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I have a collection of 2D points in Euclidean space which I want to cluster.

However, I want to ensure that in the clusters generated, they are at least a fixed distance away from one another (meaning points that are very close to one another will be guaranteed to be in a single cluster).

I have tried K-Means but it only minimises the intra-cluster sum-of-squares, rather than guaranteeing a minimum distance between clusters. Is there a variant (of K-means) or other clustering algorithms that exists?

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You want DB scan.

http://scikit-learn.org/stable/auto_examples/cluster/plot_dbscan.html

You also need to choose the number of neighbors which qualify a point as a node.

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