An online 2-dimensional clustering problem with variable sized clusters

In the 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, and we have to assign them to clusters at the time of arrival without any information about the further points. A point can be assigned to an existing cluster, or a new cluster can be opened for it. We study two-dimensional variants in the l _∞ norm, thus clusters are actually squares. The cost of a cluster is the sum of a fixed setup cost and the area of the square. The goal is to minimize the sum of the costs of the clusters used by the algorithm. We study two variants, both maintaining the properties 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 size 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. We present a 7-competitive algorithm in the strict model and an approximately 5.22-competitive algorithm for the flexible variant. We also give lower bounds on the possible competitive ratio, this bound is 2.768 in the strict model and 1.743 in the flexible model.