T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38, 1985, 293--306.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....aims to minimize the maximum diameter of all clusters while limiting the number of clusters. The diameter of cluster i is defined as the maximum distance between any pair of URLs in cluster i. It represents the worst case correlation within that cluster. We use the classical K split algorithm [22]. It is a O(NK) approximation algorithm, where N is the number of points and K is the number of clusters. It guarantees a solution within twice the optimal. The second method aims to minimize the number of clusters while limiting the maximum diameter of all clusters. This problem is NP complete, ....

T. F. Gonzalez, "Clustering to minimize the maximum intercluster distance, " Theoretical Computer Science, vol. 38, pp. 293--306, 1985.

....problem, in which we have an additional constraint that the size of each cluster should be at most c for some parameter c n=p. If p is considered as part of the input, most facility location problems are NP Hard, even in the plane or even when only an approximate solution is being sought [118, 133, 186, 216, 217, 195]. Although many of these problems can be solved in polynomial time for a fixed value of p, some of them still remain intractable. In this section we review efficient algorithms for a few specific facility location problems, to which the techniques introduced in Part I can be applied; in these ....

....p disks of radius r cover D. Feder and Green [118] showed that computing a set S of p supply points so that c(D; S) 1:822r under the Euclidean distance function, or c(D; S) 2r under the L1 metric, is NP Hard. The greedy algorithm described in Figure 3, originally proposed by Gonzalez [133] and by Hochbaum and Shmoys [154, 155] computes in O(np) time a set S of p points so that c(D; S) 2r . function procedure GREEDY COVER (D; p) D: set of n points in R for i = 1 to n do NN Dist(i) 1; for i = 1 to p do s i = d j s.t. NN Dist(j) max 1ln NN Dist(l) for j = 1 to n ....

T. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoret. Comput. Sci., 38 (1985), 293--306.

....their distance bounded below by one. This property ensures that polarity takes precedence over spatial and temporal distance in determining dissimilarity, which is important to avoiding the behavior shown in Figure 2(a) 3. 3 Clustering For clustering sinks , we adopt the K Center heuristic [15] which seeks to minimize the maximum radius (distance to the cluster center) over all clusters. K Center is just one of several potential clustering methods (e.g. bottom up matching and complete linkage) that could be used to achieve the purpose of grouping sinks with common characteristics. ....

....s i s j , pDist s i s j , b01 b0.65= 10 The remaining points are clustered to their closest seed. Let be the diameter of any set of points . For geometric instances, K Center guarantees that the maximum diameter of any cluster is within a factor of two of the optimal solution [15]. The complete description of the K Center algorithm is shown in Figure 5. Step 1 picks a random sink , then identifies the sink furthest away from , which will lie on the periphery of the data set. This step identifies as the first cluster seed, which are all contained in the set . Steps 2 5 ....

T. F. Gonzalez, "Clustering to Minimize the Maximum Intercluster Distance", Theoretical Computer Science, 38, pp. 293-306, 1985.

....centroids to different clusters. Such a set C is called a piercing set [3] We achieve this by ensuring that each point P 2 C in the set is sufficiently far from any already chosen point P 2 C i.e. Dist(P; P ) threshold for a user defined threshold. This technique, proposed by Gonzalez [54], is guaranteed to return a piercing if no outliers are present. To avoid scanning though the whole database to choose the centroids, we first construct a random sample of the dataset and choose the centroids from the sample [3, 58] We choose the sample to be large For subsequent invocations of ....

T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 1985.

....correlation within that cluster. The second one aims to minimize the number of clusters while limiting the maximum diameter of all clusters. 1) Limit the number of clusters, then try to minimize the maximal diameter of all clusters. We use the classical K split algorithm by T. F. Gonzalez [29]. It is a O(NK) approximation algorithm, where N is the number of points and K is the number of clusters. And it guarantees solution within twice the optimal. 2) Limit the diameter of each cluster, and minimize the number of clusters. This can be reduced to the problem of finding cliques in a ....

