# ML | DBSCAN reachability and connectivity

**Prerequisite :** DBSCAN Clustering in ML

Density-based clustering algorithm has played a vital role in finding nonlinear shapes structure based on the density. Density-Based Spatial Clustering of Applications with Noise (DBSCAN) is the most widely used density-based algorithm. It uses the concept of density reachability and density connectivity.

Consider a set of points in some space to be clustered using DBSCAN clustering. Let *ε* be the radius of a neighborhood with respect to some point and core objects are the objects whose ε-neighborhood contains at least *MinPts* number of objects.

### Reachability –

**Directly density reachable:**

An object (or instance) q is directly density reachable from object p if q is within the ε-Neighborhood of p and p is a core object.Here directly density reachability is not symmetric. Object p is not directly density-reachable from object q as q is not a core object.

**Density reachable:**

An object q is density-reachable from*p*w.r.t*ε*and*MinPts*if there is a chain of objects q_{1}, q_{2}…, q_{n}, with q_{1}=p, q_{n}=q such that q_{i+1}is directly density-reachable from q_{i}w.r.t*ε*and*MinPts*for all 1 <= i <= n

Here density reachability is not symmetric. As *q* is not a core point thus q_{n-1} is not directly density-reachable from q, so object p is not density-reachable from object q.

### Connectivity –

**Density connectivity:**Object q is density-connected to object*p*w.r.t*ε*and*MinPts*if there is an object o such that both p and q are density-reachable from*o*w.r.t*ε*and*MinPts*.

Here density connectivity is symmetric. If object q is density-connected to object p then object p is also density-connected to object q.

Based on the above two concepts *reachability* and *connectivity* we can define the **cluster** and **noise points**.

**Cluster:**

A cluster C w.r.t. *ε* and *MinPts* is a non empty subset of D (the whole set of objects or instances) satisfying –

*Maximality:*For all objects p, q if p ε C and if q is density-reachable from p w.r.t*ε*and*MinPts*then q ε C.*Connectivity:*For all objects p, q ε C, p is density-connected to q and vice-versa w.r.t.*ε*and*MinPts*.

**Noise:**

Objects which are not directly density-reachable from at least one core object are known as *Noise points*.

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