Amitava Datta, Hans-Peter Lenhof, Christian Schwarz, and Michiel H. M. Smid. Static and dynamic algorithms for k-point clustering problems. J. Algorithms, 19:474--503, 1995.  Home/Search   Document Not in Database   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 ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid, Static and dynamic algorithms for k-point clustering problems, J. Algorithms, 19 (1995), 474--503.

....factor of (15 33) 48(2 3) 1=19:35. While the analysis that yields this factor becomes more involved, the algorithm remains simple. Both algorithms rst determine the smallest diameter D 3 of any three point subset of the input points. This can be done in O(n log n) time [EE94, DLSS95] D 3 is needed to compute the diameter of the labels, which in both cases is a constant fraction of D 3 . The observation that no point set of more than two points can be labeled with circles of diameter greater than 3)D 3 then yields the corresponding approximation factors. Like the ....

....can label x arbitrarily. r g Lemma 4. The algorithm can be implemented such that it labels a set S of n points in O(n log n) time with linear space. Proof. Our algorithm labels S in three phases. In the rst phase, we compute D 3 in O(n log n) time with one of the algorithms suggested in [EE94, DLSS95] We need D 3 to compute the diameter d algo = 15 33) 24 D 3 0:39 D 3 of the labels we are going to place, see Section 4. In the second phase, we set up a simple data structure, which will answer a limited closest pair query; limited in the sense that we only need to know pairs of points ....

Amitava Datta, Hans-Peter Lenhof, Christian Schwarz, and Michiel H. M. Smid. Static and dynamic algorithms for k-point clustering problems. J. Algorithms, 19:474-503, 1995.

....O(n log k k) expected time. This was the motivating example considered by Matousek [39] Interestingly, as we now know, the problem is easier than finding the smallest circle enclosing k points, which has clusteringrelated applications and is believed instead to have complexity near (nk) [23, 30, 38]. minimum area annulus enclosing all but k points in k) expected time. The problem of finding all local minima of the k level and the ( k) level has been explored in several papers. By specializing our 2 d algorithm to the feasible case, we can find all O(k ) local minima of the ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. J. Algorithms, 19:474--503, 1995.

....decision tree model of computation. In fact, we can improve the performance of our variance algorithm by substituting the new neighbor algorithm for Vaidya s. The new algorithm is faster than the one described by Aggarwal et al. 2] for all values of k. Very recently, Datta et al. [12]developed new algorithms for each of the problems we discuss in Section 5, except variance, using a searchtechnique developed by Lenhof and Smid [27] that does not require the computation of nearest neighbors and follows the algebraic decision tree model. In the plane, their algorithms use the ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. Report MPI-I-93-108, Max-Planck-Institut Inform., Saarbrucken, Germany,1993. To appear in Proc. 3nd Workshop Algorithms Data Struct.

....k ) 2 cannot be the optimal solution value for the problem. As any two sites in C are at least D1 (S) distance away, we only need to consider a constant number (24) of points in S which are the closest to p i . Overall, for all p i this can be computed in O(n log n) time using standard techniques [8]. For each such point p j , we simply measure d1 (p i ; p j ) and d min (p i ; p j ) If d1 (p i ; p j ) 2 or d min (p i ; p j ) is out of the range [D1 (S) 2; D1 (S) then throw it away as a valid candidate. Eventually we have at most 2 Theta 24n 1 = O(n) number of candidates. We sort them ....

A. Datta, II.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems, In Proc. 3rd Worksh. Algorithms and Data Structures, springer-verlag, LNCS 709, pages 265-276, 1993.

....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 algorithm whose ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. J. Algorithms, 19:474--503, 1995.

....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 computing the smallest perimeter, the best known running time is O(n log n k 2 n) for algorithms by Datta et al. [DLSS95] and by Eppstein and Erickson [EE94] The algorithm of Aggarwal et al. AIKS91] can be used for both variants of the problem, the area and the perimeter, and takes time O(min(k 2 n log n, n 3 ) while Segal and Kedem s solution [SK98] for both variants runs in O(n k(n k) 2 ) time and ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. J. Algorithms, 19:474--503, 1995.