T. F. Gonzalez, "Clustering to minimize the maximum intercluster distance," Theoretical Computer Science, vol. 38, pp. 293--306, 1985.

....in. Even the simplest clustering problems are known to be NP Hard, including the Euclidean k center problem in the plane [13, 24] In fact, it is NP Hard to approximate the two dimensional k center problem within a factor of 2 even under the L1 metric [12] The greedy algorithm by Gonzalez [14] gives a 2 approximation algorithm for the k center problem in any metric space. This algorithm requires O(kn) distance computations. The running time was improved by Feder and Greene [12] to O(n log k) for any L p metric. Several efficient algorithms have been developed for Euclidean and ....

T. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoret. Comput. Sci., 38 (1985), 293--306.

....are given by B. Everitt [7] R.O. Dabes. et al 6] and J. Hartigan [20] L. Kaufman and P. Rousseeuw [26] provide detailed programming examples of six common clustering methods. An interesting, simple, efficient clustering algorithm called a greedy algorithm was proposed by T. Gonzales [17] and later improved by T. Feder and D. Greene [9] Further development of this method may be found in the work by P. Agarwal and C. Procopiuc [1] Another simple and efficient algorithm for clustering data, an extremely popular k means algorithm, was conceptually described by S.P. Lloyd [30] and ....

T. Gonzalez, Clustering to minimize the maximum inter-cluster distance, Theoretical Computer Science #38, 1985

....objective function is minimized. In this paper, we concentrate on the (unweighted) k; r) center problem [7, 20] in which the goal is to choose k centers from the given set of n points so that every point is within distance r from some center in the graph. In particular, the k center problem [24, 27] of minimizing the maximum distance to a center is exactly (k; r) center when the goal is to minimize r subject to nding a feasible solution. In addition, the r domination problem [7, 21] of choosing the fewest vertices whose r neighborhoods cover the whole graph is exactly (k; r) center when the ....

....a point. Previous results. r domination and k center are NP hard even for planar graphs [20] For r domination, the current best approximation (for general graphs) is a (log n 1) factor by phrasing the problem as an instance of set cover [7] For k center, there is a 2 approximation algorithm [24, 27] which applies more generally to the case of weighted graphs satisfying the triangle inequality. Furthermore, no (2 ) approximation algorithm exists for any 0 even for unweighted planar graphs of maximum degree 3 [35] For geometric k center in which the weights are given by Euclidean ....

T. F. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoret. Comput. Sci., 38 (1985), pp. 293-306.

....(resp. error) ratio. A k center (resp. k median) algorithm is a approximate if it computes a set of k points with radius (resp, error) ratio a. We now give a brief overview of the approximability results known for the k center and k median problems. The farthest point technique of Gonzalez [5] yields a simple greedy 2 approximate k center algorithm running in O(nk) time. Hochbaum and Shmoys [7] match this factor of 2 approximation bound (albeit with a somewhat worse running time) using a more general approximation technique that is applicable to a certain class of bottleneck problems ....

....incremental center (resp. median) problem asks us to determine a rank function r with minimum radius (resp. error) ratio. An incremental center (resp. median) algorithm is a approximate if it computes a rank function with radius (resp. error) ratio a. The farthest point technique of Gonzalez [5] provides a 2 approximate O(n ) time incremental center algorithm. Given the hardness result for the k center problem, no polynomial time incremental center algorithm can achieve a better radius ratio unless P = NP. The incremental median problem is addressed in [10] where it is motivated ....

T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293--306, 1985.

....code for M, where A 2, this procedure must in fact produce an optimal 1 competitive code. But we just argued that in this case we can compute I the largest independent set of G, which is NP hard. Remark 1 Notice that the greedy algorithm above is exactly the same as that of Gonzalez [G85] for the so called k center problem. This is a just a coincidence, since our problems and the analysis are quite di erent. Remark 2 Note, while the greedy algorithm is extremely ecient for generic metric spaces, we are mainly interested in the Hamming space F . For this space we cannot a ord ....

