David Eppstein and Jeff Erickson. Iterated nearest neighbors and finding minimal polytypes. Discrete and Computational Geometry, 11:321--350, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....See [91, 95, 97, 121, 212] for other variants of the 1 center problem. A natural extension of the 1 center problem is to find a disk of the smallest radius that contains k of the n input points. The best known deterministic algorithm runs in time O(n log n nk log k) using O(n k log k) space [116, 84] (see also [111] and the best known randomized algorithm runs in O(n log n nk) expected time using O(nk) space, or in O(n log n nk log k) expected time using O(n) space [203] Matousek [204] also showed that the smallest disk covering all but k points can be computed in time O(n log n ....

D. Eppstein and J. Erickson, Iterated nearest neighbors and finding minimal polytopes, Discrete Comput. Geom., 11 (1994), 321--350.

.... relies on the observation (see Proposition 12 in the Appendix) that the cost of the optimum solution to k TSP is at most 2 # k times larger than the size of the smallest square containing at least k points (in # d it is at most dk 1 1 d times the size of such a cube) Eppstein and Erickson [21] show how to approximate the size of this square within a factor 2 by computing for each of the n nodes its k nearest neighbors. This takes O(nk log n) time 5 . Once we have an approximation A such that OPT # A # 2 # k OPT , we place a grid of granularity A 16ck 3 2 and move each node to ....

D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. Discrete Comp. Geo., 11: 321-350, 1994.

....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 ) running time is advantageous for ....

D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. Discrete Comput. Geom., 11:321--350, 1994.

....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 points out of n use as optimization ....

.... 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 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] ....

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D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. Discrete Comput. Geom., 11:321--350, 1994.

....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 observed by ....

....No. 351. Part of this research was performed while the author was visiting at Computer Science Institute, Free University Berlin, and it was supported by the German Israeli Foundation of Scientific Research and Development (G.I.F. 1 Here are the specific running times: Eppstein and Erickson [EE93] solve the problem in O(nq log q n log n) time with O(n log n nq q 2 log q) space, and Datta et al. DLSS93] give an algorithm with the same running time and space improved to O(n q 2 log q) Efrat et al. ESZ93] achieve O(nq log 2 n) time with O(nq) space or alternatively O(nq log 2 n ....

D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. In Proc. 4th ACM-SIAM Sympos. Discrete Algorithms, pages 64--73, 1993.

....They also describe a variant with O(nk log 2 n log(n=k) running time requiring only O(n log n) space. After a preliminary version of this note was prepared, the author learned (by the kindness of A. Efrat) about two other recent works on the considered problem. Eppstein and Erickson [EE93] solve the problem in O(n log n kn log k) time with O(n log n kn k 2 log k) space, and Datta et al. DLSS93] give an algorithm with the same running time and space improved to O(n k 2 log k) The algorithm given below is still slightly better and hopefully simpler. On the other hand, the ....

D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. In Proc. 4th ACM-SIAM Sympos. Discrete Algorithms, pages 64--73, 1993.

....our rounding procedure will ensure that the size of the bounding box is poly(n) Delta dmin . First it computes in nearly linear time a crude approximation to OPT that is correct within a factor n. Algorithms to compute such approximations were known before our work; see [27] for matching and [9] for k MST and k TSP. The latter algorithm runs in O(nk log n) time. Let A OPT Delta n be this approximation. The procedure lays a grid of granularity A=8cn 2 in the plane and moves every node to its nearest gridpoint. Now dmin A=8cn 2 , and triangle inequality implies that cost of the ....

D. Eppstein and J. Erickson. Iterated Nearest Neighbors and Finding Minimal Polytopes. Discrete Comp. Geo., 11: 321-350, 1994.

....in the solution of several geometric problems. 1 Introduction One set of problems that has received considerable attention in the computational geometry literature is the placement of a polygon (or a polygonal annulus region) so that it contains a given point set (or a subset of it) see, e.g. [BBD, BBDG, BDP, BDG, DGR, DS, EE, ESZ, GMR]. Problem variants include placement of a polygon by translation only, placement by translation and rotation, or placement allowing some geometric transformation (such as scaling, offsetting, or perspective transformations) Optimization variants of the problem include maximization of the number ....

