H. Edelsbrunner and R. Waupotitisch. Computing a ham sandwich cut in two dimensions. J. Symbolic Comput. , 2:171--178, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....) and of A(L 2 ) is always odd. Several prune and search algorithms have been proposed for computing a ham sandwich cut in the plane. Megiddo [214] gave a linear time algorithm for the special case in which S 1 and S 2 are linearly separable. Modifying this algorithm, Edelsbrunner and Waupotitsch [109] gave an O(n log n) time algorithm when S 1 and S 2 are not necessarily linearly separable. A linear time, recursive algorithm for this general case is given by Lo and Steiger [193] It works as follows. At each level of recursion, the algorithm maintains two sets of lines, R and B, and two ....

H. Edelsbrunner and R. Waupotitsch, Computing a ham-sandwich cut in two dimensions, J. Symbolic Comput., 2 (1986), 171--178.

....the (separated) sets of points to which L 1 and L 2 are dual. 5 Non Separated Ham Sandwich Cut We describe the algorithm for Theorem 1. To set the stage we begin by outlining the framework, which resembles that of the (non optimal) O( m n)log(m n) algorithm of Edelsbrunner and Waupotitsch [8] which finds a ham sandwich cut for sets S 1 (of m points in R 2 )andS 2 (of n points in R 2 ) Then we show how to follow this algorithm e#ciently when S 1 and S 2 are subsets of the O( m 2 )andO( n 2 ) vertices in the arangements of m and n lines, respectively. The General ....

H. Edelsbrunner and R. Waupotitsch. Computing a Ham-Sandwich Cut in Two Dimensions. J.Symbolic Computation 2, 171-178 (1986).

....log n. 2 6.4 Other Primitives Our techniques directly apply to several other primitives including those presented in [2] Primitives that decide on the relative position of derived objects may pose a limitation to our method. Consider, for instance, the two dimensional ham sandwich algorithm in [31] with lines on the plane being the input objects and their intersection points being the derived objects. The three primitives of the algorithm are: Deciding whether a point lies above or below a line; comparing the first coordinate of two points; and comparing the distances of two points from a ....

H. Edelsbrunner and R. Waupotitsch. Computing a ham-sandwich cut in two dimensions. J. Symbolic Comput., 2:171--178, 1986.

....1 ) and of A(L 2 ) is always odd. Several prune and search algorithms have been proposed for computing a ham sandwich cut in the plane. Megiddo [184] gave a linear time algorithm for the special case where S 1 and S 2 are linearly separable. Modifying this algorithm, Edelsbrunner and Waupotitsch [94] gave an O(n log n) time algorithm when S 1 and S 2 are not necessarily linearly separable. A linear time, recursive algorithm for this general case is given by Lo and Steiger [166] 4 The level of a point p with respect to A(L) is the number of lines lying strictly below p. The k level of A(L) ....

H. Edelsbrunner and R. Waupotitsch, Computing a ham-sandwich cut in two dimensions, J. Symbolic Comput., 2 (1986), 171--178.

....carried out; there are several different approaches, see e.g. CS89, Sei91, BDS 92] Perturbation schemes are extensively compared in [ECS95] here we discuss only the most relevant approaches to our own. The first systematic approach in computational geometry is due to Edelsbrunner [Ede86, EW86, EG86] It is called Simulation of Simplicity (SoS) and was generalized by Edelsbrunner and Mucke in [EM90] Their implementation of a three and four dimensional convex hull algorithm was the basis for our own program. More recently, SoS was implemented in conjunction with three dimensional ....

H. Edelsbrunner and R. Waupotitsch. Computing a ham-sandwich cut in two dimensions. J. Symbolic Comput., 2:171--178, 1986.

