# A Distributed Execution Pipeline for Clustering Trajectories Based on a Fuzzy Similarity Relation
## 2 Literature  
Since [PERSON] initiated fuzzy logic [8], researchers have taken interest in defining fuzzy clusters, which reflect more the fuzzy nature of features, instead of crisp ones. Consequently, fuzzy clustering approaches have emerged. Research on fuzzy clustering can be categorised into approaches leveraging an objective function (fuzzy c-means) and approaches manipulating a proximity relation. The last ones can be further classified into approaches handling directly the proximity relation and approaches proposing techniques to convert the proximity relation into a similarity relation then apply a specific clustering technique to get the different partition trees [9]. Research on directed approaches include [10]. The authors proposed a new approach based on the entropy measure, which lies in \([0,1]\). The main idea is that this measure between two nodes equals 0 if they are respectively very close or very far from each other and 1 if they get close to the average distance between all nodes. The entropies are measured from a proximity relation and cluster centres defined by the lowest entropies. [PERSON] et al. [11] defined a quadratic time complexity algorithm for defining clusters from a proximity relation. The authors define a cluster as a maximum weighted clique of nodes. The weights refer to the different similarities. The obtained clusters are both compact and well separated. Both the supra works define a cluster node as a node similar to all the other nodes in the same cluster. In contrast, [PERSON] [12] defined an approach based on cluster centres. Each cluster node is similar to the centre and the centres are updated at each iteration. Different algorithms exist in the literature to identify the transitive closure. The most known ones are the smart [6] and semi-naive [7] algorithm. They are leveraged to extract a similarity relation leading to extended clusters. [PERSON] et al. [13] introduced the max-min transitivity that yields a similarity relation having equivalence relations as a resolution form. Each equivalence relation leads to non-overlapping partition trees. [PERSON] et al. [14] defined max-prod and max-\(\Delta\) as an extension of the previous work. However, the resulting relation can lead to overlapping partitions. Hence, the necessity for specific algorithms to extract non-overlapping clusters based on a specific cluster definition. Also, the directed approaches can handle these max-t transitive relations to extract non-overlapping clusters. [PERSON] et al. [15] identified clusters by considering a metric of trapezoidal numbers and leveraging the max-min transitivity. In the scope of the latter approaches, relatively few works have been conducted into evaluating the performance of computing the transitive closure in a distributed environment. In particular, studies [16; 17] have proposed new fragmentation techniques for computing the transitive closure in a parallel manner. [PERSON] [18] evaluated the smart and semi-naive algorithm in Hadoop map/reduce environment. Although Spark outperforms Hadoop in processing, none of the previous works have evaluated the transitive closure in its environment. An integration in Spark was only given in [19; 20]. Initially, we extended the approach of [11] by leveraging Spark into clustering spatial trajectories. Still, our proposition could not fully distribute the clustering process. Moreover, we have not considered the transitive closure and we did not give a detailed implementation of the distributed proximity relation construction process. In the latter, we applied the max-min transitivity over a proximity relation referring to the similarities between cyber-criminals in Twitter. Unfortunately, we did not evaluate the performance nor have we considered spatial data. Decidedly, the literature includes several works related to trajectory clustering [21] and its applications on resolving real world problems [22].