Edelsbrunner, H. and Guibas, L. J., "Topologically sweeping an arrangement," Journal of Computer and System Sciences, vol 38, 1989, pp. 165-194.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for computing statistical measures of data depth, both those already coded (e.g. 9] and those currently in development. The implementation is general enough to be used in place of any topological sweep subroutine in existing code. The topological sweep method of Edelsbrunner and Guibas [5] is one of the classical algorithms in Computational Geometry. It sweeps an arrangement of n planar lines in O(n ) time and O(n) space with a topological line and is a critical ingredient in several space and time efficient algorithms (e.g. 12] 6] 9] 7] The technique has been adapted ....

....is simple to compute and does not require special preprocessing. The modified algorithm was coded and the code was verified on different types and sizes of data sets. The code was incorporated in an implementation of an algorithm for statistical data analysis. 2 The Topological Sweep Algorithm [5] Let H be a arrangement of n lines in the plane. Vertical line sweep could report all intersection pairs sorted in order of x coordinate in Theta(n log n) time and O(n) space (e.g. 3] 4] If one only needs to report the intersection points of the lines according to a partial order related ....

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H. Edelsbrunner and Leonidas J. Guibas. Topologically sweeping an arrangement. J. Comput. Syst. Sci., 38:165--194, 1989. Corrigendum in 42 (1991), 249--251.

....linear time per pair (p; q) so O(n ) time overall. The space requirement is only linear. ut Remark. It is an interesting open question to determine if one can reduce the space complexity of the above algorithm to linear while maintaining the quadratic time bound (e.g. using topological sweep [4]) 5 Partitioning Polygons Now we consider the case in which S is a set of (pairwise disjoint) simple polygons with a total of n vertices. The size of S 1 (S 2 ) will now be the perimeter of the relative convex hull, rconv(S 1 ) rconv(S 2 ) of S 1 (S 2 ) with respect to the set S 2 (S 1 ) ....

....the maximum sum of the perimeters or areas (or more general functions of both) using only linear storage space, in the point partitioning case. It seems likely that one should be able to achieve a running time of O(n ) using O(n) storage, by, for example, methods of topological sweep [4]. We leave this as an open problem. Acknowledgement We wish to thank Esther Arkin, Robert Freimer, and Christine Piatko for helpful discussions. This research was partially supported by a grant from the Hughes Research Laboratories, Malibu, CA and by NSF Grants IRI 8710858 and ECSE 8857642. We ....

H. Edelsbrunner and L. J. Guibas. Topologically sweeping an arrangement. J. Comput. Syst. Sci., 38:165--194, 1989. Corrigendum in 42 (1991), 249--251.

....Theorem 1 Let P be a set of n points in the plane. Algorithm Right fi region solves the Right fi region Problem for P in O(n ) time. 6 Proof: Correctness follows from Lemma 3. The radial sorting of Step 1 can be done in O(n time by the algorithm of [18] or the alternative algorithm in [6]. Since in Step 2 each point receives a label Maybe at most once, this step requires linear time. Similarly Steps 3 and 4 require linear time. Since Steps 2, 3 and 4 are executed n times, Algorithm Right fi region has an O(n time complexity. 2 3.2 Computing Proximity Graphs We will now use ....

H. Edelsbrunner and L. J. Guibas. Topologically sweeping an arrangement. J. Comput. Syst. Sci., 38 165--194, 1989.

....low, but for now let us consider the running time of the algorithm for a single sample. Step (1) can be performed in O(n ) O(n) time. The computation of both the k and (k 1) trapezoids can be accomplished by a single application of topological plane sweep in O(r ) time and O(r) space [3]. Under our assumption that t 1=4, this is O( n) Clearly we can count the number of lines intersecting oe min in O(n) time, bounding the running time for step (3) To perform step (2) we apply the inverse dual transformation to return to primal space. Assuming that points are in general ....

H. Edelsbrunner and L. J. Guibas. Topologically sweeping an arrangement. J. Comput. Syst. Sci., 38:165--194, 1989. Corrigendum in 42 (1991), 249--251.

....The trivial method for constructing a visibility graph takes time O(nS) The first efficient algorithm was due to Lee[6] and Sharir and Schrr[9] and runs in time O(n 2 lag n) Later Asano et al. 1] and Welzl[10] gave an O(n 2) algorithm which is optimal in the worst case. Edeisbrunner and Guibas[2] iraproved the working storage of this method to O(n) Recently, Ghosh and Mount[3] designed an optimal O(m q n log n) algorithm where m is the size of Gs. Their method uses O(m) storage. Although the latest method is optimal in time, the authors themselves state that the method will be very ....

....based on the tech nique of [10] As a basic step it uses a solution to the following problem: Given a set of n points, for each point p compute the order of the other points by angle around p. This problem has already been solved in optimal O(n 2) time and O(n) storage by Edelsbrun her and Guibas[2] using dualization and topologically sweeping. We obtain the same complexity in a different way, without using dualization. The method is very simple. It took us less than 2 hours to program and the resulting code has less than 100 lines. It also handles all special cases, like multiple points on ....

