I have a question about popular clustering algorithm DBSCAN (https://www.aaai.org/Papers/KDD/1996/KDD96-037.pdf).

Authors of this paper propose definition of a cluster:

Let $D$ be a database of points. A cluster $C$ with respect to $\varepsilon$ and $MinPts$ is non empty subset of D satisfying the following conditions:

  1. $\forall p, q$: if $p \in C$ and $q$ is density reachable from $p$, then $q \in C$
  2. $\forall p, q \in C$: $p$ is density connected to $q$

Definitions of density-connectivity and density-reachability are stated in paper (definitions 3 and 4).

My question is: is a boarder point $p$ a cluster with one point?

It satisfies both conditions:

  1. There are no points density-reachable from $p$, so 1. is ok.
  2. $p$ is in eps-neighborhood of a core point (because it's a border point), therefore $p$ is density connected to itself.

I wouldn't have problem with that, if later in the paper authors wouldn't state that

cluster contains at least MinPts points

Do you have any ideas what might be going on here?


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