The k-means for lines is a set of k centers (points) that minimizes the sum of squared distances to a given set of n lines in R^d. This is a straightforward generalization of the k-means problem where the input is a set of n points. Related problems minimize sum of (non-squared) distances, other norms, m-estimators or ignore the t farthest points (outliers) from the k centers. We suggest the first provable PTAS algorithms for these problems that compute (1+epsilon)-approximation in time O(n (n) epsilon^2) for any given epsilon (0, 1), and constant integers k, d, t 1, including support for streaming and distributed input. Experimental results on Amazon EC2 cloud and open source are also provided.
Langebreg and Schulman @cite_35 addressed the @math -center problem (i.e., the case @math to find a single ball intersecting all input lines). From a computational point of view, the @math -center problem significantly differs from the general @math -center problem, the @math -center problem is a convex optimization problem and therefore fundamentally easier than the cases of @math -center for @math .
@cite_35
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The @math -center problem is a classical combinatorial optimization problem which asks to find @math centers such that the maximum distance of any input point in a set @math to its assigned center is minimized. The problem allows for elegant @math -approximations. However, the situation becomes significantly more difficult when constraints are added to the problem. We raise the question whether general methods can be derived to turn an approximation algorithm for a clustering problem with some constraints into an approximation algorithm that respects one constraint more. Our constraint of choice is privacy: Here, we are asked to only open a center when at least @math clients will be assigned to it. We show how to combine privacy with several other constraints. Abstract In the k -center problem for streaming points in d -dimensional metric space, input points are given in a data stream and the goal is to find the k smallest congruent balls whose union covers all input points by examining them. In the single-pass streaming model, input points are allowed to be examined only once and the amount of space that can be used to store relative information is limited. In this paper, we present a single-pass, ( 1.8 + e ) -factor, O ( d e ) -space data stream algorithm for the Euclidean 2-center problem for any d ≥ 1 . This is the first result with an approximation factor below 2 using O ( d e ) space for any d . Our algorithm naturally extends to the Euclidean k -center problem with k > 2 . We present a single-pass ( 1.8 + e ) -factor data stream algorithm for the Euclidean k -center problem for any d ≥ 1 , which uses O ( 2 k ( k + 3 ) ! d e ) space and O ( 2 k ( k + 2 ) ! d e ) update time. Abstract This paper considers the planar Euclidean two-center problem: given a planar n-point set S, find two congruent circular disks of the smallest radius covering S. The main result is a deterministic algorithm with running time O(nlog2nlog2logn), improving the previous O(nlog9n) bound of Sharir and almost matching the randomized O(nlog2n) bound of Eppstein. If a point in the intersection of the two disks is given, then we can solve the problem in O(nlogn) time with high probability. We consider the @math -Center problem and some generalizations. For @math -Center a set of @math center vertices needs to be found in a graph @math with edge lengths, such that the distance from any vertex of @math to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by [SODA 2010]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations, where the two paradigms are combined. In particular, we prove that for any @math computing a @math -approximation is W[2]-hard for parameter @math and NP-hard for graphs with highway dimension @math . The latter does not rule out fixed-parameter @math -approximations for the highway dimension parameter @math , but implies that such an algorithm must have at least doubly exponential running time in @math if it exists, unless the ETH fails. On the positive side, we show how to get below the approximation factor of @math by combining the parameters @math and @math : we develop a fixed-parameter @math -approximation with running time @math . Additionally we prove that, unless P=NP, our techniques cannot be used to compute fixed-parameter @math -approximations for only the parameter @math . We also provide similar fixed-parameter approximations for the weighted @math -Center and @math -Partition problems, which generalize @math -Center.