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Edelsbrunner, H., and L. Guibaz, Topologically sweeping in an arrangement, Proc. 18th Symp. on Theory of Computing, 1986, 389-403.

....neighbors between lines and points: a) primal, nearest point neighbor to a line# (b) dual, vertical ray shooting in a line arrangement. minimum area triangle with s is the nearest vertical neighbor of l. This observation was used to develop O(n ) algorithms for the minimum triangle problem [7, 16, 15]. We can tighten this characterization as follows. Let 4pqr be the minimum area triangle, and assume that the vertical projection of r is between those of p and q. Then as before r is the nearest neighbor of line pq, but the vertical segment connecting r and line pq actually touches segment pq. ....

....variance O(n log n 2 Table 3. New results for finding minimum measure convex k gons, given n points in the plane. Compare Table 1. Proof: The minimum circumradius convex k point set is contained in the O(4 nearest neighbors to eachofitspoints. Edelsbrunner and Guibas [15] describe an algorithm that finds, given a set of n points, the largest (cardinality) convex subset that includes a given leftmost point, in time O(n ) and space O(n) For eachpoint p and each circumcircle containing it, rotate the points within the circle so that p is leftmost, and find the ....

H. Edelsbrunner and L. J. Guibas. Topologically sweeping an arrangement. J. Comput. Syst. Sci., 38:165--194, 1989.

....sweep about q. The result is that, for the xed choice of s, the values f(p; q; s) can be tabulated in total time O(n ) The O(n) angular sorts of points about each choice of q can be done, using the standard method of computing the arrangement of dual lines, in total time O(n ) see [3]. r r r r r r r r r r r r p i 1 p i s q Fig. 2. Notation used in recursion (2) The total number of convex polygons determined by S is obtained simply by summing: f(p; q; s) 3 Counting convex k gons Now consider the problem in which we count only convex k gons, for a ....

H. Edelsbrunner and L. Guibas, Topologically sweeping an arrangement, Journal of Computer and System Sciences 38 (1989), 165-194.

....geometric optimization problems of finding a k gon minimizing or maximizing a certain objective function. A celebrated result in this area is that a minimum area triangle can be found in time O(n by using geometric duality to transform the problem into one of searching a line arrangement [7, 8]. Algorithms are also known for optimizing other functions including minimum perimeter [1, 5, 9] and maximum perimeter and area [2, 4] For some time it remained open whether the minimum area triangle result could be generalized to finding minimum area k gons. There are actually four reasonable ....

....the area of the triangle is c d(x, l) where c is half the length of the horizontal projection of s. Therefore the point in P forming the minimum area triangle with s is the nearest vertical neighbor of l. This observation was used to develop O(n algorithms for the minimum triangle problem [7, 8]. We can tighten this characterization as follows. Let xyz be the minimum area triangle, and assume that the horizontal projection of y is between those of x and z. Then as before y is the nearest neighbor of line xz, but the vertical segment connecting y and line xz actually touches segment xz. ....

H. Edelsbrunner and L.J. Guibas. Topologically sweeping in an arrangement. 18th ACM Symp. Theory of Computing (1986) 389--403.

....that improve upon the O(n ) time bound for other types of restricted polyhedra. Thirdly, our algorithms require O(n ) and O(nk log n log log(n=k) storage, respectively. It may be possible to improve this, perhaps by using the topological line sweep algorithm of Edelsbrunner and Guibas[9]. 10 Acknowledgements The authors would like to thank David Thompson for providing a stimulating research environment. ....

Edelsbrunner, H., and L. Guibas, Topologically sweeping an arrangement. J. Comp. Sys. Sciences, 38, pp. 165--194, 1989.

....(see also left part of Figure 3) properties of the ip graph of pseudotriangulations of a given point set are reported in [42, 8, 45] A challenging question is to extend these notions and or applications to 3D environments. Source Obstacles Data structures Predicate Edelsbrunner Guibas [16] points Simple left turn Pocchiola Vegter [39] disks Splittable queues 2 ; 3 Rivi ere [43] line segments Simple left turn This paper disks Simple left turn Table 1: Optimal time and linear space visibility graph or visibility complex algorithms for obstacles with constant complexity. In this ....

....r; p; q : when traversing counterclockwise the boundary of the circle through p; q and r. Our technique, specialized to point obstacles or P. Angelier and M. Pocchiola, August 2001 3 circles with an in nitesimal radius [22] subsumes the topological sweep technique of Edelsbrunner and Guibas [16] and its extensions developed in [6, 43] This last point will be explained in a di erent paper. For a bibliography on the algorithmic of visibility graphs we refer to [39, 30, 5] PSfrag replacements b (b) R(b) L(b) b t t 0 R (b) L (b) a (a; H) a; H) R (b) L (b) R (a) L (a) ....

