  Offline k-means clustering was studied extensively, and algorithms with a
constant approximation are available. However, online clustering is still
uncharted. New factors come into play: the ordering of the dataset and whether
the number of points, n, is known in advance or not. Their exact effects are
unknown. In this paper we focus on the online setting where the decisions are
irreversible: after a point arrives, the algorithm needs to decide whether to
take the point as a center or not, and this decision is final. How many centers
are needed and sufficient to achieve constant approximation in this setting? We
show upper and lower bounds for all the different cases. These bounds are
exactly the same up to a constant, thus achieving optimal bounds. For example,
for k-means cost with constant k>1 and random order, Theta(log n) centers are
enough to achieve a constant approximation, while the mere a priori knowledge
of n reduces the number of centers to a constant. These bounds hold for any
distance function that obeys a triangle-type inequality.