....circle problem has recently been studied independently, by several other researchers using different techniques. Eppstein and Erickson Introduction 3 [11] Solve this problem, among related family of problems in O(nk log k n log n) time with O(n log n nk k 2 log k) space, and Datta et al. [7] give an algorithm with the same running time and space improved to O(n k 2 log k) After the first appearance of our paper, Matousek gave an (extremely simple) randomized algorithm [18] with expected time complexity O(n log n nk) time using O(nk) space or expected time O(n log n nk log k) ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. In Proc. 3rd Workshop Algorithms Data Struct., Lecture Notes in Computer Science, 1993.

.... e, and a parameter k n, find the smallest value of d for which there exists a copy e 0 of e and a subset S 0 S of cardinality k, such that S 0 H(e 0 ; d) This problem is a natural extension of the well studied problem of computing the smallest disc enclosing k points of S; see [10, 12, 14, 21]. A recent attack on this problem is given in [2] See also [13] ffl The segment center problem is actually a special case of the more general problem of computing the one directional Hausdorff distance, under euclidean motion, between two sets of objects, which can be stated as follows (see ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. In Proc. 3rd Workshop Algorithms Data Struct., Lecture Notes in Computer Science, 1993, 265--276..

....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 D. ....

....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 log(n=q) time with O(n log n) space, and the author [Mat93b] has O(n log n nq) time with O(nq) space or time ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. In Proc. 3rd Workshop Algorithms Data Struct., Lecture Notes in Computer Science, 1993.

....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 latter authors also consider various generalizations and also a dynamic version of the problem, none of which will ....

....complexity of the above mentioned algorithms somewhat, and our algorithms are quite simple, in particular the one with linear space. Let us remark that some ingredients of our technique can most likely be also applied to improve and or simplify some of the more general results of Datta et al. [DLSS93]. With the current approach, it seems unlikely to get a running time substantially below O(nk) for the considered problem. Determining the actual complexity remains a challenging open problem. Let us begin the exposition of the algorithm by few definitions. For a point p in the plane, let B(p; ....

[Article contains additional citation context not shown here]

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. In Proc. 3rd Workshop Algorithms Data Struct., Lecture Notes in Computer Science, 1993. 5

....is to nd 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 [135] The best known deterministic algorithms are somewhat slower [58, 73, 70]. Geometric Optimization September 7, 2000 Facility Location Problems 16 Matou sek [136] 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 [36] presented a randomized algorithm for computing the discrete 1 center in R 3 whose ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid, Static and dynamic algorithms for k-point clustering problems, J. Algorithms, 19 (1995), 474-503.

....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 ....

....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 probably ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. J. Algorithms, 19:474--503, 1995.

....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 ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid, Static and dynamic algorithms for k-point clustering problems, J. Algorithms, 19 (1995), 474--503.

....to determine such an upper bound before they can actually place labels whose size depends on this upper bound. Zhu and Poon use D 3;1 , the minimumover the diameters of all 3 subsets of the input points, as an upper bound for the maximum label size. D 3;1 can be 3 computed in O(n log n) time [DLSS95] Instead of computing D 3;1 we preprocess the input by computing relevant adjacency information, i.e. for each input point we nd a constant number of rectilinear nearest neighbors (in the L1 metric) For this task a simple O(n log n) algorithm is known [For92] Of course D 3;1 can be computed ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. J. Algorithms, 19:474-503, 1995.

....have to determine such an upper bound before they can actually place labels whose size depends on this upper bound. Zhu and Poon use D 3;1 , the minimum over the diameters of all 3 subsets of the input points, as an upper bound for the maximum label size. D 3;1 can be computed in O(n log n) time [5]. Instead of computing D 3;1 we preprocess the input by computing relevant adjacency information, i.e. for each input point we nd a constant number of rectilinear nearest neighbors (in the L1 metric) For this task a simple O(n log n) algorithm is known [10] Of course D 3;1 can be computed in ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. J. Algorithms, 19:474-503, 1995.

