## 4 Results
### Vulnerability Evaluation  
In this subsection, some additional experiments based on a real dataset with parameters presented in Table 11 and Figures 9-11, are conducted. The \(k\)-anonymity set computation is utilized with the aid of \(R\)-Tree indexing. As GeoSpark works with spatial data in two-dimensional space, our focus is on three different cases for the \(k\)-anonymity set computation in order to evaluate the vulnerability under the formulated anonymity set in each space, which are:  
1. Euclidean coordinates \((x,y)\)
2. Polar coordinates \((u,\theta)\)
3. Hough-\(X\) coordinates of \((x,y)\) denoted as \((U_{x},a_{x})\) and \((U_{y},a_{y})\)  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Time Results (Minutes) for Different Spatial Partitioning Methods**} \\ \hline Mobile Objects & \(R\)-Tree & \(KDB\)-Tree & Completed Jobs \\ \hline N = 80 & 1.4 & 1.5 & 766 \\ \hline N = 500 & 3.6 & 3.7 & 4546 \\ \hline N = 2000 & 12 & 12 & 18,046 \\ \hline \multicolumn{4}{c}{**Data Distribution for 2 Different Spatial Partitioning Methods**} \\ \hline Mobile Objects & R-Tree & KDB-Tree & \\ \hline N = 80 & 40 40 0 & 30 25 25 \\ \hline N = 500 & 250 250 0 & 236 132 132 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Impact of Spatial Partitioning in Time Performance.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Anonymity Set Size \(k\)** & **Anonymity Set Attributes** \\ \hline
5, 10 & \((x,0)\) \\
5, 10 & \((0,y)\) \\
5, 10 & \((x,y)\) \\
5, 10 & \((U_{x},a_{x})\) \\
5, 10 & \((U_{y},a_{y})\) \\ \hline \hline \end{tabular}
\end{table}
Table 11: Parameters when using \(N=500\) Trajectories, \(L=100\) Timestamps.  
Figure 9: (**a**) Euclidean Space and (**b**) Polar Space for \(N=500\) Trajectories, \(L=100\) Timestamps.  
The experimentation has been carried out on a dataset of size \(N=500\) Trajectories (or moving objects).  
Actually, Figure 9 on the left depicts that assuming only the \(x\) feature (or \(y\) one) for the anonymity set formulation, the vulnerability in both cases is about 2 times higher than in the case when both attributes \((x,y)\) are considered. Similarly, in Figure 9 on the right, the vulnerability is measured in polar space, where the performance is worse than in Euclidean one. This is attributed to the fact that polar coordinates are linearly dependent on spatial data \((x,y)\), revealing the curse of dimensionality. Also, this figure reveals that the anonymity set varies from timestamp to timestamp, meaning that for given objects angle direction and velocity, more than one out of \(k\) nearest neighbors change.  
Finally, Figures 10 and Figures 11 show the vulnerability performance in Hough-\(X\) space, which is compared with the corresponding one in Euclidean space. Obviously, the results are better in Euclidean space as Hough-\(X\) space is linearly dependent on the former. Finally, focusing on the case where \(k=10\) in Figure 12, the vulnerability varies slowly around 0.4 and 0.3 (red and orange dashed lines respectively) in Hough-\(X\) space, while in Euclidean space, it achieves values very close to 0.1 (purple solid lines).  
Figure 11: Hough-\(X\) Space of \(y\) for (**a**) \(k=5\) and (**b**) \(k=10\) for \(N=500\) Trajectories, \(L=100\) Timestamps.  
Figure 10: Hough-\(X\) Space of \(x\) for (**a**) \(k=5\) and (**b**) \(k=10\) for \(N=500\) Trajectories, \(L=100\) Timestamps.