C. Papadimitriou, "Worst-case and probabilistic analysis of a geometric location problem," SIAM Journal on Computing, vol. 10, pp. 542--557, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....It is known that the solution for the Fermant Weber problem is unique and algebraic provided that all points of D are not collinear. Several numerical approaches have been proposed to compute an approximate solution. See [56, 271] for the history of the problem and for the known algorithms, and [230] for some heuristics for the p median problem that work well for a set of random points. Recently, Arora et al. 40] described an (1 ) approximation algorithm for the p median problem in the plane whose running time is n . For d 2, the running time of their algorithm is n O( log n= ....

C. H. Papadimitriou, Worst-case and probabilistic analysis of a geometric location problem, SIAM J. Comput., 10 (1981), 542--557.

.... version are presented in [4, 14] Notice that the MAX GAIN problem is a special case of the k median one (while in the facility location problem there is no such restriction for the number of new facilities to open) Noticeably, the Euclidean version of the k median problem is already NP hard [17]. The main difference between the above (more general) problems and our problem(s) concerns the re striction on the possible locations for the new stations. Indeed, in practice it is quite unlikely that the radius associated with a settlement crosses several tracks far apart from one another: if ....

C. H. Papadimitriou. Worst-case and probabilistic analysis of a geometric location problem. SIAM J. Cornput., 10, 1981.

....problem in graph theory: given a graph with N nodes, each node i with a request rate r(i) pick K( N) nodes as servers and assign each node to one of these servers so that the total weighted distance between all nodes i and their servers, weighted by r(i) is minimized. This problem is shown in [6, 8] to be NP hard for general graphs. Subsequent efforts have been to find polynomial algorithms to solve special cases and to find approximation algorithms to solve the general case. These algorithms, however, are not applicable to large scale self organizing CDN we envision, for two reasons. First, ....

C. Papadimitriou. Worst-case and probabilistic analysis of a geometric location problem. SIAM J. Comput., 10:542--557, 1981.

....problem in graph theory: given a graph with N nodes, each node i with a request rate r(i) pick K( N) nodes as servers and assign each node to one of these servers so that the total weighted distance between all nodes i and their servers, weighted by r(i) is minimized. This problem is shown in [15, 21] to be NP hard for general graphs. Subsequent efforts have been to find polynomial algorithms to solve special cases and to find approximation algorithms to solve the general case. For tree graphs, 15] presents an O(N ) algorithm to solve it optimally. This is improved to an O(N K) ....

C. Papadimitriou. Worst-case and probabilistic analysis of a geometric location problem. SIAM J. Comput., 10:542--557, 1981.

.... version are presented in [4, 14] Notice that the MAX GAIN problem is a special case of the k median one (while in the facility location problem there is no such restriction for the number of new facilities to open) Noticeably, the Euclidean version of the k median problem is already NP hard [17]. The main difference between the above (more general) problems and our problem(s) concerns the re striction on the possible locations for the new stations. Indeed, in practice it is quite unlikely that the radius associated with a settlement crosses several tracks far apart fi om one another: ....

C. H. Papadimitriou. Worst-case and probabilistic analysis of a geometric location problem. SIAM J. Cornput., 10, 1981.

....The goal is to choose a subset U of size s of the vertices that minimizes the sum of distances from each vertex to its nearest neighbour in U . Call U the median set. The s median problem arises in data compression, network location, and clustering. It is NP hard even in the Euclidian space [48, 57]. Lund and Yanakakis s lower bounds for the set covering problem imply that it is NP hard to find ffl approximate solutions of size o(s log jV j) to the s median problem for an ffl sufficiently small [41, 46] We generalize the analysis of Cornuejols et al. 21] to account for approximate ....

C.H. Papadimitriou. Worst-case and probabilistic analysis of a geometric location problem. SIAM Journal on Computing, 10:542--557, 1981.

....It is known that the solution for the Fermant Weber problem is unique and algebraic provided that all points of D are not collinear. Several numerical approaches have been proposed to compute an approximate solution. See [38, 177] for the history of the problem and for the known algorithms, and [151] for some heuristics for the p median problem that work well for a set of random points. Recently, Arora et al. 22] described an approximation algorithm for the p median problem in the plane whose running time is n O(1= For d 2, the running time of their algorithm is n O( log n= d ....

C. H. Papadimitriou, Worst-case and probabilistic analysis of a geometric location problem, SIAM J. Comput., 10 (1981), 542-557.

.... 6 5 fflM Y 4K fflM Y 20K fflM Y 20K fflM Y K : The rest of the proof follows from the Lipschitz bound. 2 4.3. 2 Approximate Clustering Unfortunately, finding optimal clusters is NP hard even in Euclidean space (Kariv and Hakimi [34] Garey and Johnson [35] Papadimitriou [36]; Megiddo [37] However, as shown by Lin and Vitter [2] we have approximate clustering algorithms with provably good performance guarantees. We may use these approximate clustering algorithms for learning: 18 4 MEMORY EFFICIENT LEARNING OF SMOOTH FUNCTIONS Algorithm LC2 (learning by approximate ....