T. Gonzalez. Clustering to minimize the maximum inter-cluster distance. Theoretical Computer Science, (38)293-306, 1985.

....cluster. Hence we use a solution to the closely related k center problem, where we want to choose k vertices of V , such that the longest distance from V to these k centers is minimized. These fundamental problems arise in many areas and have been widely investigated in several papers (see e.g. Go85] and [HS86] Unfortunately, both problems are NP hard, and it has been shown in [Go85] and [HS86] that unless P=NP there does not exist a (2 #) approximation What is needed is only a su#cient (even if not necessary) condition that vertices are close. 7 algorithm for any fixed # 0. ....

....to choose k vertices of V , such that the longest distance from V to these k centers is minimized. These fundamental problems arise in many areas and have been widely investigated in several papers (see e.g. Go85] and [HS86] Unfortunately, both problems are NP hard, and it has been shown in [Go85] and [HS86] that unless P=NP there does not exist a (2 #) approximation What is needed is only a su#cient (even if not necessary) condition that vertices are close. 7 algorithm for any fixed # 0. Nevertheless, there are various fast and simple 2 approximation algorithms for these problems. ....

T. Gonzalez, "Clustering to Minimize the Maximum Inter-Cluster Distance ", Theoretical Computer Science 38 (1985), 293--306.

....DRMS(ci,cj) is the square root of the mean of the squared distances of the corresponding atoms of ci and cj, after ci is transformed to cj. This transformation is computed using a basis of three predefined atoms al, a2, and a3. The clustering performed minimizes the maximum inter cluster distance [49]. Figure 5 shows two of the clusters obtained with the randomized approach of [36, 37] for CDP. At the end of the clustering step, a representative per cluster can be retained. The method de scribed above borrows ideas from randomized techniques for path planning in high dimensional configuration ....

T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293-306, 1985.

....of BIC criterion. This gives a much better initial guess to the next iteration and covers a range of admissible k. The tremendous popularity of k means has brought to life many extensions and modifications. Mao Jain [MJ96] used Mahalanobis distance to handle hyper ellipsoidal clusters. Gonzales [Gon85] used the maximum of intra cluster variances instead of the sum. Almost every industrial implementation of k means somehow resolves the issue of categorical attributes. Huang [Hua98] described a possible generalization from k means to k prototypes that incorporates categorical attributes. In some ....

Gonzalez. T.F. Clustering to minimize the maximum intercluster distance. Theoretical Comp. Science, 38,293-306, 1985.

....have been obtained yet. Many of heuristics for problems in VLSI CAD can be easily analyzed and theoretically understood. This may lead to ability to obtain an algorithm closer to optimum solution. As an example we analyze the performance of a simple clustering algorithm proposed by Gonzalez[16]. Gonzalez proposed a O(kn) algorithm for a clustering problem. k is the number of clusters and n is the number of nodes to be clustered. The algorithm is simple and can be easily analyzed. He has shown that the proposed heuristic gives a solution within two times the optimal solution. The ....

....solution. The problem is formally stated as: Given an undirected weighted graph G, an objective function f, and integer k, partition the graph into k clusters such that the value of objective function corresponding to the partition is minimized. Clustering problem formulated above is NP hard [16, 17]. This problem is studied well in detail in [16] The graph is assumed to be a complete graph. The objective is minimizing the maximum weight of an edge that is inside a cluster. Also another important assumption is that weights of the edges in graph G satisfy the triangle inequality. The ....

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T. F. Gonzalez, "Clustering to Minimize the Maximum Intercluster Distance", in Theoretical Computer Science, 38:293-306, 1985.

