Approximation algorithms for facility location problems (extended abstract)
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Approximation schemes for Euclidean k-medians and related problems
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Rounding via trees: deterministic approximation algorithms for group Steiner trees and k-median
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
On approximating arbitrary metrices by tree metrics
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Approximating geometrical graphs via “spanners” and “banyans”
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
A constant-factor approximation algorithm for the k-median problem (extended abstract)
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Greedy strikes back: improved facility location algorithms
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Nearly linear time approximation schemes for Euclidean TSP and other geometric problems
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Polynomial time approximation schemes for Euclidean TSP and other geometric problems
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Probabilistic approximation of metric spaces and its algorithmic applications
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
The analysis of a simple k-means clustering algorithm
Proceedings of the sixteenth annual symposium on Computational geometry
A local search approximation algorithm for k-means clustering
Proceedings of the eighteenth annual symposium on Computational geometry
An Efficient k-Means Clustering Algorithm: Analysis and Implementation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Embeddings and non-approximability of geometric problems
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation Schemes for Geometric NP-Hard Problems: A Survey
FST TCS '01 Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science
Clustering Data Streams: Theory and Practice
IEEE Transactions on Knowledge and Data Engineering
Bypassing the embedding: algorithms for low dimensional metrics
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
On coresets for k-means and k-median clustering
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
A local search approximation algorithm for k-means clustering
Computational Geometry: Theory and Applications - Special issue on the 18th annual symposium on computational geometry—SoCG2002
Smaller coresets for k-median and k-means clustering
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
On k-Median clustering in high dimensions
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
On efficient deployment of sensors on planar grid
Computer Communications
Approximating TSP on metrics with bounded global growth
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Clustering for metric and non-metric distance measures
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for clustering uncertain data
Proceedings of the twenty-seventh ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Approximating k-hop minimum spanning trees in Euclidean metrics
Information Processing Letters
Efficient approximation algorithms for clustering point-sets
Computational Geometry: Theory and Applications
On Euclidean vehicle routing with allocation
Computational Geometry: Theory and Applications
Approximation schemes for degree-restricted MST and red-blue separation problem
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
FSTTCS'07 Proceedings of the 27th international conference on Foundations of software technology and theoretical computer science
Quantization-based clustering algorithm
Pattern Recognition
Clustering for metric and nonmetric distance measures
ACM Transactions on Algorithms (TALG)
Linear time algorithms for clustering problems in any dimensions
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Bregman clustering for separable instances
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
On euclidean vehicle routing with allocation
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
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In the k-median problem we are given a set S of n points in a metric space and a positive integer k. The objective is to locate k medians among the points so that the sum of the distances from each point in S to its closest median is minimized. The k-median problem is a well-studied, NP-hard, basic clustering problem which is closely related to facility location. We examine the version of the problem in Euclidean space. Obtaining approximations of good quality had long been an elusive goal and only recently Arora, Raghavan and Rao gave a randomized polynomial-time approximation scheme for the Euclidean plane by extending techniques introduced originally by Arora for Euclidean TSP. For any fixed 驴 0; their algorithm outputs a (1 + 驴)-approximation in O(nknO(1/驴) log n) time.In this paper we provide a randomized approximation scheme for points in d- dimensional Euclidean space, with running time O(21/驴d n log n log k); which is nearly linear for any fixed 驴 and d. Our algorithm provides the first polynomialtime approximation scheme for k-median instances in d-dimensional Euclidean space for any fixed d 2: To obtain the drastic running time improvement we develop a structure theorem to describe hierarchical decomposition of solutions. The theorem is based on a novel adaptive decomposition scheme, which guesses at every level of the hierarchy the structure of the optimal solution and modifies accordingly the parameters of the decomposition. We believe that our methodology is of independent interest and can find applications to further geometric problems.