D. Eppstein and J. Erickson, Iterated nearest neighbors and finding minimal polytopes, Discrete & Computational Geometry, 11 (1994), 321--350.

....in Section 4. Consider now the Smallest k point Clustering (SC k ) problem: given a set P of n points in R d and 1 k n, find a subset of P of size k that is contained in a d dimensional L t sphere of smallest radius. The SC k problem for d 2 has been studied by various researchers [3, 11, 14]. In [11] the most recent of these, the authors partition the original problem into O(n=k) subproblems, and each subproblem is solved in O(k d=2 log 2 k) time (the authors treat d as a constant) This time is prohibitive even for moderate d. Previous results suggest that the SC k problem is ....

D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. Discrete Comput. Geom., 11:321--350, 1994.

....See [78, 82, 84, 102, 182] for other variants of the 1 center problem. A natural extension of the 1 center problem is to find a disk of the smallest radius that contains k of the n input points. The best known deterministic algorithm runs in time O(n log n nk log k) using O(n k 2 log k) space [100, 72] (see also [96] and the best known randomized algorithm runs in O(n log n nk) expected time using O(nk) space, or in O(n log n nk log k) expected time using O(n) space [174] Matousek [175] also showed that the smallest disk covering all but k points can be computed in time 2 O(n log n ....

D. Eppstein and J. Erickson, Iterated nearest neighbors and finding minimal polytopes, Discrete Comput. Geom., 11 (1994), 321--350.

.... relies on the observation (see Proposition 12 in the Appendix) that the cost of the optimum solution to k TSP is at most 2 # k times larger than the size of the smallest square containing at least k points (in # d it is at most dk 1 1 d times the size of such a cube) Eppstein and Erickson [20] show how to approximate the size of this square within a factor 2 by computing for each of the n nodes its k nearest neighbors. This takes O(nk log n) time 5 . Once we have an approximation A such that OPT # A # 2 # k OPT , we place a grid of granularity A 16ck 3 2 and move each node to ....

D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. Discrete Comp. Geo., 11: 321-350, 1994.

....is to find a disk of the smallest radius that contains k of the n input points. The best known randomized algorithm runs in O(n log n nk) expected time using O(nk) space, or in O(n log n nk log k) expected time using O(n) space [141] The best known deterministic algorithms are somewhat slower [62, 77, 74]. Matousek [142] also showed that the smallest disk covering all but k points can be computed in time 3 O(n log n k 3 n ) Chan [38] presented a randomized algorithm for computing the discrete 1 center in R 3 whose expected running time is O(n log n) There are several other extensions ....

D. Eppstein and J. Erickson, Iterated nearest neighbors and finding minimal polytopes, Discrete Comput. Geom., 11 (1994), 321--350.

....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. Eppstein and J. Erickson, "Iterated Nearest Neighbors and Finding Minimal Polytopes ", Proceedings of the 4th Annual ACM-SIAM Syposium on Discrete Algorithms, (1993), pp. 64-73.

....dimensions. These algorithms are based on the construction of p th order Voronoi diagrams [Lee82, PS85] Problems involving the placement of p facilities so as to minimize suitable cost measures have been studied extensively in the literature (see [BE95, AI 91, DL 93, KP93, RKM 93, EE94] and the references cited therein) These problems can roughly be divided into two main categories. The first category of problems involves selecting a set of p facilities so as to minimize (or maximize) the cost (distance) from the unselected sites to the selected sites. Problems that can be ....

J. Erickson and D. Eppstein, Iterated nearest neighbors and finding minimal polytopes. Discrete and Computational Geometry, (11) 1994, pp. 321--350.

....optimum set must be maintained as new points are inserted. Our methods further generalize to higher dimensional versions of these problems. Our techniques apply to all of the problems cited above. Portions of this paper were presented at the 3rd and 4th ACM SIAM Symposia on Discrete Algorithms [17, 19]. This paper includes work done while Jeff Erickson was at the Universityof California at Irvine. David Eppstein s researchwas partially supported by NSF grant CCR 9258335. Department of Information and Computer Science, University of California, Irvine, CA 92717 Computer Science Division, ....

D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. In 4th ACM-SIAM Symp. Discrete Algorithms, pages 64--73, 1992.

....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 points with the ....

