How to Assign Points to the Clusters?

DBSCAN's core principle of density reachability is what dictates cluster assignments. Points are assigned to clusters based on the following:

• Directly density-reachable: a point $p$ is directly density-reachable from a core point $q$ if $p$ is within the ε-radius of $q$. All directly density-reachable points from a core point automatically belong to the same cluster as the core point;
• Density-reachable: a point $p$ is density-reachable from a core point $q$ if there is a chain of points $p_1$, $p_2$, ..., $p_n$, where $p_1 = q$, $p_n = p$, and each $p_{i+1}$ is directly density-reachable from $p_i$. This means that even if a point is not directly within the ε-radius of a core point, it can still belong to the same cluster if it is connected to that core point through a chain of directly density-reachable points;
• Density-connected: two points $p$ and $q$ are density-connected if there exists a core point $o$ such that both $p$ and $q$ are density-reachable from $o$. All density-connected points belong to the same cluster.

Border points are assigned to the cluster of the first core point from which they are found to be reachable. Noise points, by definition, are not density-reachable from any core point and therefore are not assigned to any cluster. They are treated as outliers.

In essence, DBSCAN builds clusters by connecting core points that are density-reachable from each other, and border points inherit the cluster assignment of the core points they are associated with.