DOBKIN, D. P., DRYSDALE, R. L., , AND GUIBAS, L. J. Finding smallest polygons. In Computational Geometry, F. P. Preparata, Ed., vol. 1 of Advances in Computing Research. JAI Press, 1983, pp. 181--214.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a generalization of depth peeling that we call depth picking. In this problem, we wish to select the k level of a scene, rather than just the next one, as in depth peeling. The operation of selecting a particular level in an arrangement is a key operation in a variety of geometric algorithms[2]. For example, picking the median level (k = n 2) of a set of planes has applications to problems in clustering and data fitting[17] Depth peeling can be used to solve this problem: to pick the k level, we peel off k layers of the scene, and this can be done in k passes using the two sided ....

DOBKIN, D. P., DRYSDALE, R. L., , AND GUIBAS, L. J. Finding smallest polygons. In Computational Geometry, F. P. Preparata, Ed., vol. 1 of Advances in Computing Research. JAI Press, 1983, pp. 181--214.

....a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions. 1 Introduction Anumber of recent papers have discussed problems of selecting, from a set of n points, the k points optimizing some particular criterion [2, 14, 20]. Criteria that have been studied include diameter [2] variance [2] area of the convex hull [20] convex hull perimeter [14,20] and rectilinear diameter and perimeter [2] Such problems are useful in clustering, line detection, statistical data analysis, and other geometric applications. We ....

....dimensions. 1 Introduction Anumber of recent papers have discussed problems of selecting, from a set of n points, the k points optimizing some particular criterion [2, 14, 20] Criteria that have been studied include diameter [2] variance [2] area of the convex hull [20] convex hull perimeter [14,20], and rectilinear diameter and perimeter [2] Such problems are useful in clustering, line detection, statistical data analysis, and other geometric applications. We study and improveknown algorithms for these problems. Wealsointroduce dynamic versions of these problems, in which the optimum set ....

[Article contains additional citation context not shown here]

D. P. Dobkin, R. L. Drysdale, and L. J. Guibas. Finding smallest polygons. In F. P. Preparata, editor, Computational Geometry,volume 1 of Advances in Computing Research, pages 181--214. JAI Press, 1983.

....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 ways of generalizing this: one could search for (1) a minimum area k gon, 2) a minimum area ....

D.P. Dobkin, R.L. Drysdale and L.J. Guibas. Finding smallest polygons. Adv. Computing Research, Vol. 1, JAI Press (1983) 181--214.

....cross section at height z equals the FCVD for aspect ratio z. However, we present a more direct approach which has some similarities to the computation of the smallest rectangle or polygon containing at least k of n points, see the articles by Agarwal et al. 2] Datta et al. 4] Dobkin et al. [5], or Eppstein and Erickson [6] for example. Our algorithm constructs the smallest color spanning rectangle in time O(n(n k)log 2 k) using a technique by Overmars and van Leeuwen [13] for dynamically maintainig maximal elements. We also give a simple algorithm whose O(n(n k) 2 ) ....

D. P. Dobkin, R. L. Drysdale, III, and L. J. Guibas. Finding smallest polygons. In F. P. Preparata, editor, Computational Geometry, volume 1 of Adv. Comput. Res., pages 181--214. JAI Press, London, England, 1983. 3

....optimization problems for a set S of n points have already been studied in the literature, with motivations from statistical clustering or pattern recognition. For example, the convex polygon with minimum perimeter containing k points of S can be found by using the methods of Dobkin et al. [DDG83], Aggarwal et al. AIKS91] or finally Eppstein and Erickson [EE94] the last one in time O(n log n k 3 n) The minimum area convex polygon containing k points of S can be determined in time O(n 2 log n kn 2 min(k 2 ,n) combining results of [EE94] and Eppstein et al. EORW92] Similar ....

.... 2 min(k 2 ,n) combining results of [EE94] and Eppstein et al. EORW92] Similar problems for selecting k points out of n use as optimization criterion the diameter or the variance of the k set, or they ask for the smallest circle with respect to a certain metric containing at least k points [AIKS91, DDG83, ESZ94, EORW92, Mat95, Smi92], this latter problem is of course very closely related to the Voronoi diagram of order k. Other very natural optimization criteria are the perimeter and the area of the axis parallel rectangle enclosing a k point set, these criteria are sometimes briefly called the L# perimeter and L# area. For ....

David P. Dobkin, Robert L. Drysdale, III, and Leonidas J. Guibas. Finding smallest polygons. In Franco P. Preparata, editor, Computational Geometry, volume 1 of Adv. Comput. Res., pages 181--214. JAI Press, Greenwich, Conn., 1983.

