What is the time complexity of k-means?

In this answer, note that i used in the k-means objective formula and i used in the analysis of the time complexity of k-means (that is, the number of iterations needed until convergence) are different.

It depends on what you call k-means.

The problem of finding the global optimum of the k-means objective function

is NP-hard, where Si is the cluster i (and there are k clusters), xj is the d-dimensional point in cluster Si and μi is the centroid (average of the points) of cluster Si.

However, running a fixed number t of iterations of the standard algorithm takes only O(t*k*n*d), for n (d-dimensional) points, where kis the number of centroids (or clusters). This what practical implementations do (often with random restarts between the iterations).

The standard algorithm only approximates a local optimum of the above function, and so do all the k-means algorithms that I've seen.