structure in O(log n) amortized time (amortized over the sequence of n deletions) Consequently, ....

B. Chazelle. On the convex layers of a planar set. IEEE Trans. on Inf. Theory, IT31: 509-517, 1985.

....n) However, when computing (r; 1) triangles, we do not need to re insert points into the convex hull maintenance structure in order to compute l c . Since, we do not need to insert points into the convex hull but simply delete them; we opt for a deletion only convex hull maintenance structure [Cha85, HS92], which provides better amortized time complexities for point deletions than Overmars and van Leeuwen s method. In [HS92] the point deletion operation removes a point from the convex hull maintenance structure in O(log n) amortized time (amortized over the sequence of n deletions) Consequently, ....

B. Chazelle. On the convex layers of a planar set. IEEE Trans. on Inf. Theory, IT-31:509--517, 1985.

....after presorting is linear because T max has linear complexity. In general, an order of points and the corresponding triangulation can be found using an algorithm that maintains the convex hull of points. It can be done in O(n log n) time since the only update operation is the deletion of a point [4]. The algorithm maintains a data structure representing a triangulation. We assume that the edges triangles incident to a triangle edge and the vertices incident to an edge can be accessed in constant time. We assume also that the edges incident to a vertex are stored in a doubly linked list ....

B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT-31(4):509--517, July 1985.

....the input set S of n points in the Euclidean plane into a set of convex polygons defined as follows: i) compute the convex hull of S and remove its points from S, ii) then repeat instruction (i) until no point remains in S. This problem is a natural extension of the convex hull problem. Chazelle [2] proposed an optimal sequential algorithm for the convex layers problem which runs in O(n log n) time. The sequential algorithm is time optimal because the computation of a convex hull, which is the first hull of the convex layers, requires Omega Gamma n log n) time [13] Dessmark et al. 4] ....

....in O(1) time using n processors if S is stored in an array. In Step 2, the size of S i is n 1 ffl 2 (1 i n 1 Gammaffl 2 ) The convex layers of each subset S i , CL(S i ) CL 1 (S i ) CL 2 (S i ) CL k i (S i ) can be computed by the known Chazelle s sequential algorithm [2] in O(n 1 ffl 2 log n) time. Therefore, all the convex layers of S 1 ; S 2 ; S n 1 Gammaffl 2 can be computed in O(n 1 ffl 2 log n) time using n 1 Gammaffl 2 processors. Then we store each CL j (S i ) 1 j k i ) 1 i n 1 Gammaffl 2 ) in two balanced trees BT 1 [CL j (S i ) ....

B. Chazelle: "On the convex layers of a planar set", IEEE Transactions on Information Theory, 31(4):509-517 (1995).

....a point s on some edge of A. 2. 2 The data structure Since the vertices edges of the dual polygons A and B correspond to the edges vertices of the convex hull of P , we can maintain A and B by a known dynamic convex hull structure; in the decremental case, the required time bound is O(n log n) [7, 12]. In addition, we need structures for the vertices of A and for the edges of B: 1. We maintain a set V A of O(n) points that contains all vertices of polygon A. We permit extra non vertex points in the set, as long as they are inside A. Lemma 2.1, when translated to the dual, allows us to ....

B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT-31:509-517, 1985.

....we need not insert it into any other dynamic data structure, because it cannot again be crossed by within I II III IV Fig. 4. Quadrants determined by starting and ending positions of . this sweep. The deletions only convex hull data structure of Hershberger and Suri [5] see also Chazelle [2]) provides a suitable data structure for quadrants II and IV; it supports tangent finding in O(log n) time and point deletion in O(log n) amortized time. Points in quadrant III, specifically those in the shaded region, pose a difficulty. At any instant of the sweep, the points in quadrant III ....

B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inf. Theory, IT-31 (1985), 509--517.

....input set S of n points in the Euclidean plane into a set of convex polygons defined as follows: i) compute the convex hull of S and remove its points from S, ii) then repeat instruction (i) until no point remains in S. This problem is a natural extension of the convex hull problem. Chazelle [1] proposed an optimal sequential algorithm for the convex layers problem which runs in O(n log n) time. Dessmark et al. 2] proved that the problem is P complete. In [3] Fujiwara et al. considered an EP algorithm for the problem under a very strong constraint. If all points of S lie on d ....

