R. Seidel, Constructing higher-dimensional convex hulls at logarithmic cost per face, Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing (Berkeley, California), ACM Press, 1986, pp. 404--413.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on Problem 1. Many algorithms for the Vertex Enumeration Problem in fact compute the whole face lattice of the polytope. Swart [60] analyzing an algorithm of Chand and Kapur [10] proved that there exists a polynomial total time algorithm for this problem. For a faster algorithm see Seidel [56]. Fukuda, Liebling, and Margot [22] gave an algorithm which uses working space (without space for the output) bounded polynomially in the input size, but it has to solve many linear programs. For xed dimension, the size of the output is polynomial in the size of the input; hence, a polynomial ....

R. Seidel, Constructing higher-dimensional convex hulls at logarithmic cost per face, in Proc. 18th Ann. ACM Sympos. Theory Comput., 1986, pp. 404-413.

....log f) time method for both 2 d and 3 d convex hulls. In dimensions 3, the fastest output sensitive algorithms currently known (excluding the results of this paper) are an improvement of the gift wrapping method [21, 35] by Chan [3] and an improvement of Seidel s beneath beyond algorithm [34] by Matousek [26] The former runs in O(n log f (nf) n) time and the latter runs in O(n n f log n) time. In this paper we give a convex hull algorithm in four dimensions that runs in O( n f)log f) time and uses linear space. Our basic strategy is divide and conquer. In order to ....

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. In Proceedings of the 18th Annual ACM Symposium on Theory of Computing, pages 404--413, 1986.

....years of unsuccessful researchs on finding an optimal algorithm for any dimensions, B. Chazelle has recently given such an algorithm [Cha91] An algorithm whose complexity is O(n f log n) where f is the size of the output, that is the number of facets of the convex hull, is described in [Sei86] In the plane, deterministic on line algorithms are known, with optimal complexity O(log n) per insertion (see for example [AES85] 1.2 Voronoi diagram The Voronoi diagram of a set S of n points, called sites, in IE is a geometric structure used to solved proximity queries, such as the ....

....triangulation may have a quadratic size. In [Boi88, BCDT91] we obtain an optimal algorithm O(n log n t) n is the number of point sites, and t the number of tetrahedra in the triangulation) for the restricted case when the points are assumed to lie on two planes. The algorithm by R. Seidel [Sei86] for the computation of convex hulls in any dimension runs in f log n) where f is the numbers of faces of the hull. Too few results exist in this field, and it can be supposed that this subject will be much studied in future. ....

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. In ACM Symposium on Theory of Computing, pages 404--413, 1986.

....of the boundary complexes of polytopes. Shellability is important both in combinatorial and computational geometry, for example, it was essential for the proof of the upper bound of the number of faces of polytopes [11] or has been used for efficient convex hull construction of polytopes [14]. Shellability has also been studied from algebra through the Stanley Reisner ring of simplicial complexes [8] 15] A pure simplicial complex is extendably shellable if any sequence of a subset of facets satisfying the condition of being pasted nicely can be continued to a shelling. This means ....

R. Seidel, Constructing higher dimensional convex hulls at logarithmic cost per face, in Proc. 18th Annual ACM Symposium on Theory of Computing (STOC), pp.404--413, 1986.

.... enumeration, permanent, polytope, fan, spherical polytope, oriented matroid, face numbers, f vector, h vector, #P 1 Introduction The extensively studied notion of shellability is of basic importance for the combinatorial, enumerative and algorithmic study of simplicial complexes and posets [1, 2, 4, 6, 16]. However, there are important classes of complexes which are not shellable, but do possess the somewhat weaker property of partitionability. This property often suffices to lead to efficient computation of the face numbers of such complexes, and to derive properties of these numbers and ....

R. Seidel. Constructing higher dimensional convex hulls at logarithmic cost per face. Proc. Ann. ACM Symp. Theory Comput., 18:404--413, 1986.

....using the above mentioned next v algorithm or it can be found in form of a table in [27] There are plenty of algorithms for point set problems which are based on computing the orientation of a sequence of points. Prime examples are algorithms for the construction of convex hulls; see for example [66, 72, 73] or [21,54, 67] References to other applications can be found in [27] which also extends the concept of orientation to points with homogeneous coordinates in arbitrary dimensions. Sphere test. The crucial primitive operation for constructing Delaunay triangulations is to check if a triangle is ....

R Seidel. Constructing higher-dimensional convex hulls in logarithmic cost per face. In Proceedings of the 13th Annual ACM Symposium on Theory of Computing, pages 484 507, 1986.

