SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for clustering to minimize the sum of diameters
Nordic Journal of Computing
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Clustering to minimize the sum of cluster diameters
Journal of Computer and System Sciences - STOC 2001
Incremental Clustering and Dynamic Information Retrieval
SIAM Journal on Computing
Incremental algorithms for facility location and k-Median
Theoretical Computer Science - Approximation and online algorithms
A primal-dual algorithm for online non-uniform facility location
Journal of Discrete Algorithms
Online unit clustering: Variations on a theme
Theoretical Computer Science
An Improved Algorithm for Online Unit Clustering
Algorithmica
A Randomized Algorithm for Online Unit Clustering
Theory of Computing Systems
Variable-Size Rectangle Covering
COCOA '09 Proceedings of the 3rd International Conference on Combinatorial Optimization and Applications
A generalized minimum cost k-clustering
ACM Transactions on Algorithms (TALG)
On the online unit clustering problem
ACM Transactions on Algorithms (TALG)
Memoryless facility location in one pass
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
Better bounds on online unit clustering
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
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In online clustering problems, the classification of points into sets (called clusters) is done in an online fashion. Points arrive one by one at arbitrary locations, to be assigned to clusters at the time of arrival. A point can be assigned to an existing cluster, or a new cluster can be opened for it. We study a one dimensional variant on a line, where there is no restriction on the length of a cluster, and the cost of a cluster is the sum of a fixed set-up cost and its diameter. The goal is to minimize the sum of costs of the clusters used by the algorithm. We study several variants, all maintaining the essential property that a point which was assigned to a given cluster must remain assigned to this cluster, and clusters cannot be merged. In the strict variant, the diameter and the exact location of the cluster must be fixed when it is initialized. In the flexible variant, the algorithm can shift the cluster or expand it, as long as it contains all points assigned to it. In an intermediate model, the diameter is fixed in advance while the exact location can be modified. We give tight bounds on the competitive ratio of any online algorithm in each of these variants. In addition, for each one of the models, we consider also the semi-online case, where points are presented sorted by their location.