D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. Disc. and Comp. Geometry, 11:321--350, 1994.

....) the k nearest neighbors graph of V , by introducing k edges from a vertex to its k nearest neighbors. In any constant dimension #, one can compute k NNG(V ) in time O(kn log n) 16] or even O(kn n log n) 4, 5, 9] The k nearest neighbors graphs are useful for certain clustering problems [12]. However at present they have not been studied extensively, and few of their combinatorial properties are known. 3 Monotone Logical Grid Boris [3] proposed a data structure, called the Monotone Logical Grid (MLG) as a way of storing and indexing a set of points in R # for n body simulation. ....

D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. Discrete & Computational Geometry 11 (1994) 321--350.

....undergoes update operations such as insertions or deletions of single points. Dynamic algorithms have been studied for many geometric optimization problems, including closest pairs [7, 23, 25, 26] diameter [7, 26] width [4] convex hulls [15, 22] linear programming [2, 9, 18] smallest k gons [6, 11], and minimum spanning trees (MSTs) 8] Many of these algorithms suffer under a restriction that, if deletions are allowed at all, they may only occur at certain prespecified times the algorithms are not fully dynamic. A number of other papers have considered dynamic computational geometry ....

D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. In Proc. 4th Symp. Discrete Algorithms, pages 64--73, 1993.

....problem in lower dimensions (e.g. pointlocation [31, 30] or computing the area of rectangles [29] or for a di#erent problem (e.g. convex layers [6] or hidden surface removal [4] As a result, a lot of attention has been paid to studying dynamic geometric algorithms. For some examples see [3, 4, 16, 18, 20, 23, 28, 32, 33, 34], or see [11] for an excellent survey on this subject. Researchers have also studied the special cases when objects are only allowed to be inserted (deletions are not allowed) or when the sequence of insertions and deletions is known in advance. If the problem is decomposable (i.e. the solutions ....

D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. Manuscript, 1992.

....2 and 4. Problem 3 is a generalization of the well known nearest neighbors problem. For classification problems, it is more robust than a simple nearest neighbors search. The graph of k nearest neighbors to each point has certain interesting theoretical properties [18, 19] Eppstein and Erickson [13] showed how a variety of clustering problems such as those of finding k points with minimum diameter, circumradius, or variance, could all be solved efficiently using algorithms for Problem 3 as a subroutine, improving previous techniques based on kth order Voronoi diagrams. Problem 3 has also ....

....improved that result to O(k 2 n n log n) time. Dickerson et al. 11] presented an asymptotically faster algorithm requiring O(n log n kn log k) time for the planer case; as with their solution to Problems 2 and 1, it searches the standard Delaunay triangulation. Eppstein and Erickson [13] solved the planar problem for the simpler L # metric in time O(n log n kn) Once again, however, these approaches were not efficient in higher dimensions. Vaidya [26] gives an alternate approach based on a modified form of quadtrees; his algorithm works in any dimension and requires O(kn ....

D. Eppstein and J. Erickson, "Iterated nearest neighbors and finding minimal polytopes", Proc. 4th Annual ACM-SIAM Symposium on Discrete Algorithms, Jan. 1993, 64-- 73.

....metric is also not critical. However we will use the k MST formulation for simplicity. The k MST problem was introduced independently by Zelikovsky and Lozevanu [116] and by Ravi et al. 97] Many similar k point selection problems with other optimization criteria can be solved in polynomial time [42, 58] but the k MST problem is NP complete [97, 116] as are obviously the k TSP and k Steiner tree variants) so one must resort to some form of approximation. In a sequence of many papers, the approximation ratio was reduces to O(k 1 4 ) 97] O(log k) 64, 87] O(log k log log n) 54] O(1) 21] ....

D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. Disc. Comp. Geom., vol. 11, 1994, pp. 321--350.

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David Eppstein and Jeff Erickson. Iterated nearest neighbors and finding minimal polytypes. Discrete and Computational Geometry, 11:321--350, 1994.

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D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. Discrete and Computational Geometry, 11(3):321--350, 1994.

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David Eppstein and Jeff Erickson. Iterated nearest neighbors and finding minimal polytypes. Discrete and Computational Geometry, 11:321--350, 1994.

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D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. Discrete & Computational Geometry, 11: 321--350, 1994.

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