....that S i (1 i n 1 Gammaffl 2 ) contains n 1 ffl 2 points and the x coordinates of the points of S i are less then the x coordinates of the points of S i 1 . ii) In parallel, for each subset S i (1 i n 1 Gammaffl 2 ) we compute the convex layers of S i by Chazelle s sequential algorithm [1]. iii) Then we merge the convex layers of S 1 ; S 2 ; S n 1 Gammaffl 2 into those ones of S as follows. Let x = 1 and repeatedly do the following instructions until S is not empty: a) Find the xth convex layer of S, denoted as CL x , from the outermost layers of S 1 ; S 2 ; ....

B. Chazelle: "On the convex layers of a planar set", IEEE Transactions on Information Theory, 31(4):509-517 (1995).

....to v i . It takes time proportional to the number of edges in E(i) In general an order of points and the corresponding triangulation can be found using an algorithm that maintains the convex hull of points. It can be done in O(n log n) since the only update operation is the deletion of a point [3]. The algorithm maintains a data structure representing a triangulation. We assume that the edges triangles incident to a triangle edge and the vertices incident to an edge can be accessed in constant time. We assume also that the edges incident to a vertex are stored in doubly linked list ....

B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT-31(4):509--517, July 1985.

....6. For each degree of symmetry compute invariants of the corresponding sequence. 7. The minimum distance of two sets of the invariants in Euclidean feature space expresses the measure of equivalence of the point sets. The convex layers can be computed either the Chazelle s fast algorithm [7] with the computing complexity O(n log n) or much simpler Jarvis march [8] with running time O(n 2 ) If we know the convex layers, then the left tangents can be found with the computing complexity O(n) The tree is created from the left tangents and transcribed into the left list by the ....

Chazelle B.: On the Convex Layers of a Planar Set, In IEEE Trans. on Information Theory, Vol. IT-31 (1985), No. 4, pp. 509-517

....case (d = 2) was solved optimally by Chazelle, Guibas, and Lee [28] Their data structure takes linear space and answers a query in O(log n k) time, where k is the number of reported points. The preprocessing can be accomplished in O(n log n) time using Chazelle s algorithm for convex layers [22]. Unfortunately, the approach does not generalize to higher dimensions. A preliminary version of this paper appeared in Proc. 39th IEEE Sympos. Found. Comput. Sci. 1998. This work was done while the author was at the Department of Mathematics and Computer Science, University of Miami. y ....

B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT-31:509--517, 1985.

....improvement to this output sensitive running time was reported, except for special cases. Cole, Sharir, and Yap [18] described a modification when k is much smaller than n; their algorithm takes O(n log n (n m) log 2 k) time (see also [25] using a convex layers subroutine by Chazelle [13]. The author [10] noted a modification for another case where the output size m is much smaller than n; the time bound becomes O(n log m m log 2 n) We were still no closer to the known lower bound Omega Gamma n log m m) in terms of the number of logarithmic factors. Important progress have ....

....the best deterministic result for k levels of curves. Finally, it prepares for the description of our next solution. The best solution we get for kinetic priority queues is based on the author s dynamic hull structure [12] which involves dynamization of known semi dynamic methods by Chazelle [13] and Hershberger and Suri [30] which are modifications of Overmars and van Leeuwen s structure) This gives our fastest deterministic algorithm for planar k level construction. Except for the semi dynamic subroutine, we provide a self contained description in Section 2.3, with pseudocodes for the ....

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B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT-31:509--517, 1985.

....case (d = 2) was solved optimally by Chazelle, Guibas, and Lee [25] Their data structure takes linear space and answers a query in O(log n k) time, where k is the number of reported points. The preprocessing can be accomplished in O(n log n) time using Chazelle s algorithm for convex layers [19]. Unfortunately, the approach does not generalize to higher dimensions. A preliminary version of this paper appeared in Proc. 39th IEEE Sympos. Found. Comput. Sci. 1998. y Department of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33124 4250, tchan cs.miami.edu ....

B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT-31:509--517, 1985.

.... 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 ....

....in the form of special case results. First, if deletions are not allowed, then it is relatively easy to achieve optimal O(log n) insertion time [32] If instead insertions are not allowed after preprocessing, then a modification of Overmars and van Leeuwen s structure, as described by Chazelle [11] and Hershberger and Suri [24] can perform n deletions on an n point set in O(n log n) total time, which is optimal in the amortized sense. Portions of this work were performed while the author was at Department of Mathematics and Computer Science, University of Miami. y Department of ....

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B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT-31:509--517, 1985.

....if one allows the output triangulation to have degenerate (zero area) triangles. Figure 12: Updating a Hamiltonian triangulation with a new interior point. a b v u Q A Figure 13: Traversing an onion layer. Onion Method Assume now that S is an arbitrary point set. We compute (in time O(n log n) [6]) the onion of S, obtaining the convex hull (conv(S) then the convex hull of those points not on the boundary of conv(S) etc. In this way, conv(S) is partitioned into convex annuli, and a core convex polygon. Thus, we are done once we prove the following simple claim, since it will allow us ....

