I was going through the k-means Wikipedia page. Based on the algorithm, I think the complexity is O(n*k*i) (n = total elements, k = number of cluster iteration)
So can someone explain me this statement from Wikipedia and how is this NP hard?
If
kandd(the dimension) are fixed, the problem can be exactly solved in timeO(ndk+1 log n), wherenis the number of entities to be clustered.
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.
The problem is NP-Hard because there is another well known NP hard problem that can be reduced to (planar) k-means problem. Have a look at the paper The Planar k-means Problem is NP-hard (by Mahajan et al.) for more info.
