Computational Geometry Questions Long
In Computational Geometry, there are several types of geometric clustering algorithms used to group geometric objects based on their spatial relationships. These algorithms aim to partition the input data into clusters or groups, where objects within the same cluster are more similar to each other than to those in other clusters. Some of the commonly used geometric clustering algorithms are:
1. Hierarchical Clustering: This algorithm builds a hierarchy of clusters by iteratively merging or splitting existing clusters based on a similarity measure. It can be agglomerative (bottom-up) or divisive (top-down). Agglomerative hierarchical clustering starts with each object as a separate cluster and merges the closest pairs until a desired number of clusters is obtained. Divisive hierarchical clustering starts with all objects in a single cluster and recursively splits them until each object forms its own cluster.
2. K-means Clustering: This algorithm aims to partition the data into K clusters, where K is a user-defined parameter. It iteratively assigns each object to the nearest cluster centroid and updates the centroids based on the mean of the objects assigned to each cluster. The process continues until convergence.
3. DBSCAN (Density-Based Spatial Clustering of Applications with Noise): This algorithm groups objects based on their density. It defines clusters as dense regions separated by sparser regions. It starts with an arbitrary object and expands the cluster by adding nearby objects that satisfy a density criterion. Objects that do not meet the density criterion are considered noise or outliers.
4. OPTICS (Ordering Points To Identify the Clustering Structure): This algorithm is an extension of DBSCAN that provides a hierarchical clustering result. It computes a reachability distance for each object, which represents the density-based connectivity to other objects. The algorithm orders the objects based on their reachability distances, allowing the identification of clusters at different density levels.
5. Mean Shift Clustering: This algorithm iteratively shifts the centroids of clusters towards the regions of higher object density. It starts with initial centroids and updates them by shifting towards the mean of the objects within a certain radius. The process continues until convergence, resulting in clusters centered around density peaks.
6. Spectral Clustering: This algorithm uses the eigenvectors of a similarity matrix to perform clustering. It represents the data as a graph, where objects are nodes and edges represent pairwise similarities. By computing the eigenvectors of the Laplacian matrix associated with the graph, the algorithm identifies clusters based on the spectral properties of the matrix.
These are just a few examples of geometric clustering algorithms used in Computational Geometry. Each algorithm has its own strengths and weaknesses, and the choice of algorithm depends on the specific problem and data characteristics.