Fast (< n^2) clustering algorithm

Question

I have 1 million 5-dimensional points that I need to group into k clusters with k << 1 million. In each cluster, no two points should be too far apart (e.g. they could be

Answer 1

Put the data into an index such as an R*-tree (Wikipedia), then you can run many density-based clustering algorithms (such as DBSCAN (Wikipedia) or OPTICS (Wikipedia)) in O(n log n).

Density-based clustering (Wikipedia) seems to be precisely what you want ("not too far apart")