....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 ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid, Static and dynamic algorithms for k-point clustering problems, J. Algorithms, 19 (1995), 474--503.

....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 points ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. In Proc. 3rd Workshop on Algorithms and Data Structures, pages 265--276. LNCS 709, Springer-Verlag, 1993.

....the points in the set P . The underlying strategy for their algorithms is that a k point subset satisfying any of the closeness measures must be in the set of nearest neighbours for a particular point. The best known sequential algorithms for many of these problems were presented by Datta et al. [11]. They improved upon the strategy of Eppstein and Erickson [14] by imposing a grid on the point set and solving the problems for smaller subsets of points. The applications in pattern recognition and cluster analysis typically involve large sets of data. Also, in many cases, it is important to ....

.... In the first problem, we compute a minimum L1 diameter k 2 point subset (or the minimum side length square) In the second problem, we compute the minimum L1 perimeter k point subset (or the minimum perimeter rectangle) The best known sequential algorithms for these two problems were given in [11]. The sequential algorithm for the minimum L1 diameter k point subset problem runs in O(n log n n log 2 k) time. We present a parallel algorithm which runs in O(log 2 n log 2 k log log k log k) time and O(n log 2 n) work. So, the work done by our algorithm (processortime product) matches ....

[Article contains additional citation context not shown here]

A. Datta, H. P. Lenhof, C. Schwarz and M. Smid. "Static and dynamic algorithms for k-point clustering problems", Proc. Workshop on Algorithms and Data Structures (WADS), Lecture Notes in Computer Science, Vol. 709, (1993), pp. 265-276. A full version will appear in Journal of Algorithms.

....and this does indeed turn out to be the case. Note that the floor function was only used to compute the grid box containing a given point. Therefore, we will modify the algorithm Theorem 4.1 by using a degraded grid for which we only need algebraic functions. The method we use already appears in [9] and [11] We sketch the structure here and refer to these papers or [18] for details. Consider a standard grid of mesh size #. Fixing the origin as a lattice point, we divide the space into slabs of width # in each dimension. Since we can identify a slab using the floor function, this gives rise ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid, Static and dynamic algorithms for k-point clustering problems, J. Algorithms, 19 (1995), pp. 474--503.

....an algorithm that does not use the floor function. Note that this function was only used to compute the grid box containing a given point. Therefore, we will modify the algorithm of Theorem 4 by using a degraded grid for which we only need algebraic functions. The method we use already appears in [DLSS93] and [GRSS93] We sketch the structure here and refer to these papers or [Sch93] for details. Consider a standard grid of mesh size ffi . Fixing the origin as a lattice point, we divide the space into slabs of width ffi in each dimension. Since we can identify a slab using the floor function, ....

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. In Proc. 3rd Workshop on Algorithms and Data Structures, volume 709 of Lecture Notes in Computer Science, pages 265--276. Springer-Verlag, 1993.

No context found.

Amitava Datta, Hans-Peter Lenhof, Christian Schwarz, and Michiel H. M. Smid. Static and dynamic algorithms for k-point clustering problems. J. Algorithms, 19:474--503, 1995.

No context found.

A. Datta, H.-P. Lenhof, C. Schwarz, and M. Smid. Static and dynamic algorithms for k-point clustering problems. J. Algorithms, 19:474-503, 1995.

No context found.

A. Datta, H. P. Lenhof, C. Schwarz, M. Smid. Static and dynamic algorithms for k-point clustering problems. Lecture Notes in Computer Science, vol. 70, Springer, Berlin, 1993, 265--276.

No context found.

Amitava Datta, Hans-Peter Lenhof, Christian Schwarz, and Michiel H. M. Smid. Static and dynamic algorithms for k-point clustering problems. J. Algorithms, 19:474-503, 1995.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC