*3.2. OPTICS-xi*

The OPTICS-xi [27] algorithm is rooted in the concepts of reachability described for the DBSCAN, but it exploits some derived properties to build an ordered structure for the dataset containing information about every value in a given range, and it uses this structure to generate a proper clustering. The OPTICS indexing structure is based on the assumption that given a constant *min*\_*pts* value, density-based clusters with respect to a higher density (i.e., a lower value for ) are completely contained in density-connected sets with respect to a lower density (i.e., a higher value for ). For each point, the structure stores the *core distance* and the *reachability distance*. Given a parameter *min*\_*pts*, the *core distance* of a point *p* is the distance to its *minPtsth* nearest neighbor (it is undefined whether *p* has less than *minPts* neighbors). The *reachability distance* of point *p* with respect to a point *o* is, intuitively, the smallest distance such that *p* is directly density-reachable from *o* if *o* is a core point. By exploiting these above-introduced concepts, the OPTICS-xi algorithm is capable of generating an indexing structure of the dataset that keeps the cluster hierarchy for a variable neighborhood radius. Now, if a specific value for is chosen, by exploiting the structure, it is possible to perform a clustering that is very similar to the DBSCAN one. Given the generated values of reachability distance stored in the OPTICS indexing structure, the algorithm first generates the related *reachability plot*, and then it looks at the steep slopes within the graph to find clusters. The *ξ* (0 < *ξ* < 1) parameter is exploited to define what counts as a steep slope. The results of the *xi* clustering extraction method are very sensitive to the tuning of the *ξ* parameter. The OPTICS-xi time complexity is *<sup>O</sup>*(*nlogn*).