....space. The algorithm is incremental but its complexity is amortized over the n insertions. This result is optimal in the worst case for d larger than 2 and even. An output sensitive algorithm with time complexity O(n 2 f log n) where f is the number of faces of the convex hull is reported in [Sei86] Randomized algorithms have also been recently proposed by Clarkson and Shor [CS89] and by Mulmuley [Mul89] These algorithms are incremental and optimal but static. A more on line algorithm for d 3 can be found in [GKS92] however the analysis is only amortized over the n insertions. We ....

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. In ACM Symposium on Theory of Computing, pages 404--413, 1986.

....that a consistent depth order for the simplices in a Delaunay triangulation, with respect to any viewpoint x, can be obtained by sorting the distances of their circumcenters from x. This is precisely the order in which the Delaunay tetrahedra are computed by Seidel s shelling convex hull algorithm [55]. We can easily extract a consistent depth order for the Delaunay edges from this simplex order. The next lemma describes sucient (but not necessary) conditions for three mutually orthogonal segments to form a screw. Lemma 2.3. Let c and C be concentric axis aligned cubes of width 1 and w, ....

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. Proc. 18th Annu. ACM Sympos. Theory Comput., 404-413, 1986.

.... the facets of a polytope is a shelling if it satisfies some topological condition (defined below at (3) Shellings have many applications both in combinatorial and computational geometry: for example, they are crucial for the upper bound theorem [18] 73] and are used in convex hull construction [91]. A total order of the facets of a polytope corresponds to a total order of the vertices of the polar polytope. Such order of vertices is a polar shelling. A line shelling is some special shelling, and its polar becomes an ordering of the vertices of the polar polytope by a sweep of parallel ....

.... polytopes was shown by Bruggesser Mani [18] Shellability is important both in combinatorial and computational geometry, for example, it was essential for the proof of the upper bound of the number of faces of polytopes [73] or has been used for efficient convex hull construction of polytopes [91]. Shellability has also been studied from algebra through the Stanley Reisner ring of simplicial complexes [34] 92] A pure simplicial complex is extendably shellable if any sequence of a subset of facets satisfying the condition of being pasted nicely can be continued to a shelling. This means ....

Raimund Seidel, Constructing higher dimensional convex hulls at logarithmic cost per face, in: Proc. 18th Annual ACM Symposium on Theory of Computing (STOC), pp.404--413, 1986.

....a simple polygon of decreasing size to triangulate. Unfortunately, this solution is wrong, but we will show in this paper how to correct it. Overview In this paper, we provide a very simple and efficient O(k log k) algorithm to delete a vertex in a planar Delaunay triangulation based on shelling [BM71, Sei86] and duality [DMT92, Ped70] We also discuss the effective complexity of this algorithm and of a few others for small values of k. We will study the different kinds of geometric predicates necessary for these algorithms. This algorithm generalizes well in higher dimensions: its time complexity ....

....to C qrs and the signed vertical distance between p and P qrs . 1 1 If C is a circle of center x and radius r, p is a point and l is a line through p intersecting C in t and u, then power(p;C) jxpj 2 Gamma r 2 = 2. 3 Shelling Convex hulls may be computed by the shelling algorithm [Sei86]. The shelling [BM71] of a convex polyhedron P is the enumeration of its faces in some appropriate order. Imagine an observer is moving along a line l going through the polyhedron, starting at the intersection of P and l. At the starting position, the observer can only see one face of P (the face ....

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. In Proc. 18th Annu. ACM Sympos. Theory Comput., pages 404--413, 1986.

.... the facets of a polytope is a shelling if it satisfies some topological condition (defined below at (3) Shellings have many applications both in combinatorial and computational geometry: for example, they are crucial for the upper bound theorem [1] 7] and are used in convex hull construction [10]. A total order of the facets of a polytope corresponds to a total order of the vertices of the polar polytope. Such order of the vertices is a polar shelling. A line shelling is some special shelling, and its polar becomes an ordering of the vertices by a sweep of hyperplanes, 3 Extended ....

Raimund Seidel, Constructing higher dimensional convex hulls at logarithmic cost per face, in Proc. 18th Annual ACM Symposium on Theory of Computing (STOC), 1986, 404--413.

