Chazelle, Bernard. (1991). An optimal convex hull algorithm and new results on cuttings. Pages 29-38 of: Focs.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of hyperplanes, using a divide and conquer approach. The efficiency of the resulting algorithms and data structures depends heavily on the size of the cutting, that is, its number of simplices. Therefore, much research in the past few years has been devoted to constructing cuttings of small size [2, 5, 7, 8, 9, 16, 17, 18]. These days there are several methods for constructing cuttings of optimal size O(r i) 5, 7, 18] The concept of cuttings is readily generalized to sets of other geometric objects than hyperplanes: a (1 r) cutting for a set H of geometric objects in E s is a partitioning of E s into elementary ....

....depends heavily on the size of the cutting, that is, its number of simplices. Therefore, much research in the past few years has been devoted to constructing cuttings of small size [2, 5, 7, 8, 9, 16, 17, 18] These days there are several methods for constructing cuttings of optimal size O(r i) [5, 7, 18]. The concept of cuttings is readily generalized to sets of other geometric objects than hyperplanes: a (1 r) cutting for a set H of geometric objects in E s is a partitioning of E s into elementary shapes which we call boxesqsuch that each box is intersected by at most n r objects. See ....

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B. Chazelle, An Optimal Convex Hull Algorithm and New Results on Cuttings, Proc. 32nd IEEE S!mp. on Foundations of Computer Science, 1991, pp. 29-38.

....log rain x, y ) time algorithm computing the incidences between families of z points and families of y hyperplanes in E d where one is a restricted family. This algorithm is an extension of the algo rithm developed by Agarwal [1] and it is based on the following results presented by Chazelle [7]: A ( cutting of size O(r d) for a family of n hyperplanes in E d can be computed in O(nr d ) time, for any rd r n; A family of n hyperplanes in E d can be preprocessed in O(n d) time to allow for point location in O(1og n) time per query. Let P be a family of x points and H be a family ....

....in Pi and the hyperplanes in Hi plus the incidences between the points in B and all the hyperplanes in hr. The time complexity of this algorithm is determined by the time to construct the subproblems plus the time to solve them. The (l; cutting C of hr can be computed in O(yr a ) time (see [7]) During the construction of the cutting, the hyperplanes are distributed among the simplices. To divide the points among the subproblems, we preprocess the O(r) hyperplanes used to construct C for point location. This can be done in O(r a) C O( lr a ) time. For each point p P, locate a cell of ....

B. Chazelle. An optimal convex hull algorithm and new results on cutting. In Proc. o.f the $'2th IEEE Found. o.f Comp. Science, 1991.

.... minimizing , in time r 1 n r f(m) We know of no fully polynomial time algorithm to find minimum volume or boundary measure sets, except in the plane [2,14,20] A naive algorithm runs in time O( k Delta k bd=2c ) by explicitly computing the convex hull of every k point subset [5]. We use this algorithm as a subroutine. Throughout this section, weletjAj and denote the volume and boundary measure of the convex hull of A. The following lemma relates the volumes of bounding boxes and extremal simplices. d (A)j =2 d jS d (A)j. Proof: The volume of a ....

B. Chazelle. An optimal convex hull algorithm and new results on cuttings. In 32nd IEEE Symp. Found. Comput. Sci., pages 29--38, 1991.

....of i 2 , each of which determines at most one matching of P in S. Since every matching has complexity k, the assertion follows. K The sequential time necessary to find all the O(D 2 (n) line segments of length b with endpoints in S is denoted by A 2 (n) We have the following. Proposition 5. 4 [Agar90, Chaz91]. For any fixed 0, A 2 (n) O(n 4#3 ) Theorem 5.5 [G6K92] The point set pattern matching problem in R 2 can be solved sequentially in O(A 2 (n) kD 2 (n) log n) time. K Our CGM algorithm for solving the point set pattern matching problem in R 2 is based on finding which rigid ....

B. Chazelle, An optimal convex hull algorithm and new results on cuttings, in Proc. 32nd IEEE Symposium on Foundations of Computer Science, 1991," pp. 29#38.

....b5=2c ) O(n 2 ) It is not dicult to nd con gurations that attain this bound. Since is of degree two, intersecting a polytope with can only increase the above complexity by a constant factor. Finally, in order to compute the polytopes L 0 and R 0 we can use an algorithm of Chazelle [21] that constructs the representation of a polytope by the incidence graph of their faces [20] in O(n 2 ) time. Therefore, we have the following: Theorem 5.4. Given a simple polygonal chain of n vertices, computing all the lines with respect to which it is strictly monotonic can be done in O(n ....

Chazelle, An optimal convex hull algorithm and new results on cuttings, Procd. 32nd Symp. on Foundations of Computer Science, pp 29-38, 1991.

.... 16] The method for identifying the points on the convex hull in two and three dimensions cannot be extended to dimensions greater than three since the number of facets of the convex hull of P has an exponential growth in higher dimensions, i.e. there can be up to Omega Gamma n bd=2c ) facets [5]. Deciding whether a point in P is an extreme point or not can be formulated as a linear programming problem. There are many deterministic and randomized algorithms that solve a linear program with n constraints in time O(n) 5, 8, 10, 18, 22, 23] Unfortunately, the constants are exponential ....