....of the following type. Given a point set P with n points, the problem is to compute a subset of k points such that some closeness measure is minimized. As an example, we may want to minimize the perimeter of the convex hull of the k point subset. This measure was considered by Dobkin et al. [13]. Aggarwal et al. [4] considered closeness measures like diameter, side length of the enclosing square, the perimeter of enclosing rectangle, etc. They gave algorithms for these problems on the plane based on higher order Voronoi diagrams. Eppstein and Erickson [14] gave a general framework for ....

D. P. Dobkin, R. L. Drysdale and L. J. Guibas. "Finding smallest polygons", In: F. P. Preparata (ed.), Advances in Computing Research, Vol. 1, Computational Geometry, J. A. I. Press, London, (1983), pp. 181-214.

....clustering and pattern recognition minimization problems tend to play a more important role than maximization problems. Minimization problems seem to be computationally harder than maximization problems in this context. Finding minimum perimeter k gons was studied by Dobkin, Drysdale and Guibas [6]. Their O(k 2 n log n k 5 n) algorithm was recently improved to O(n log n k 4 n) by Aggarwal, Imai, Katoh, and Suri [2] This recent paper also studies problems like finding minimum diameter k gons and minimum variance k gons. In this paper we will concentrate on the problem of finding ....

D.P. Dobkin, R.L. Drysdale and L.J. Guibas, Finding Smallest Polygons, In: Advances in Computing Research, Vol. 1, JAI Press, 1983, pp. 181-- 214.

....with a degree constraint on an optimal solution. This allows us to apply dynamic programming to find the exact solution. Several researchers in computational geometry have presented exact algorithms for choosing k points that minimize other objectives such as diameter, perimeter, area and volume [3, 16, 17, 18]. 1.4 Short trees Keeping the longest path in a network small is often an important consideration in network design. We investigate the problem of finding networks with small diameter. Recall that the diameter of a tree is the maximum distance (path length) between any pair of nodes in the tree. ....

D. P. Dobkin, R. L. Drysdale and L. J. Guibas, "Finding Smallest Polygons," in Advances in Computing Research, Vol. 1, JAI Press, 1983, pp 181-214.

....example, we may want to minimize the diameter of the k points, its smallest enclosing circle, it smallest enclosing axes parallel cube, etc. Clearly, different closeness measures lead to different optimal k point subsets. We mention some of the results in this area. Dobkin, Drysdale and Guibas [56] use the following technique to find a k point subset for which the perimeter of their convex hull is minimized. First, they give a polynomial time algorithm for solving this problem. Then, they give an improved algorithm that first constructs the (ck) order Voronoi diagram of the points of S, for ....

....are inefficient for large values of k. In [91] Matousek gives algorithms that are especially efficient if k is close to n. These are based on generalizing LP type problems [114] to optimization problems with k violated constraints. Eppstein and Erickson [62] improve the general framework of [56, 5]. Their main idea is to replace the expensive O(k) order Voronoi diagram by sets of O(k) nearest neighbors to each of the points of S. In this way, the number of O(k) size subsets to which an expensive algorithm is applied is reduced from O(n) in [5] to only O(n=k) The framework was further ....

D.P. Dobkin, R.L. Drysdale, III, and L.J. Guibas. Finding smallest polygons. In: Computational Geometry, Advances in Computing Research, Vol. 1, JAI Press, London, 1983, pp. 181--214.

....a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions. 1 Introduction A number of recent papers have discussed problems of selecting, from a set of n points, the k points optimizing some particular criterion [2, 14, 20]. Criteria that have been studied include diameter [2] variance [2] area of the convex hull [20] convex hull perimeter [14, 20] and rectilinear diameter and perimeter [2] Such problems are useful in clustering, line detection, statistical data analysis, and other geometric applications. We ....

.... 1 Introduction A number of recent papers have discussed problems of selecting, from a set of n points, the k points optimizing some particular criterion [2, 14, 20] Criteria that have been studied include diameter [2] variance [2] area of the convex hull [20] convex hull perimeter [14, 20], and rectilinear diameter and perimeter [2] Such problems are useful in clustering, line detection, statistical data analysis, and other geometric applications. We study and improve known algorithms for these problems. We also introduce dynamic versions of these problems, in which the optimum ....

[Article contains additional citation context not shown here]

D. P. Dobkin, R. L. Drysdale, and L. J. Guibas. Finding smallest polygons. In F. P. Preparata, editor, Computational Geometry, volume 1 of Advances in Computing Research, pages 181--214. JAI Press, 1983.

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