....objects in a cluster are more similar to each other than to objects in different clusters. According to Jain et al. [24] the general problem is known to be NP Hard. Some heuristics in use are the k means algorithm by MacQueen [27] k medioid by Kaufman et al. [26] and one algorithm by Gonzalez [17] which instead of minimizing the sum of the partitions, minimizes the maximum inter cluster distance. According to Fasulo, There are a few weaknesses in these methods. They favor spherical clusters, not general shapes and do not deal with noise. These weaknesses have been addressed by Agrawal et ....

Teofilo F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293-- 306, 1985.

....problem tractable, heuristics are employed. These heuristics construct a representative object for a cluster and use this for the distance calculations. Some heuristics in use are the k means algorithm by MacQueen [15] the k medioid algorithm by Kaufman et al. [13] and an algorithm by Gonzalez [7]. 4 These all belong to a class of clustering methods that rely on iterative partitional clustering. These algorithms operate in a similar fashion to a gradient descent search. They begin with an initial partition, on each iteration they make slight modifications to the partition to improve a ....

Teofilo F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293--306, 1985.

....all clusters through gradient descent on the cluster likelihood space. A di erent class of clustering methods, originating in the computational theory community, uses a sequential search and works by successively partitioning the instance space so as to maximize the distance between clusters [52]. Model Scaling for HMMs A hidden Markov model comprises a set of K hidden states, a matrix of K 2 transition probabilities between those states, a set of K prior probabilities for the states, and a symbol generation distribution for each state. The model structure, or topology, is the number of ....

T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293-306, 1985.

....approximation factors above can be guaranteed for a restricted class of non convex polygons. We first develop a heuristic we called canonical Voronoi insertion which approximately solves a certain extreme packing problem for point sets within P . The method is similar to the one used in Gonzalez [9] and Feder and Greene [8] developed for clustering problems. We then show how to modify the heuristic, in order to produce a set of n points whose Delaunay triangulation within P constitutes a constant approximation for the problems stated above. Note that the solution we construct is neccessarily ....

....addressed in Section 1. The algorithm determines the location of the point set Sn in a greedy manner. Namely, starting with an empty set S, it repeatedly places a new point inside P at the position which is farthest from the set V # S. The idea of the algorithm originates with Gonzalez [9] and Feder and Greene [8] and was developed for approximating minimax k clusterings. Comparable insertion strategies are also used for mesh generation in Chew [6] and in Ruppert [15] there called Delaunay refinement . Their strategies aim at di#erent quality measures, however, and insertion does ....

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T. Gonzalez, "Clustering to minimize the maximum intercluster distance", Theoretical Computer Science 38 (1985), 293-306.

....activity weighing and is a hybrid between the MinMax and LindeBuzo Gray (LBG) algorithms. Section 5 shows that the quality of quantized images produced using the weighted MinMax algorithm is superior to the best version of Xiang s MinMax algorithm. 2. THE MINMAX ALGORITHM 1 In 1985, Gonzalez [6] proposed an approximation algorithm that seeks to minimize the maximum intercluster distance. One of the interesting aspects of the MinMax algorithm is that Gonzalez [6] proved it constructs a K split whose maximum intercluser distance is less than or equal to two times optimal solution value. ....

....MinMax algorithm is superior to the best version of Xiang s MinMax algorithm. 2. THE MINMAX ALGORITHM 1 In 1985, Gonzalez [6] proposed an approximation algorithm that seeks to minimize the maximum intercluster distance. One of the interesting aspects of the MinMax algorithm is that Gonzalez [6] proved it constructs a K split whose maximum intercluser distance is less than or equal to two times optimal solution value. The application of Gonzalez s general purpose MinMax clustering algorithm to color image 1 This algorithm is commonly referred to as either the MaxMin or MinMax algorithm; ....

TEOFILO F. GONZALEZ, Clustering to Minimize the Maximum Intercluster Distance, Theoretical Computer Science, 38(2-3), 1985, 23-306.