B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT-31:509--517, 1985.

....O(1) time whether v is supporting, reflex, or inflex with respect to p. Hence, finding the supporting vertices and updating the convex hull takes O(log n) time. Some deletion only data structures with linear size and O(log n) amortized deletion time for planar convex hulls are given by Chazelle [29] and by Hershberger and Suri [77] p p p (a) v i v i 1 v i 1 v i 1 v i 1 v i v i v i 1 v i 1 (b) c) Figure 11: Classification of vertex v i with respect to p: a) left supporting; b) reflex; c) inflex. 7.2 Fully Dynamic Maintenance of Planar Convex Hulls Now we describe the ....

B. Chazelle, "On the Convex Layers of a Planar Set," IEEE Trans. Inf. Theory IT-31 (4) (1985), 509--517.

....4 shows the convex layer of a point set. This onion peeling process of a point set is central in the study of robust estimators in statistics, in which the outliers, points lying on the outermost convex layers should be removed. In this section we describe an efficient algorithm due to Chazelle[11] that runs in optimal O(n log n) time. As described in Section 2.1, each convex layer of C(S) can be decomposed into two convex polygonal chains, called upper and lower hulls (Figure 5) Let l and r denote the points with the minimum and maximum x coordinate respectively in a convex layer. The ....

....of the subtree rooted at node v of T and let U(v) denote its upper hull of the convex hull of S(v) Thus U(ae) where ae denotes the root of T is the upper hull of the convex hull of S in the outermost layer. The union of all the upper hulls U(v) for all nodes v is a tree, called hull graph[11]. A similar graph is also computed for the lower hull of the convex hull. To minimize the amount of space, at each internal node v we store the bridge (common tangent) connecting a point in U(v l ) and a point in U(v r ) where v l ; v r are the left and right children of node v. Figures 6(a) ....

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B. Chazelle, "On the Convex Layers of a Planar Set," IEEE Trans. Inform. Theory, IT-31 (1985), 509--517.

.... alternative proof that red black trees support constant time amortized finger updates (which is a fact known to folklore) We show the utility of the biased finger tree structure by giving an optimal O(n log n) time spacesweeping algorithm for the well known 3 dimensional layers of maxima problem [2, 7, 15, 27]. We also give improved methods for dynamic point location in a convex subdivision [35, 8] and present a method for dynamic point location in staircase subdivision with logarithmic query and update times. We note that although the staircase subdivision we consider is a very restricted form of ....

....of M from S, and repeats this process until S is empty. The iteration number in which a point p is removed from S is called p s layer, and we denote it by l(p) and the layers of maxima problem is to determine the layer of each point p in S. This is related to the well known convex layers problem [7], and it appears that it can be solved for a 3 dimensional point set S in O(n log n log log n) time [1] using the dynamic fractional cascading technique of Mehlhorn and Naher [32] We show how to solve the 3 dimensional layers of maxima problem in O(n log n) time, which is optimal 3 . We solve ....

B. Chazelle, "On the convex layers of a planar set," IEEE Trans. Inform. Theory, IT-31 1985, 509--517.

....is possible if p was not the deepest point of P , obtained by repeatedly discarding points on the convex hull. Finally, Ikebe et al. 10] showed that there was always such an embedding. All three algorithms use quadratic time. We show that one can use a deletion only convex hull structure [4, 9] to obtain Prosenjit Bose was partially supported by an NSERC and a Killam Postdoctoral Fellowship. Michael McAllister was partially supported by an NSERC Postgraduate Fellowship. Jack Snoeyink was partially supported by an NSERC Research Grant and a B.C. Advanced Systems Institute Fellowship. ....

....access to the convex hull of points. Moreover, we need the ability to delete points from the convex hull as we embed tree vertices at them. Overmars and van Leeuwen s [19] dynamic convex hull structure permits arbitrary insertion into and deletion from the set of points. We opt for hull trees [4, 9], which provide better amortised time complexities for point deletions. This section provides a brief introduction to hull trees. A hull tree of a set of n points P stores the upper or lower convex hull of P ; the entire convex hull of P can be represented by two hull trees. A hull tree for P s ....

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B. Chazelle. On the convex layers of a planar set. IEEE Trans. Info. Theory, IT-31:509--517, 1985.

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B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT-31:509-517, 1985.

No context found.

B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT31: 509--517, 1985.

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B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT-31:509-517, 1985.

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