....dimensions, i.e. there can be up to Omega Gamma n bd=2c ) facets [5] Deciding whether a point in P is an extreme point or not can be formulated as a linear programming problem. There are many deterministic and randomized algorithms that solve a linear program with n constraints in time O(n) [5, 8, 10, 18, 22, 23]. Unfortunately, the constants are exponential functions of d. The best known deterministic algorithm has a running time of O(2 O(d log d) n) 5] while one of the fastest randomized algorithms is a combination of the approach of Clarkson [8] and Sharir and Welzl [23] with a running time of O(d ....

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B. Chazelle. An optimal convex hull algorithm and new results on cuttings. In Proc. 32nd IEEE Symp. on Foundations of Computer Science, pages 29--38, 1991.

....the two red Plucker points in the arrangement of the blue Plucker hyperplanes. From the Upper Bound Theorem [Ede87] we have an O(n 2 ) bound on the complexity of C. Using Seidel s algorithm [Sei86] we construct C in deterministic time O(n 2 log n) using a recent algorithm of Chazelle [Cha91] C can be computed in O(n 2 ) deterministic time) We subdivide C into O(n 2 ) simplices and we compute the components of s Pi for each simplex s. For any pair of adjacent simplices s and s 0 , if a component of Pi s and a component of Pi s 0 have a common boundary point we ....

B. Chazelle. An optimal convex hull algorithm and new results on cuttings. In Proceedings of the 32th IEEE Symposium on Foundations of Computer Science, pages 29--38, 1991.

.... 27] Randomization has become a fundamental design tool for geometric algorithms; often the simplest or most efficient algorithm known for a problem is one that uses random coin flips to guide its execution[17] Derandomizing randomized geometric algorithms has become a favorite pastime[13, 63], motivated in part by complexity theoretic questions about the fundamental role of randomness in algorithm analysis. A strong influence on the field has been the interplay between computational geometry and combinatorial geometry. Combinatorial geometry is concerned with combinatorial properties ....

....a point from the triangulation of the convex hull; the resulting triangulation is the same as the triangulation that would have been obtained by inserting the points in the original order, omitting the deleted point. Clarkson[17] and Seidel[85] survey randomized geometric algorithms. Chazelle[13] shows how to derandomize the convex hull algorithm, that is, how to choose the insertion order in a deterministic fashion that guarantees the same asymptotic running time. Unfortunately, the algorithm to choose the order is quite complicated. Empirical performance. The algorithm has been ....

B. Chazelle, An optimal convex hull algorithm and new results on cuttings, Proc. 32nd Ann. Symp. Found. Comp. Sci. 29--38, 1991.

....of convexhull algorithms is output sensitivity. The complexity of an output sensitive convex hull algorithm depends on the number of faces, F , in the convex hull. However, optimal parallel output sensitive Delaunay diagram construction in high dimensions is still an open problem. Chazelle [9], gave the first optimal deterministic convex hull algorithm. This algorithm is not output sensitive and thus is optimal only in the worst case sense, i.e. it runs in O(n log n n bd=2c ) which is the worst case number of faces possible in d 3 dimensions. Recently, Amato, Goodrich and Ramos ....

B. Chazelle. An optimal convex hull algorithm and new results on cutting. In 32th Annual Symposium on Foundations of Computer Science, 29--38. IEEE, 1991.

....covering, we can quickly find O(c) sized covers. 1 Introduction A set system (X; R) is a set X along with a collection R of subsets of X , which are sometimes called ranges [25] Such entities have also been called hypergraphs and range spaces in the computational geometry literature (e.g. see [5, 10, 11, 12, 13, 14, 15, 16, 20, 24, 25, 34, 36, 35, 41, 37, 39, 40]) and they can be used to model a number of interesting computational geometry problems. There are a host of NP hard problems defined on set systems, with one of the chief such problems being that of finding a set cover of minimum size (e.g. see [21, 23] where a set cover is a subcollection C ....

....test whether there is a facet that belongs to P . If so, we can return H p where p is, e.g. the centroid of this face. Otherwise, Y is certainly contained in P . We can find a (1=r) net in O i r d log d (dr)jQj j time and verify k halfspaces in O(k jP j) b d 2 c time (if d 4) [11] or in O( k jP j) log(k jP j) time (for d = 2; 3) 43] Plugging those bounds into our main theorem, we get: Theorem 5.1 Let Q P be two convex nested polytopes in R d (d 4) with a total of n facets. It is possible to find a separator of size within O(d 2 logc) of the optimum c, in a ....

B. Chazelle. An optimal convex hull algorithm and new results on cuttings. In Proc. 32nd Annu. IEEE Sympos. Found. Comput. Sci., pages 29--38, 1991.