....yet they provide no certainty. Dantzig s [Da] symmetry breaking rules in linear programming is regarded as the precursor of current systematic approaches. Edelsbrunner and Mucke generalize in [EdMu] a technique called Simulation of Simplicity (SoS for short) already presented in [EdGu] Ed] and [EdWa]. Every input parameter p i;j is perturbed into p i;j (ffl) p i;j ffl 2 iffi Gammaj ; where ffi d and d is the dimension of the geometric space where the input objects lie. The perturbation is infinitesimal due to symbolic variable ffl which, although never evaluated, is assumed ....

....as mentioned in [EdMu] That paper also lists several predicates for each one of which one of our schemes is valid. Predicates that decide on the relative position of derived objects may pose a limitation to our method. Consider, for instance, the two dimensional ham sandwich algorithm in [EdWa] with lines on the plane being the input objects and their intersection points being the derived objects. Here are the three predicates called by the algorithm, assuming that the lines passed to a test or those defining the points in a test are all distinct. Deciding whether a point lies above or ....

Edelsbrunner H. and R. Waupotitsch, Computing a ham-sandwich cut in two dimensions, J. Symbolic Comput. 2, pp. 171-178, 1986.

....the existence of such a cut. Several prune and search algorithms have been proposed for computing a ham sandwich cut in the plane. For the special case when A 1 and A 2 are linearly separable, Megiddo [91] gave a linear time algorithm. Modifying his algorithm, Edelsbrunner and Waupotitsch [46] gave an O(n log n) time algorithm when A 1 and A 2 are not linearly separable. The running time was then improved to linear by Lo and Steiger [77] Efficient algorithms for higher dimensions are given by Lo et al. 76] 5.4 Placement and Intersection Polygon placement. Let P be a polygonal ....

H. Edelsbrunner and R. Waupotitsch, Computing a ham-sandwich cut in two dimensions, J. Symb. Comput. 2 (1986), 171--178.

....d 2 log n) 1 ff (ds d 2 log n)MM(d) log d) The overhead now follows from s = Omega (log n) 2 6.4 Limitations Primitives that decide on the relative position of derived objects may pose a limitation to our method. Consider, for instance, the two dimensional ham sandwich algorithm in [28], where lines are the input objects and their intersection points are the derived objects. The three primitives of the algorithm are: deciding whether a point lies above or below a line; comparing the first coordinate of two points; and comparing the distances of two points from a line. Applying ....

H. Edelsbrunner and R. Waupotitsch. Computing a ham-sandwich cut in two dimensions. J. Symbolic Comput., 2:171--178, 1986.

....The planar case of the well known discrete Ham Sandwich Theorem [16] states that, for finite sets of red and blue points in the plane, there exists a line dividing both red and blue points into sets of equal size. The Ham Sandwich problem is well studied from an algorithmic point of view [2, 4, 5, 6, 7, 11, 12, 13, 15, 17]. An optimal algorithm of Lo et al. 12] finds a Ham Sandwich cut in linear time. Very recently Kaneko and Kano [10] considered balanced partitions of two sets in the plane. They gave the following conjecture. Conjecture 1 Let m 2; n 2 and g be positive integers. Let R and B be two disjoint ....

H. Edelsbrunner and R. Waupotitsch. Computing a ham-sandwich cut in two dimensions. J. Symbolic Comput., 2:171--178, 1986.

....n. 2 6.5.4 Other Primitives Our techniques directly apply to several other primitives including those presented in [EM90] Primitives that decide on the relative position of derived objects may pose a limitation to our method. Consider, for instance, the two dimensional ham sandwich algorithm in [EW86] with lines on the plane being the input objects and their intersection points being the derived objects. The three primitives of the algorithm are: deciding whether a point lies above or below a line; comparing the first coordinate of two points; and comparing the distances of two points from a ....

H. Edelsbrunner and R. Waupotitsch. Computing a ham-sandwich cut in two dimensions. J. Symbolic Comput., 2:171--178, 1986.