....clustering problem where the goal is to minimize the maximum diameter or radius of a cluster. In the location theory literature, the problem of minimizing the maximum radius is also known as the k center problem. For the metric version of the problem of minimizing the maximum diameter, Gonzalez [Go85] presented a simple greedy heuristic that runs in O(nk) time and provides a performance guarantee of 2. He also showed that, unless P = NP, the performance guarantee cannot be improved. Using a general technique for approximating bottleneck problems, Hochbaum and Shmoys [HS86] also presented a ....

T. F. Gonzalez, "Clustering to Minimize the Maximum Intercluster Distance ", Theoretical Computer Science, Vol. 38, No. 2-3, Jun. 1985, pp. 293-- 306.

....which uses points in the original data set to serve as surrogate centers for clusters during their creation. Such points are referred to as medoids. An well known method called CLARANS space was proposed by Ng and Han [21] for clustering in full dimensional space. We combine the greedy method of [14] with the local search approach of the CLARANS algorithm [21] to generate possible sets of medoids, and use some original ideas in order to find the appropriate dimensions for the associated clusters. The overall pseudocode for the algorithm is given in Figure 2. The algorithm proceeds in three ....

....dimensional algorithms, one of the techniques for finding a piercing set of medoids is based on a greedy method. In this process medoids are picked iteratively, so that the current medoid is well separated from the medoids that have been chosen so far. The greedy technique has been proposed in [14] and is illustrated in Figure 3. In full dimensionality, if there are no outliers and if the clusters are well enough separated, this method always returns a piercing set of medoids. However, it does not guarantee a piercing set for the projected clustering problem. In our algorithm we will use ....

T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, Vol. 38, pp. 293-306, 1985.

....Figure 4: Splitting of correlated clusters due to initial spatial clustering. this by ensuring that each point P 2 C in the set is sufficiently far from any already chosen point P 0 2 C i.e. Dist(P; P 0 ) threshold for a user defined threshold. 4 This technique, proposed by Gonzalez [10], is guaranteed to return a piercing if no outliers are present. To avoid scanning though the whole database to choose the centroids, we first construct a random sample of the dataset and choose the centroids from the sample [2, 12] We choose the sample to be large enough (using Chernoff bounds ....

T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 1985.

....the maximum distance that an unselected location is from its nearest center. Hochbaum Shmoys [13] and subsequently Dyer Frieze [10] gave 2 approximation algorithms for the metric case problem (which is best possible unless P = NP) and also gave extensions for weighted variants. Gonzalez [11] considered the variant in which the objective is to minimize maximum distance between a pair of points in the same cluster, and independently gave a 2 approximation algorithm (which is also best possible) The k median problem is closely related to the uncapacitated facility location problem, ....

T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoret. Comput. Sci., 38:293--306, 1985.

....problem, in which we have an additional constraint that the size of each cluster should be at most c for some parameter c n=p. If p is considered as part of the input, most facility location problems are NP Hard, even in the plane or even when only an approximate solution is being sought [76, 88, 120, 130, 143, 144]. Although many of these problems can be solved in polynomial time for a xed value of p, some of them still remain intractable. In this section we review ecient algorithms for a few speci c facility location problems, which can be solved using randomization. In these applications, p is usually a ....

T. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoret. Comput. Sci., 38 (1985), 293-306.

....# 1 n V # (R) But by induction, since the sets G 1 , G 2 , Gm 1 are the result of a valid execution of the partitioning algorithm on V # , it follows that m 1 # n V # (R) and so the claim follows. Note that the claim also follows from the standard analysis for the p center problem [12, 13, 14]. Thus, the total cost of connecting each g i to some u j is at most 2R 0 n(R) # 2rR n(R) The expected cost of the tree, therefore, can be seen as bounded by the integral: # 2rR n(R)d Here, is the probability measure of the algorithm using balls of radius 2R. Now, recall that once the ....

T.E. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293--306, 1985.

