DBSCAN defines clusters as contiguous regions of high density, which are separated by regions of lower density. It categorizes each data point in the dataset into one of three distinct types based on the density of its local neighborhood:

1. Core Point: A data point is classified as a core point if there are at least MinPts (a user-defined parameter, representing the minimum number of points required to form a dense region) within a specified radius of eps (another user-defined parameter, epsilon, which defines the maximum distance for one point to be considered in the neighborhood of another point). Core points are the "heart" of a cluster.
2. Border Point: A data point is classified as a border point if it is within the eps radius of a core point but does not itself satisfy the MinPts criterion (i.e., it doesn't have enough neighbors within its own eps radius to be a core point). Border points are on the "edge" of a cluster.
3. Noise Point (Outlier): A data point is classified as a noise point (or outlier) if it is neither a core point nor a border point. These points are considered to be in low-density regions and are isolated from dense clusters.

1. Start with an Unvisited Point: The algorithm begins by selecting an arbitrary, unvisited data point from the dataset.
2. Check Neighborhood Density: It then examines the eps-neighborhood of this point.
3. If the point has at least MinPts neighbors within its eps radius, it is classified as a core point. A new cluster is initiated around this core point. All data points within its eps-neighborhood (including other core points and any border points) are immediately added to this new cluster.
4. If the point does not have MinPts neighbors within its eps radius, it is temporarily marked as noise (it might later be identified as a border point if it falls within another core point's neighborhood, but for now, it's considered low-density).
5. Expand the Cluster (Reachability): If a new cluster was formed (i.e., the starting point was a core point), the algorithm then recursively expands this cluster. It visits each newly added point in the cluster:
6. If a newly added point is also a core point, its eps-neighborhood is also added to the current cluster.
7. If a newly added point is a border point, it is added to the current cluster but its neighborhood is not expanded further.
8. This expansion process continues until no more points can be added to the current cluster (i.e., all reachable core and border points have been included).
9. Mark as Visited: All data points that were successfully included in the newly formed cluster are marked as "visited" to prevent re-processing.
10. Move to Next Unvisited Point: The algorithm then selects another unvisited data point from the dataset and repeats the entire process from step 1.
11. Final Noise Identification: Any data points that were temporarily marked as noise in step 2 and were never subsequently added to any cluster during the expansion phase (meaning they are not core points and are not reachable from any core points) remain classified as noise points (outliers) in the final output.

The performance and the resulting clusters from DBSCAN are highly dependent on the careful selection of two fundamental parameters:

1. eps (epsilon or maximum distance): This parameter defines the maximum distance between two samples for one to be considered as in the neighborhood of the other. It essentially sets the radius of the neighborhood around each data point.
2. Impact: Choosing an appropriate eps is critical. If eps is too small, many data points that should be in the same cluster might be classified as noise. If eps is too large, distinct clusters might be merged into a single, large cluster.
3. Selection Strategy: A common heuristic for selecting eps is to use a K-distance graph. This involves plotting the distance to the k-th nearest neighbor for each point (where k is MinPts minus 1), sorted in ascending order. Look for the "knee" or "elbow" in this graph, which suggests a good eps value where the density significantly drops.
4. MinPts (minimum points): This parameter specifies the minimum number of data points required to form a dense region (i.e., for a point to be considered a core point).
5. Impact: A larger MinPts makes the definition of a "dense region" stricter, leading to fewer but potentially more robust clusters and more points being classified as noise. A smaller MinPts can lead to more, smaller, and potentially noise-sensitive clusters.
6. Selection Strategy: A common rule of thumb for 2-dimensional data is to set MinPts = 4 (or 2 * dimensionality). For higher-dimensional datasets, larger values for MinPts are often used, as density becomes sparser in higher dimensions.

Advantages of DBSCAN:

1. Discovers Arbitrary Cluster Shapes: One of its most significant advantages is its ability to identify and form clusters of complex, non-linear, and arbitrary shapes. This is a major improvement over K-Means, which is limited to roughly spherical clusters.
2. Does Not Require Pre-specifying K: Unlike K-Means, DBSCAN does not require you to provide the number of clusters in advance. The algorithm determines the number of clusters based on the data's inherent density structure.
3. Robust Outlier Detection: It naturally identifies and separates "noise" points (outliers) from actual clusters, labeling them explicitly. This is extremely valuable in applications where anomaly detection is important.
4. Resistant to Noise (to an extent): Compared to hierarchical clustering methods like single linkage, DBSCAN is generally more robust to noise because it requires a minimum number of points to form a dense region.

Disadvantages of DBSCAN:

1. High Parameter Sensitivity: DBSCAN is highly sensitive to the choice of its two parameters, eps and MinPts. Finding optimal values for these parameters can be challenging and often requires iterative experimentation, domain knowledge, or specific heuristic methods.
2. Struggles with Varying Densities: It can have difficulty finding clusters effectively when the densities of the clusters vary significantly within the same dataset. A single pair of eps and MinPts might not be suitable for clusters of vastly different densities.
3. Curse of Dimensionality Impact: As the dimensionality of the data increases, the concept of density becomes less meaningful. In high-dimensional spaces, distances between points tend to become more uniform, making it very difficult to find appropriate eps values and effectively distinguish dense regions from sparse ones. This phenomenon is known as the "curse of dimensionality."
4. Border Points Ambiguity: Border points that are reachable from multiple clusters might be arbitrarily assigned to one of them, which can sometimes lead to slightly different results on repeated runs (though core points will be consistently assigned).