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H. Edelsbrunner and L. J. Guibas. Topologically sweeping an arrangement. J. Comput. Syst. Sci., 38:165-194,

....of size function may recall other techniques. For instance, in [23] Kupeev and Wolfson propose a method of estimating shape similarity based on scanning a 2D closed contour along a direction #. Also, there maybesome resemblance with the topological sweep studied by Edelsbrunner and Guibas in [7], thus leading to topics of computational geometry. In both cases, however, the di erences are greater than the similarities. Let us now consider the example in Figure 2. The set # is depicted on Figure 2 the left and the chosen measuring function is the distance of each point from the ....

Edelsbrunner, H., Guibas, L.J.: Topologically sweeping an arrangement. Journal of Computer and System Sciences ##, 165-194 (1989)

....d 2 R, the same facets of P require support, and so the total contact area, C area (d) for all such d is a constant C area (R) Thus we need to nd a face, R, of A area for which C area (R) is minimum. In [16] we show how to do this in O(n 2 ) time using a planar (topological) sweep algorithm [10]. 2.3 Minimizing volume of supports For convex P, the combinatorial structure of the supports for a direction d is determined by the back facets and by the vertex of P that is farthest away in direction d. This is the vertex touching the platform when P is built along d; we call it the extreme ....

Edelsbrunner, H. and Guibas, L. (1989). Topologically sweeping an arrangement, Journal of Computer and System Sciences, 38, 165-194.

....Simp Med computes the bivariate simplicial median of n points in O(n 4 log n) time and O(n 2 ) space or alternatively in O(n 4 ) time and space. Instead of performing step 3 in algorithm Simp Med, we can perform a topological sweep, which takes O(n 2 ) time and O(n) space for n lines [EG89] Instead of processing each segment sequentially, we essentially process all O(n 2 ) segments in parallel. Every time the sweep curve encounters an intersection point on some segment, we process the point in the same way as step 3d. Thus we have the following theorem: Theorem 7 The simplicial ....

H. Edelsbrunner and L. Guibas. Topologically sweeping an arrangement. Journal of Computer and System Sciences, 38:165-- 194, 1989.

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H. Edelsbrunner and Leonidas J. Guibas, "Topologically sweeping an arrangement," J. Comput. Syst. Sci., vol. 38, 1989, pp. 165-194.

....boundary of the half plane shadow, then P 4 can be turned off as long as none of P 1 , P 2 and P 3 senses a transition. 2. 2 Line Arrangements The arrangement of lines in the dual space and the cells they create can be computed by using the topological sweep algorithm of Edelsbrunner and Guibas [2], as modified by Rafalin, Souvaine and Streinu to deal with degeneracies [6] The details of topological sweep algorithm are beyond the scope of this paper; we only describe how the sweep results are used. A typical topological sweep algorithm computes the segments created by the intersections of ....

H. Edelsbrunner and Leonidas J. Guibas, "Topologically sweeping an arrangement," J. Comput. Syst. Sci., vol. 38, 1989, pp. 165-194.

....Of course we do want to guarantee that all events affecting any particular object happen in the correct sequence. This then would give us a local clock for each object and we would not require global synchronization between all these clocks an approach reminiscent of a topological sweep [26], but now in the time domain. ffl What if too many events happen in a short time interval and we do not have the computational resources to process them all in that interval Can we batch the processing of nearby events for efficiency Given a description of a class of motions for the objects ....

H. Edelsbrunner and L. J. Guibas. Topologically sweeping an arrangement. J. Comput. Syst. Sci., 38:165--194,

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Edelsbrunner, H. and Guibas, L. J., "Topologically sweeping an arrangement," Journal of Computer and System Sciences, vol 38, 1989, pp. 165-194.

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H. Edelsbrunner, L. Guibas, Topologically sweeping an arrangement, Journal of Computer and System Sciences 38 (1989) 165-194.

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H. Edelsbrunner and L. Guibas, Topologically sweeping an arrangement, Journal of Computer and System Sciences, 38 (1989), 165--194.

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H. Edelsbrunner and Leonidas J. Guibas. Topologically sweeping an arrangement. J. Comput. Syst. Sci., 38:165194, 1989. Corrigendum in 42 (1991), 249251.

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H. Edelsbrunner and Leonidas J. Guibas. Topologically sweeping an arrangement. In Proc. 18th Annu. ACM Sympos. Theory Comput., pages 389403, 1986.

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H. Edelsbrunner, L. Guibas, Topologically sweeping an arrangement, Journal of Computer and System Sciences 38 (1989) 165-194.

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Edelsbrunner, H. and Guibas, L.J., Topologically sweeping an arrangement, Stanford Universi- ty Tech. Rept., Stanford, CA 1985).

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H. Edelsbrunner, L. Guibas, Topologically sweeping an arrangement, J. Comp. Syst. Sc. 38 (1989), pp. 165--194.

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H. Edelsbrunner and L. J. Guibas. Topologically sweeping an arrangement. J. Comput. Syst. Sci., 38:165194,

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