# The Efficient Processing of Moving \(k\)-Farthest Neighbor Queries in Road Networks +
## 6 Evaluation of the Example M&FN Query Using MOFA
### Computing the Valid Segments for the Query Segment \(\overline{v_{1}v_{2}}\)  
Figure 12 shows the changes in the distance from each candidate facility \(f\) to a query point \(q\) in \(\overline{v_{1}v_{2}}\), where \(f\in\{f_{1},f_{2},f_{3}\}\). The distance from \(f_{1}\) to \(v_{1}\) (\(v_{2}\)) is \(d(f_{1},v_{1})=21\) (\(d(f_{1},v_{2})=17\)). Figure 12a demonstrates the change in the distance from \(f_{1}\) to \(q\) in \(\overline{v_{1}v_{2}}\). The shortest path from \(f_{1}\) to \(q\) in \(\overline{v_{1}v_{2}}\) is either \(f_{1}\to v_{1}\to q\) or \(f_{1}\to v_{2}\to q\). The distance from \(f_{1}\) to \(q\) is evaluated as \(d(f_{1},q)=\{min(d(f_{1},v_{1})+len(\overline{v_{1}q}),d(f_{1},v_{2})+len( \overline{v_{2}q}))|q\in\overline{v_{1}v_{2}}\}=\{min(21+len(\overline{v_{1} q}),17+len(\overline{v_{2}q}))|q\in\overline{v_{1}v_{2}}\}\), where \(len(\overline{v_{1}q})+len(\overline{v_{2}q})=5\). Similarly, the change in the distance from \(f_{2}\) to \(q\) in \(\overline{v_{1}v_{2}}\) can be drawn (Figure 12b), where \(d(f_{2},v_{1})=22\) and \(d(f_{2},v_{2})=18\). In the same manner, the change in the distance from \(f_{3}\) to \(q\) in \(\overline{v_{1}v_{2}}\) can also be drawn (Figure 12c), where \(d(f_{3},v_{1})=21\) and \(d(f_{3},v_{2})=16\).  
Figure 13 plots \(d(q,f_{1})\), \(d(q,f_{2})\), and \(d(q,f_{3})\) together, which are illustrated in Figure 12a-c, respectively, to compute the valid segments in \(\overline{v_{1}v_{2}}\). Recall that the query points in the valid segment have the same \(k\) FNs. The example M&FN query requires two facilities farthest from the query point \(q\) in \(\overline{v_{1}v_{2}}\); hence, the second FN must be identified from  
\begin{table}
\begin{tabular}{c l l l} \hline \(q_{b}\) & \(\{\overline{f_{1}}\overline{f_{2}}\overline{f_{3}}\}\) & \(\{\overline{f_{4}}\overline{f_{5}}\overline{f_{6}}\}\) & \(d_{cond}(q_{b})\) & \(\Sigma(q_{b})\) \\ \hline \(v_{1}\) & \(d_{max}(v_{1},\{\overline{f_{1}}\overline{f_{2}}\overline{f_{3}}\})=23.5\) & \(d_{max}(v_{1},\{\overline{f_{4}}\overline{f_{5}},\overline{f_{5}}\overline{f_ {6}}\})=12\) & \(d_{cond}(v_{1})=16\) & \(\Sigma(v_{1})\)=\(\{f_{1},f_{2},f_{3}\}\) \\ \(v_{2}\) & \(d_{max}(v_{2},\{\overline{f_{1}}\overline{f_{2}}\overline{f_{3}}\})=19\) & \(d_{max}(v_{2},\{\overline{f_{4}}\overline{f_{5}},\overline{f_{5}}\overline{f_ {6}}\})=7\) & \(d_{cond}(v_{2})=12\) & \(\Sigma(v_{2})\)=\(\{f_{1},f_{2},f_{3}\}\) \\ \hline \end{tabular}
\end{table}
Table 2: Summary of the computation of the candidate facilities for \(\overline{v_{1}v_{2}}\).  
Figure 11: Arrangement of the facility clusters based on their largest distance to \(v_{2}\).  
Figure 10: Arrangement of the facility clusters based on their largest distance to \(v_{1}\).  
each query point \(q\). Figure 13 shows that facility \(f_{1}\) is the second FN for the entire query segment \(\overline{v_{1}v_{2}}\), and facility \(f_{2}\) is farther from the query point \(q\) than the second FN \(f_{1}\). Therefore, a valid segment can be found in \(\overline{v_{1}v_{2}}\), and the query result is \(\Phi(\overline{v_{1}v_{2}})=\{\langle\overline{v_{1}v_{2}},\{f_{1},f_{2}\}\rangle\}\).