....set is polynomially solvable. It can be solved by searching on the distinct distance values, d, and for each test value considering only edges of length larger than d and testing for bipartition of the resulting graph (see [3] 16] The 3 partition problem is NP hard even in the plane [12] [5]. The one dimensional k partition problem, where the elements are points on a line, is however polynomially solvable [3] Assuming that the distance matrix obeys the triangle inequality, a 2 approximation for the k partition problem can be obtained in time O(jV jk) 5] see [9] for a different ....

.... even in the plane [12] 5] The one dimensional k partition problem, where the elements are points on a line, is however polynomially solvable [3] Assuming that the distance matrix obeys the triangle inequality, a 2 approximation for the k partition problem can be obtained in time O(jV jk) [5] (see [9] for a different method to obtain a 2 approximation) Assuming P 6= NP , a 2 Gamma ffl approximation for the single choice k partition problem (when k is an input to the problem) cannot be obtained in polynomial time for any ffl 0 ( 5] 9] This is true even when the elements are ....

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T. Gonzalez, "Clustering to minimize the maximum intercluster distance", Theoretical Computer Science 38, 293-306, 1985.

....can be done with two different goals. Either we seek to maximize the distances between two different clusters or we try to minimize the distances that occur within one cluster. The optimization of most of such criteria is NP complete (for example the Maximum k Cut [5] or Minimum k Clustering [4]) We developed the Max k Separation criterion which has only quadratic time complexity. This criterion aims to maximize the minimal distance between any two clusters. So given a completely connected graph G with nodes V we look for k clusters fC 1 ; C k g (V = U i=1; k C i ) for ....

Teofilo F. Gonzalez: "Clustering to minimize the maximum intercluster distance" Theoretical Computer Science 38, North-Holland, pp. 293-306, 1985

....two vertices. Thus the minimum number of rectangles needed to cover M is 4. We show that the 2RCRn problem is NP hard by reducing a restricted version of the exact cover by three sets to it. We shall refer to this problem as the BXC3 problem. The RXC3 problem was shown to be NP complete in [6]. Restricted exact cover by three sets (BXC3) Given a finite set of ele ments X = x, xsq and a collection S = St ll ( t ( r, St C X and l St [ 3) of r = 3q S sets (3 element subsets of X) in which each element of X appears in exactly three S sets and no element of X appears more than ....

T. Gonzalez, Clustering to Minimize the Maximum Inter-Cluster Distance, The- oretical Computer Science, 38, October 1985, 293-306.

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T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38, 1985, 293--306.

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T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Comput. Sci., 38:293--306, 1985.

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T. F. Gonzalez, "Clustering to Minimize the Maximum Intercluster Distance", Theoretical Computer Science, Vol. 38, No. 2-3, June 1985, pp. 293--306.

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T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293--306, 1985.

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T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293-- 306, 1985.

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Gonzalez, T. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science 38 (1985), 293--306.

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T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293-306, 1985.

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T. F. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoret. Comput. Sci., 38 (1985), pp. 293--306.

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Gonzalez, T. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science 38 (1985), 293--306.

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T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoret. Comput. Sci., 38:293--306, 1985.

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T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoret. Comput. Sci., 38:293-306, 1985.

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T. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoret. Comput. Sci. 38 (1985), 293-- 306.

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T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoret. Comput. Sci., 38:293-306, 1985.

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T. F. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoret. Comput. Sci., 38 (1985), pp. 293--306.

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T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38(2-3):293-306, June 1985.

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T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38, 1985, 293--306.

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T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoret. Comput. Sci., 38:293-306, 1985.

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T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293-- 306, 1985.

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T. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoret. Comput. Sci., 38 (1985), 293--306.

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T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293--306, 1985.

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T. Gonzalez. Clustering to minimize the maximum inter-cluster distance. Theoretical Computer Science, NorthHolland, 38:293--306, 1985.

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Teofilo F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293--306, 1985.

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