C. H. Papadimitriou, "Worst-case and probabilistic analysis of a geometric location problem", SIAM Journal on Computing, vol. 10, pp. 542--557, 1981.

....problem, and it goes back to the 17th century. It is known that the solution for the Fermant Weber problem is unique and algebraic. Several numerical approaches have been proposed to compute an approximate solution. See [48, 233] for the history of the problem and for the known algorithms, and [197] for some heuristics for the p median problem that work well for a set of random points. 7.4 Segment center Given a segment e, we wish to find a translated and rotated copy of e so that the maximum distance from each point of the given set D of demand points to this copy is minimized. This ....

C. H. Papadimitriou, Worst-case and probabilistic analysis of a geometric location problem, SIAM J. Comput., 10 (1981), 542--557.

....it can see the next request. Related work The static version of our problem is known under the name of the k median problem and has been extensively studied in the past. Kariv and Hakimi [9] proved that the k median problem is NP complete even for planar graphs of maximum degree 3. Papadimitriou [12] proved that the k median problem in the plane is NPcomplete. Megiddo and Supowitz [11] proved that it is NP hard to approximate the k median problem to within some constant factor both in the Euclidean and the Rectilinear metric. The approximation of the k median problem is also investigated in ....

C. Papadimitriou, Worst-Case and Probabilistic Analysis of a Geometric Location Problem, SIAM Journal on Computing, 10 (1981), pp. 542--557.

....It is known that the solution for the Fermant Weber problem is unique and algebraic provided that all points of D are not collinear. Several numerical approaches have been proposed to compute an approximate solution. See [40, 187] for the history of the problem and for the known algorithms, and [159] for some heuristics for the p median problem that work well for a set of random points. Recently, Arora et al. 23] described an approximation algorithm for the p median problem in the plane whose running time is n O(1= For d 2, the running time of their algorithm is n O( log n= ....

C. H. Papadimitriou, Worst-case and probabilistic analysis of a geometric location problem, SIAM J. Comput., 10 (1981), 542--557.

....to some given probability density (p) then we may be interested in minimizing the expected distance R p2D d(c; p) p)dp for a feasible center location c 2 F . To the best of our knowledge, there are only few references that discuss k median problems with continuous demand: See the papers [45, 59] for a discussion of continuous demand that arises probabilistically by considering a discrete demand in an unbounded environment with a large number of demand points, leading to a heuristic for optimal placement of many center points. Drezner [17] describes in Chapter 2 of his book that normally ....

C. Papadimitriou. Worst case and probabilistic analysis of a geometric location problem. SIAM J. Computing, 3:542-557, 1981. 17

.... in an optimal solution must contain the points of their respective clusters) Apparently, attention drifted away from the TWGD problem as the NP hardness results emerged for the graphical and geometric (Euclidean and even bidimensional, i.e. m = 2) versions of representative clustering problems [4, 22, 27, 28, 30, 39]. Work concentrated on suitable approximation algorithms for them [8, 41] Others concentrated on special cases where polynomial algorithms can be found (for example, the case p = 2 [23] More recently, theoretical results have concentrated on polynomial approximation schemes for the ....

C.H. Papadimitriou. Worst-case and probabilistic analysis of geometric location problems. SIAM J. of Computing, 10:542--557, 1981.

....the classical well studied p median problem which has been object of research since the early 60 s. The p median problem is NP hard, even if (V; E) is a planar graph of maximum degree 3 as shown by Kariv and Hakimi [11] The problem remains NPhard in the Euclidean plane as shown by Papadimitriou [15]. Even to approximate the problem within a constant factor turns out to be NP hard in the case of the Euclidean This research has been supported by the Spezialforschungsbereich F003 Optimierung und Kontrolle , Projektbereich Diskrete Optimierung. y Technische Universitat Graz, Institut fur ....

C. Papadimitriou, Worst-case and probabilistic analysis of a geometric location problem, SIAM J. Comput. 10, 1981, 542--557.

....in an optimal solution must contain the points of their respective clusters. Apparently, attention drifted away from the TWGD problem as the NP hardiness results emerged for the graphical and geometric (Euclidean and even bidimensional, i.e. m = 2) versions of representative clustering problems [4, 22, 27, 28, 30, 39]. Work concentrated on suitable approximation algorithms for them [8, 41] Others concentrated in special cases where polynomial algorithms can be found (for example, the case p = 2 [23] More recently, theoretical results have concentrated on polynomial approximation schemes for the ....

C.H. Papadimitriou. Worst-case and probabilistic analysis of geometric location problems. SIAM J. of Computing, 10:542--557, 1981.

