A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. J. Algorithms, 12:38--56, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 ....

....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 study and improveknown algorithms for these problems. ....

[Article contains additional citation context not shown here]

A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. J. Algorithms, 12:38--56, 1991.

....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 ....

....to prove that the minimum k gon can be found in a small set of points, the nearest vertical neighbors of a segment determined by two of the input points. The k gon can then be found by applying the algorithm of Eppstein et al. to these small sets. This is similar to the approach of Aggarwal et al. [1], who find small sets for minimum perimeter problems using high order Voronoi diagrams. We also solve the related problem of finding a set of k points with minimum area convex hull, in time O(n ) We again use nearest vertical neighbors, but fewer of them, and the proof no longer needs ....

A. Aggarwal, H. Imai, N. Katoh and S. Suri. Finding k points with minimum diameter and related problems. 5th ACM Symp. Comput. Geom. (1989) 283--291.

....for this problem, where a horizontal 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 ....

A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. J. Algorithms, 12:38--56, 1991.

....z is in T 1 then T e (z,y) is better than T e (x, y) But T e (x, y) is the best spanning tree containing (x, y) therefore if (x, y)isinanyofthek best spanning trees, there can be at most k 1 possible sites z, and C contains at most k 1 sites including x and y. # Lemma 13. [1, 2] There are O(rn) pairs of sites (x, y) that belong to a common region of an order r Voronoi diagram. The diagram can be found, and all such pairs can be enumerated, in time O(r 2 n n log n) Theorem 3. The k best spanning trees for a set of points in the plane can be found in time O(k 2 n ....

....same as in lemma 17 and theorem 4, using the bound of lemma 18 instead of that of lemma 16. # Acknowledgements This research was performed at the Xerox Palo Alto Research Center. I would like to thank Frances Yao for suggesting the Euclidean version of this problem and directing me to reference [2]. ....

A. Aggarwal, H. Imai, N. Katoh, and S. Suri, Finding k Points with Minimum Diameter and Related Problems, J. Algorithms, to appear.

....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 problems for selecting k ....

.... 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 ....

[Article contains additional citation context not shown here]

A. Aggarwal, H. Imai, N. Katoh, and Subhash Suri. Finding k points with minimum diameter and related problems. J. Algorithms, 12:38-- 56, 1991.

....New York University x Faculty of Computer Science, The Technion; Current address: School of Mathematical Sciences, Tel Aviv University. 1 Introduction 2 that optimizes some cost function, among all possible k subsets. This problem was studied for a variety of cost functions. Aggarwal et al. [3] solve this problem when the parameter to be optimized is the diameter of the k subset (in time O(k 2:5 n log k n log n) the variance of the k subset (in time O(k 2 n n log n) the size of an axis parallel enclosing square, or the perimeter of an axis parallel enclosing rectangle (both ....

A. Aggarwal, H. Imai, N. Katoh, and S. Suri, Finding k points with minimum diameter and related problems, J. Algorithms, 12:38--56, 1991.

....problems. 1 Introduction Smallest enclosing circles. We begin by discussing the following geometric problem: Given a set P of n points in the plane and an integer q, find the smallest circle enclosing at least q points of P . This problem has recently been investigated in several papers [AIKS91] EE93] DLSS93] ESZ93] Mat93b] and it motivated also this paper. In these previous works, algorithms were obtained with roughly O(nq) running time 1 . There seems to be little hope at present at improving these bounds substantially in the full range of values of q. In particular, as ....

.... 1 this question was raised e.g. by Efrat et al. ESZ93] One of the first methods coming to mind for solving the problem is to construct the qth order Voronoi diagram for the point set P , and find the required circle by inspecting all its cells (this approach was pointed out by Aggarwal et al. AIKS91] It is known that the combinatorial complexity of the qth order Voronoi diagram is Theta( n Gamma q)q) and it can be constructed in time O(n (n Gamma q)q) see [Cla87] AM91] In our setting, this says that the smallest circle enclosing all but k points can be found in O(n 1 (k ....

A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. J. Algorithms, 12:38--56, 1991. 17

.... a fringe benefit, our algorithm slightly improves on the best solution known for a special case of another problem in the family: for n input points instead of rectangles, a minimum perimeter rectangle that contains k points can be found with higher order Voronoi diagrams in time O(k 2 n log n) AIKS89] whereas our algorithm solves this problem in time O(n 3 ) an improvement at least for k = Theta(n) We describe the enclosing problem formally in chapter 2 where we also explain the most important ideas of our solution. In chapter 3 we propose an efficient algorithm for the problem and ....

