Online clustering with variable sized clusters

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.