## 2 Related Work
### Efficient Privacy Preserving \(k\)-NN Queries  
In spatial databases, the processing of \(k\)-NN query over stationary objects has be extensively studied. The last decade, thanks to technological advances, real-time spatio-temporal data of moving objects can be monitored and in following processed. Hence, the Continuous \(k\)-NN querying in real-time and dynamic environments has attracted the attention of researchers [21]. In [22], the authors attempt to develop an efficient algorithm to process the \(k\)-NN queries on uncertain locations of objects. A probability model is designed to quantify the possibility of each object being one of the \(k\) nearest neighbors. The uncertainty of the object's location lies in the fact that their position is monitored by a sensor-based tracking system, instead of the GPS technique. In a recent work, the authors study the problem of \(CkNN\) queries on moving objects to retrieve the \(k\)-NNs of all points along a query trajectory in spatial road networks [23]. A novel direction-aware \(CkNN\) algorithm, the so-called \(DACKNN\), is recommended. To ensure an efficient query, the algorithm excludes from the analysis moving objects that are far away from the query point. In [24] a fast continuous query privacy-preserving framework in road networks is recommended, based on the concepts of both \(k\)-anonymity and \(l\)-diversity. The authors in [25] propose a method to make location cloaking less vulnerable to query tracking attacks. The proposed method is applied on road networks, such as subways, railways, and highways, where the road network is known and fixed, except for the trajectories. It is called adaptive-fixed \(k\)-anonymization and generates smaller cloaking regions without compromising the privacy of the query issuer's location. Futhermore, the authors in [26] study the problem of location disclosure adopting the \(k\)-anonymity method in the centralized architecture based on a single trusted anonymizer. However, this strategy may compromise user privacy involving continuous LBSs. A dual-\(K\) mechanism (DKM) is suggested to protect the users' trajectory privacy for continuous LBSs. The proposed method firstly inserted multiple anonymizers between the user and the location service provider (LSP). The \(k\) anonymization is achieved by sending \(k\) query locations to different anonymizers. To improve user trajectory privacy, the dynamic pseudonym mechanism is combined with the location selection one. Hence, the user trajectory (spatio-temporal points) cannot be obtained by neither the LSP nor the anonymizer. Note that in our previous work [9], a pseudonyms system is recommended to ensure and/or reinforce the privacy of mobile objects. Actually, a mobile user can initiate a discrete number of \(k\)-NN queries in each spatio-temporal point with different pseudonyms for each one. The recommended protocols provide unlinkability, thus, the service provider cannot connect to the IP and validate that they belong to the same user. The main characteristic of this approach is that the \(k\)-NN queries can be issued not only in Euclidean space but also in Hough-X/Y space. As the analysis and results show, Hough space is an appropriate solution as it preserves a user's privacy and provides storage efficiency as well, assuming that the initial non-linear trajectory of a moving object is splitted into a set of linear sub-trajectories. In this work, the problem of \(k\)-anonymity for privacy preserving in spatio-temporal databases is evaluated in a distributed environment. Although traditional privacy preserving solutions have been designed in Euclidean space, our framework assumes the concept of \(k\)-anonymity in Hough space as well. Due to the constant evolution of the mobile objects' location information in time, it is required to evaluate massive single query point \(k\)-NN queries for massive mobile objects per timestamp. The spatio-temporal \(k\)-NN queries are issued with the aim to formulate the \(k\)-anonymity set of moving objects.