....another factor of 2 by the way we set Z(i) The rest of the proof follows from the Lipschitz bound. 4.3 LEARNING BY CLUSTERING 19 4.3. 2 Approximate Clustering Unfortunately, finding optimal clusters is NP hard even in Euclidean space (Kariv and Hakimi [20] Garey and Johnson [12] Papadimitriou [32]; Megiddo [26] However, as shown by Lin and Vitter [24] we have approximate clustering algorithms with provably good performance guarantees. We may use these approximate clustering algorithms for learning: Algorithm LC2 (learning by approximate clustering) 1. Let m = Omega Gamma ks ffl log ....

C. H. Papadimitriou, "Worst-case and Probabilistic Analysis of a Geometric Location Problem," SIAM Journal on Computing 10 (1981), 542--557.

....can see the next request. Related work The static version of our problem is known under the name of the k median problem and has been extensively studied in the past. Kariv and Hakimi [9] proved that the k median problem is NP complete even for planar graphs of maximum degree 3. Papadimitriou [12] proved that the k median problem in the plane is NP complete. Megiddo and Supowitz [11] proved that it is NPhard to approximate the k median problem to within some constant factor both in the Euclidean and the Rectilinear metric. The approximation of the k median problem is also investigated in ....

C. Papadimitriou, Worst-Case and Probabilistic Analysis of a Geometric Location Problem, SIAM Journal on Computing, 10 (1981), pp. 542--557.

....The problem of computing the optimal placement of resources is known in the literature as the k median problem and its static version has been very well studied. Kariv and Hakimi [3] proved that the k median problem is NP complete even for planar graphs of maximum degree 3. Papadimitriou [6] proved that the k median problem in the plane is NP complete. Megiddo and Supowitz [5] proved that it is NP hard to approximate the k median problem to within some constant factor both in the Euclidean and the Rectilinear metric. The approximation of the k median problem is also investigated in ....

C. Papadimitriou, Worst-Case and Probabilistic Analysis of a Geometric Location Problem, SIAM Journal on Computing, 10 (1981), pp. 542--557. Algorithm Inc-Line(k) ffl Initialize k empty instances of OL Cost A 1 ; \Delta \Delta \Delta ; A k and k empty instances of OL PseudoCost

....in many areas of science and engineering. Among others we mention here: the (capacity and bottleneck) assignment problem [14] the (bottleneck and capacity) quadratic assignment problem [8, 12] the minimum spanning tree, the minimum weighted k clique problem [10] geometric location problems [11], and some others not directly related to optimization such as the height and depth of digital trees, the maximum queue length and hashing with lazy deletion [1] pattern matching [3] and so forth. We analyze this class of problems in a probabilistic framework which assumes that the weights w i ....

..... On the other hand, it is easy to prove that for the exponential distribution of weights Z max n log n (pr. while for the normally distributed weights Z max n p 2 log n (pr. cf. 10, 13] 3. 2 Location Problem on Graphs A general location problem can be formulated as follows (see also [11]) Let x 1 ; x 2 ; x n be a given set of points. The median problem selects L points c 1 ; c 2 ; c L so as to minimize (maximize) the distance between these points and the points x 1 ; x 2 ; x n . To formulate the problem in terms of our general optimization problem (1) ....

Papadimitriou, C., Worst-Case and Probabilistic Analysis of a Geometric Location Problem, SIAM J. Computing, 10, 542--557, 1981.

....It is known that the solution for the Fermant Weber problem is unique and algebraic provided that all points of D are not collinear. Several numerical approaches have been proposed to compute an approximate solution. See [57, 273] for the history of the problem and for the known algorithms, and [231] for some heuristics for the p median problem that work well for a set of random points. Recently, Arora et al. 40] described an (1 ) approximation algorithm for the p median problem in the plane whose running time is n O(1= For d 2, the running time of their algorithm is n O( log ....

C. H. Papadimitriou, Worst-case and probabilistic analysis of a geometric location problem, SIAM J. Comput., 10 (1981), 542--557.

....of arbitrary degree. Related works. The static version of our problem is known under the name of the k median problem and has been extensively studied in the past. Kariv and Hakimi [8] proved that the k median problem is NP complete even for planar graphs of maximum degree 3. Papadimitriou [11] proved that the k median problem in the plane is NP complete. Megiddo and Supowitz [10] proved that it is NP hard to approximate the k median problem to within some constant factor both in the Euclidean and the Rectilinear metric. The approximation of the k median problem is also investigated ....

C. Papadimitriou, Worst-Case and Probabilistic Analysis of a Geometric Location Problem, SIAM Journal on Computing, 10 (1981), pp. 542--557.

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C. Papadimitriou, "Worst-case and probabilistic analysis of a geometric location problem," SIAM Journal on Computing, vol. 10, pp. 542--557, 1981.

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