....) and of A(L 2 ) is always odd. Several prune and search algorithms have been proposed for computing a ham sandwich cut in the plane. Megiddo [215] gave a linear time algorithm for the special case in which S 1 and S 2 are linearly separable. Modifying this algorithm, Edelsbrunner and Waupotitsch [110] gave an O(n log n) time algorithm when S 1 and S 2 are not necessarily linearly separable. A linear time, recursive algorithm for this general case is given by Lo and Steiger [194] It works as follows. At each level of recursion, the algorithm maintains two sets of lines, R and B, and two ....

H. Edelsbrunner and R. Waupotitsch, Computing a ham-sandwich cut in two dimensions, J. Symbolic Comput., 2 (1986), 171--178.

....dioeerent approaches for incrementing a partial hull, see e.g. CS89, Sei91, BDS 92] Perturbation schemes are extensively compared in [ECS97] here we discuss only the most relevant approaches to our own. The rst systematic approach in computational geometry is due to Edelsbrunner [Ede86, EW86, EG86] It is called Simulation of Simplicity (SoS) and was generalized by Edelsbrunner and M#cke in [EM90] Their implementation of a three and four dimensional convex hull algorithm was the basis for our own program. More recently, SoS was implemented in conjunction with three dimensional ....

H. Edelsbrunner and R. Waupotitsch. Computing a ham-sandwich cut in two dimensions. J. Symbolic Comput., 2:171178, 1986.

....d 2 log n) 1 ff (ds d 2 log n)MM(d) log d) The overhead now follows from s = Omega (log n) 2 6.4 Limitations Primitives that decide on the relative position of derived objects may pose a limitation to our method. Consider, for instance, the two dimensional ham sandwich algorithm in [34], where lines are the input objects and their intersection points are the derived objects. The three primitives of the algorithm are: deciding whether a point lies above or below a line; comparing the first coordinate of two points; and comparing the distances of two points from a line. Applying ....

H. Edelsbrunner and R. Waupotitsch. Computing a ham-sandwich cut in two dimensions. J. Symbolic Comput., 2:171--178, 1986.

....DEGENERACIES 7 This forces the perturbed constants to be strictly positive and eliminates the degenerate case of having b i = 0, for some i in f1; mg. Edelsbrunner and Mucke systematize in [12] a scheme called Simulation of Simplicity (SoS for short) already presented in [9] 11] [13] and [10] It applies to algorithms that accept n input objects, each specified by d parameters, and whose tests are determinants in the nd parameters, just as our deterministic perturbation (1) of the next section. SoS perturbs every input parameter p i;j into p i;j (ffl) p i;j ffl 2 ....

H. Edelsbrunner and R. Waupotitsch, Computing a ham-sandwich cut in two dimensions, J. Symb. Comput., 2 (1986), pp. 171--178.

....used to solve degenerate linear programs. This leads to the implementation of the simplex algorithm referred to as the lexicographical method (see [Ch52] DOW55] Da63] or [Ch83] for details) In computational geometry, this technique has been used in a couple of papers, including [Ed86] and [EW86], to avoid the otherwise necessary Simulation of Simplicity 3 discussion of degenerate cases. This paper presents the theoretical foundations of SoS as well as details of its implementation. The basic idea of SoS is to perturb the given objects slightly which amounts to changing the numbers that ....

H. Edelsbrunner and R. Waupotitsch. Computing a Ham-Sandwich Cut in Two Dimensions. Journal of Symbolic Computation, 2(2):171--178, June 1986. Simulation of Simplicity 32

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H. Edelsbrunner and R. Waupotitisch. Computing a ham sandwich cut in two dimensions. J. Symbolic Comput. , 2:171--178, 1986.

No context found.

H. Edelsbrunner and R. Waupotitisch. Computing a ham sandwich cut in two dimensions. Journal of Symbolic Computing, 2:171--178, 1986.

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Edelsbrunner, H., and Waupotitsch, R. (1986). Computing a ham-sandwich cut in two dimensions. J. Symb. Comput. 2 171-178.

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