A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. Proc. of the 5th Annual ACM Symposium on Computational Geometry, 5:283--291, 1989.

....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 minimum area polygons. For the case k = 3 the problem asks for the minimum area empty triangle. An O(n 2 ) time and O(n 2 ) ....

....technique is generalized to finding convex k gons (and solving the other two problems) that minimize or maximize some general weight criterion. In this way we obtain solutions to e.g. the minimum perimeter problem with the same time bounds as stated above, which is better than previous solutions [2] for large k. Another application finds the convex k gon containing the smallest or largest number of points. Finally, in Section 6 we give some concluding remarks and directions for further research. The following notations will be used throughout this paper. By l(p 1 ,p 2 ) we denote the ....

A. Aggarwal, H. Imai, N. Katoh and S. Suri, Finding k points with minimum diameter and related problems, Proc. 5th ACM Symp. on Computational Geometry, 1989, pp. 283--291.

....We now consider a different sort of clustering problem, in which we seek the best single cluster containing k points of S, rather than the best partition of S into k groups. If the quality measure is radius or diameter (or area or perimeter of convex hull) this problem is polynomially solvable [AIKS91, DLSS93, EE94] We note in passing that diameter in IR d for d 2 is still open. Another reasonable quality measure, however, leads to an NP hard problem. The Euclidean k minimum spanning tree problem measures the cost of interconnecting the cluster: given n points S in the plane, find k ....

A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k-points with minimum diameter and related problems. J. Algorithms, 12:38--56, 1991.

....) expected time. 1 Introduction The notion of k levels [2, 20, 26, 34] has proved to be an important one in computational geometry, exploited directly or indirectly in various algorithms for computing higher order Voronoi diagrams [1, 11, 14] used, for instance, in finding clusters of points [3, 23]) designing data structures for halfspace range searching [16, 17] and solving hyperplane partitioning problems (such as hamsandwich cuts [32] and weak line separators [25] The concept is also fundamental in the combinatorics of arrangements (e.g. in bounding the complexity of lower envelopes ....

....improve an early algorithm of Chazelle and Edelsbrunner [14] for higher order Voronoi diagrams [2, 20, 34] which originally ran in O(n 2 log 2 n) worst case time. These diagrams are used to solve a number of problems on a planar point set, such as finding a k point subset of minimal variance [3, 23], and finding the smallest circle enclosing all but k points [33] Corollary 5.1 The order k Voronoi diagram of n points in IR 2 can be constructed in O(n 2 log 1 n) time deterministically. Proof: First, by a standard lifting map [20] the problem reduces to constructing the k level ....

A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. J. Algorithms, 12:38--56, 1991.

....all squares and rectangles are implicitly assumed to be axis parallel. Given two rectangles R 1 ; R 2 , we let R 1 R 2 denote the smallest rectangle enclosing their union. 4. 1 Minimal k Point Subsets Motivated by applications in clustering and statistical analysis, a number of researchers [6, 10, 28, 33, 35, 50] have looked at problems of the type: given an n point set P , compute a minimal k point subset. We illustrate our technique on one specific case, where the point set S is planar, and the measure of minimality is the L1 diameter (another case will be examined in Section 6.2) This particular ....

A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. J. Algorithms, 12:38--56, 1991.

....subgraph problem is the lightest induced subgraph problem where we wish to pick k vertices such that the subgraph induced is of the minimum weight. Once again this problem is NP hard (the k independent set problem can be reduced to it) the geometric version of this problem has been studied in [1] and the graph version in [11] 3 Lower bounds An important issue in the design of approximation algorithms for optimization problems is that of the bounding technique used. For the purpose of analysing the algorithm we need to compare the cost of its solution to the optimum. But since, computing ....

A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. Journal of Algorithms, 12(1):38--56, 1991.

....of the points of S, for some suitable constant c. They show that the optimal k point subset is contained in the set of ck points corresponding to a region in this diagram. Hence, for each of the O(kn) regions, they apply their first algorithm to the corresponding subset of size ck. Aggarwal et al. [5] improved this technique, by showing that it suffices to consider only O(n) regions of the (ck) order Voronoi diagram. In [5] also closeness measures such as the diameter, enclosing square, or perimeter of the enclosing rectangle are considered. In Smid [122] a simple plane sweep algorithm is ....