....unbounded case (polyhedron) by adding a face at in nity, but for simplicity we assume P is bounded. First of all the answer does not depend on how P is given. The problem for H polytopes is equivalent to the one for V polytopes by duality. See Sections 2.11 and 2.4. There are algorithms (e.g. [Rot92, Sei86, FLM97] ) that can generate all faces from a Vrepresentation or from a H rerepsentation. Perhaps the backtrack algorithm [FLM97] is easiest 7 to implement and works directly for the unbounded case. It is also a compact polynomial algorithm (see 2.15) and thus needs little space to run. Algorithms that ....

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. In 18th STOC, pages 404-413, 1986.

....hyperplanes. If this is not the case we answer negatively. Otherwise, we construct the cell C containing 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 ....

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. In Proceedings of the 18th annual Symposium on Theory of Computing, pages 404--413, 1986.

....of the boundary complexes of polytopes. Shellability is important both in combinatorial and computational geometry, for example, it was essential for the proof of the upper bound of the number of faces of polytopes [11] or has been used for efficient convex hull construction of polytopes [14]. Shellability has also been studied from algebra through the Stanley Reisner ring of simplicial complexes [8] 15] A pure simplicial complex is extendably shellable if any sequence of a subset of facets satisfying the condition of being pasted nicely can be continued to a shelling. This means ....

R. Seidel, Constructing higher dimensional convex hulls at logarithmic cost per face, in Proc. 18th Annual ACM Symposium on Theory of Computing (STOC), pp.404--413, 1986.

....is only one possible system of inequalities generated by concatenating the equations of the single cones in one system. Theorem 1 The existence problem for a set of triangles lying on a family of parallel planes can be solved in O(n 2 log n) time. Proof. Using Seidel s convex hull algorithm [Sei86] we compute the Plucker polytope R corresponding to the solution of the unique system of linear inequalities in O(n 2 log n) The second step is to check that the Plucker hypersuperface Pi intersects the polytope R, R Pi 6= The test proceeds as follows 7 : 1. Compute the sign of Pi ....

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. In Proceedings of the 18th annual Symposium on Theory of Computing, pages 404--413, 1986.

....facet of the extended convex hull. 2.1 Shelling Ordering A shelling [8] of a polytope is a certain linear order induced on the facets of the polytope by a line. This property of numbering facets and vertices has been used in algorithms for vertex enumeration [3, 11, 13] and facet enumeration [14, 15]. To understand the shelling principle, we first need a few definitions. Let P be a polytope. A shelling of P is a sequence (F 1 , Fm ) of all facets of P such that, for all i, 1 i m, # i 1 j=1 F j ) # F i is a topological (d 2) ball. This implies that for each facet F i ....

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. In Proc. 18th Annu. ACM Sympos. Theory Comput., Berkeley, CA, pages 404--413, 1986.

....matrix inversion, determinant of a matrix, rank of a matrix) for matrices with integer entries. Many checks are trivial 5 , 2 Algorithms based on randomized incremental construction [5, 6, 4] run in time related to the size of the output and the size of intermediate hulls and the algorithm of [15] is guaranteed to construct the hull in logarithmic time per face. 3 The linear program has d variables corresponding to the coefficients of a linear function. For each vertex of F there is a constraint stating that the function value at the vertex is negative. For each non vertex consider the ....

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. In Proc. 18th Annu. ACM Sympos. Theory Comput., pages 404--413, 1986. 9

No context found.

R. Seidel, Constructing higher-dimensional convex hulls at logarithmic cost per face, Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing (Berkeley, California), ACM Press, 1986, pp. 404--413.

No context found.

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. In Proc. 18th Annu. ACM Sympos. Theory Comput., pages 404--413, 1986.

No context found.

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. In Proc. 18th Annu. ACM Sympos. Theory Comput., pages 404413, 1986.

No context found.

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. In Proc. 18th Annu. ACM Sympos. Theory Comput., pages 404-413, 1986.

No context found.

R. Seidel, Constructing Higher-Dimensional Convex Hulls at Logarithmic Cost Per Face, Proc. 18th Annu. ACM Sympos. Theory Comput. 1986, 404--413. 18

No context found.

R. Seidel, Constructing higher-dimensional convex hulls at logarithmic cost per face, Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing (Berkeley, California), ACM Press, 1986, pp. 404--413.

No context found.

R. Seidel, "Constructing higher-dimensional convex hulls at logarithmic cost per face", Proceeding of the 18 Annual ACM Symposium of Theory of Computing, ACM press, 404 -- 413, 1986.

No context found.

R. Seidel. Constructing higher-dimensional convex hulls at logarithmic cost per face. In Proc. 18th Annu. ACM Sympos. Theory Comput., pages 404413, 1986.

First 50 documents  Next 50

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