....for the worst case insertion order are about a factor of n worse. For uniform data there is a double expectation, over both insertion order and input distribution) With additional algorithmic complexity, it is possible to obtain deterministic algorithms with the same worst case running times[9]. s Figure 4: The addition of site s deletes four triangles and adds six (shown dashed) The plane sweep algorithm The plane sweep algorithm computes a planar Delaunay triangulation using a horizontal line that sweeps upwards across the plane. The algorithm discovers a Delaunay triangle when ....

B. Chazelle, An optimal convex hull algorithm and new results on cuttings, Proc. 32nd Ann. Symp. Found. Comp. Sci. pp. 29--38, 1991.

....to increase efficiency; the memoization associated with lazy evaluation is a good example. However, these effects are only benign in the sequential case; in the parallel case synchronization overhead can mitigate or even negate, the efficiency gains of using these techniques. the literature [13, 4, 8, 7], but all use ephemeral data structures. We present a new incremental algorithm to compute the 3d convex hull of a set of points (and by reduction a 2d Delaunay triangulation) based on a fully persistent representation of its hull, which is a triangulated surface. By using the persistent ....

....the input set of points to the hull algorithm has the property that no four points are coplanar. A Omega Gamma n lg n) lower bound on the problem of constructing the convex hull of a set of points in three dimensions is well known [4] Asymptotically optimal algorithms for the problem are known [8, 7]. We will be concerned with incremental methods that extend the convex hull of a subset of the set of points to include a point not in that subset. Many algorithms, including our own, are based on the tent construction. Given a point p exterior to the hull of a set of points, we may extend the ....

Bernard Chazelle. An optimal convex hull algorithm and new results on cuttings. In FOCS, pages 29--38, 1991.

....of i 2 , each of which determines at most 1 matching of P in S. Since every matching has complexity k, the assertion follows. Xi The sequential time necessary to find the O(D 2 (n) line segments of length b with endpoints in S is denoted by A 2 (n) We have the following. Proposition 8. 4 [Agar90, Chaz91] For any fixed ffi 0, A 2 (n) O(n 4 3 ffi ) Xi Theorem 8.5 [G K92] The Point Set Pattern Matching Problem in R 2 can be solved sequentially in O(A 2 (n) kD 2 (n) log n) time. Xi Theorem 8.6 The Point Set Pattern Matching Problem in R 2 can be solved in pT sort (n; p) A 2 (n) ....

B. Chazelle, An optimal convex hull algorithm and new results on cuttings, Proc. 32nd IEEE Symposium on Foundations of Computer Science (1991), 29-38.

....covering, we can quickly find O(c) sized covers. 1 Introduction A set system (X; R) is a set X along with a collection R of subsets of X , which are sometimes called ranges [25] Such entities have also been called hypergraphs and range spaces in the computational geometry literature (e.g. see [5, 10, 11, 12, 13, 14, 15, 16, 20, 24, 25, 34, 36, 35, 41, 37, 39, 40]) and they can be used to model a number of interesting computational geometry problems. There are a host of NP hard problems defined on set systems, with one of the chief such problems being that of finding a set cover of minimum size (e.g. see [21, 23] where a set cover is a subcollection C ....

B. Chazelle. An optimal convex hull algorithm and new results on cuttings. In Proc. 32nd Annu. IEEE Sympos. Found. Comput. Sci., pages 29--38, 1991.

.... in time O(n r 1 n r f(m) 2 We know of no fully polynomial time algorithm to find minimum volume or boundary measure sets, except in the plane [2, 14, 20] A naive algorithm runs in time O( Gamma n k Delta k bd=2c ) by explicitly computing the convex hull of every k point subset [5]. We use this algorithm as a subroutine. Throughout this section, we let jAj and j Aj denote the volume and boundary measure of the convex hull of A. The following lemma relates the volumes of bounding boxes and extremal simplices. Lemma 9.2. For all A ae IR d , jB d (A)j = 2 d d jS d (A)j. ....

B. Chazelle. An optimal convex hull algorithm and new results on cuttings. In 32nd IEEE Symp. Found. Comput. Sci., pages 29--38, 1991.

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Chazelle, Bernard. (1991). An optimal convex hull algorithm and new results on cuttings. Pages 29-38 of: Focs.

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B. Chazelle, An optimal convex hull algorithm and new results on cuttings, Proc. 32nd Annu. IEEE Sympos. Found. Comput. Sci., 1991, pp. 29--38.

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B. Chazelle, An optimal convex hull algorithm and new results on cuttings, Proceedings 32nd Symp. on Fund. of Comp. Sci. 1991, pp. 29-38.

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B. Chazelle, An optimal convex hull algorithm and new results on cuttings, Proc. 32nd Ann. Symp. Found. of Comp. Sci. 29--38, 1991.

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B. Chazelle. An optimal convex hull algorithm and new results on cuttings. In Proc. 32nd Annu. IEEE Sympos. Found. Comput. Sci., pages 29--38, 1991.

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B. Chazelle, An optimal convex hull algorithm and new results on cuttings, Proc. 32nd IEEE Annual Symp. on Foundations of Computer Science, 1991, 29--38.

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