....corresponding to a region in this diagram. Hence, for each of the O(kn) regions, they apply their first algorithm to the corresponding subset of size ck. Aggarwal et al. 5] improved this technique, by showing that it suffices to consider only O(n) regions of the (ck) order Voronoi diagram. In [5], also closeness measures such as the diameter, enclosing square, or perimeter of the enclosing rectangle are considered. In Smid [122] a simple plane sweep algorithm is given for finding the k points in a set of n planar points whose enclosing axes parallel square is minimal. Efrat, Sharir and ....

[Article contains additional citation context not shown here]

A. Aggarwal, H. Imai, N. Katoh and S. Suri. Finding k points with minimum diameter and related problems. Journal of Algorithms 12 (1991), pp. 38--56.

....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 ....

....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 study and improve known algorithms for these problems. ....

[Article contains additional citation context not shown here]

A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. J. Algorithms, 12:38--56, 1991.

....follows, all squares and rectangles are implicitly assumed to be axis parallel. Given two rectangles R 1 ; R 2 , we let R 1 R 2 denote the smallest rectangle enclosing their union. Minimal k point subsets. Motivated by applications in clustering and statistical analysis, a number of researchers [5, 7, 22, 26, 28, 40] have recently looked at problems of the type: given an n point set P , select a minimal k point subset. We illustrate our technique on one specific case, where the point set S is planar, and the measure of minimality is the L1 diameter (another case will be mentioned in Section 7) This ....

A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. J. Algorithms, 12:38--56, 1991.

....p of facilities on the nodes of G, with at most one facility per node, so as to minimize some measure of the distances between facilities. This problem has been studied for both diameter and sum objectives [RKM 93] Some geometric versions of this problem have also been studied [AI 91] Consider the following extension of the compact location problem. Suppose we are given two weight functions ffi c ; ffi d on the edges of the network. Let the first weight function ffi c represent the cost of constructing an edge, and let the second weight function ffi d represent the actual ....

A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with Minimum Diameter and Related Problems. J. Algorithms, 12(1):38--56, March 1991.

....j=1 0 2(s j ) 2 2(s j 0 t j ) 2 1 s 0 2 d X j=1 (s j ) 2 = d X j=1 i (2k 0 n)x 2 j ( S 1 ) 0 2x j ( S 1 ) 2 j : Since S 1 is optimal to MINSUM( k) T 3 is also optimal to MINSUM( k) Thus, the theorem follows. 2 8 This theorem is similar to the one given in [1], and can be seen as a special case of more general theorems in [15] If is given, the parametric problem MINSUM(k; can be solved in a greedy manner by choosing k smallest values among P d j=1 0 (2k 0 n)x 2 ij 0 4 j x ij 1 (i = 1; n) Let us first consider the case of k 6= n=2. ....

A. Aggarwal, H. Imai, N. Katoh and S. Suri, Finding k points with minimum diameter and related problems, Proceedings of the 5th Annual ACM Symposium on Computational Geometry, 1989, pp.283--291.

....perimeter enclosing polygon) of the subset has been studied in the literature [2, 3, 6] However, the authors know no previous results on computing subsets maximizing other minimum structures. Problems of nding subsets minimizing the minimum weight of a combinatorial structure are more common [1, 9, 23, 13]. In particular, the problem of nding the k set minimizing the weight of the minimum MST was recently studied by Ravi et al. 23] who proved NP hardness and gave the rst approximations. The performance ratios have recently been improved to the best possible 3 for general graphs [13] and 1 ....

A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k Points with Minimum Diameter and Related Problems. J. Algorithms 12 (1991), 3856.

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A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. J. Algorithms, 12:38--56, 1991.

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A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. J. Algo., 12:38--56, 1991.

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A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. Journal of Algorithms, 12:38--56, 1991.

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A. Aggarwal, H. Imai, N. Katoh, and S. Suri, Finding k points with Minimum Diameter and Related Problems. J. Algorithms, 12(1):38--56, March 1991.

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A.Aggarwal, H.Imai, N.Katoh, S.Suri. Finding k Points with minimum diameter and related problems. Journal of Algorithms, 12 (1991), 38--56.

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A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with Minimum Diameter and Related Problems. J. Algorithms, 12(1):38--56, March 1991.

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