# Selecting Remote Driving Locations for Latency Sensitive Reliable Tele-Operation
## 3 Selecting Driving Locations for Multiple Paths
### Selecting the Smallest Number of Remote Drivers under Latency Constraint  
In this subsection, we propose an algorithm called Extended Remote Driver Selection for Multiple Paths (ERDSMP) for Objective I. Actually, ERDSMP tries to improve the performance of RDSMP of [7]. Let \(L\) be the latency constraint, which is the largest allowable latency between a remote driving location and the vehicle. If the latency is larger than \(L\), the safety of the remote driving degrades. We try to find a smallest set of driving locations (or simply drivers) that satisfy the latency constraint. Furthermore we want to minimize the average latency between the selected driving locations and the sequence of the base stations of the path. Since at every moment, the distance between a driver and the vehicle should be within \(L\), we define the distance between a selected set of driving locations, \(D\subseteq\mathcal{F}\), and the set of base stations, \(B\subseteq\mathcal{B}\) in Equation (1). \[Dist_{max}(D,B)=\max_{b\in B}\Bigl{(}\min_{d\in D}dist(d,b)\Bigr{)} \tag{1}\]  
Thus, the algorithms for Objective I are to solve the following optimization problem. In other words, the longest distance between \(D\) and \(B\), the set of base stations for the given paths \(P\), should be less than or equal to \(L\). By this way, the safety of the remote control system is obtained. \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Definition** \\ \hline \(\mathcal{F}\) & \(\{d_{1},d_{2},\ldots,d_{m}\}\), the set of candidate driving locations. Thus, there are \(m\) candidate locations, from which we select a set of driving locations. \\ \hline \(\mathcal{B}\) & \(\{B_{1},B_{2},\ldots,B_{n}\}\), the set of base stations in the area. Thus, there are \(n\) base stations in the area. \\ \hline \(P\) & \(\{P_{1},P_{2},\ldots,P_{l}\}\), the set of paths given to the algorithms. Thus, there are \(l\) paths. \\ \hline \(dist(d,b)\) & the distance (or latency) between a driving location \(d\in\mathcal{F}\) and a base station \(b\in\mathcal{B}\). \\ The distance is the network distance. We use distance and latency interchangeably. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notations. \[ \begin{array}{ll}\text{minimize}&|D|\\ \text{such that}&D\subseteq\mathcal{F}\\ &Dist_{max}(D,B)\leq L\end{array}  \tag{2}\]  
Now, we describe ERDSMP in detail. ERDSMP actually uses RDSMP in it. So, we briefly describe RDSMP [7]. As we define before, there are \(l\) origin-destination pairs in \(P\). \(P_{i}\) is the \(i\)th path, which has a sequence of base stations \((B_{i_{1}},B_{i_{2}},\ldots,B_{i_{l_{l}}})\) of length \(l_{i}\). The objective of RDSMP is to find a reasonably small set of driving locations \(D\subseteq\mathcal{F}\) that can cover all the base stations in \(P\). The first step of RDSMP is to reduce the search space by filtering out the driving locations that are not within the latency of \(L\) from any of the base stations in \(P\). It is clear that those driving locations cannot be used to cover any of the base stations. Let \(C\) be the remaining driving locations, which are the reduced set of candidate driving locations. For each driving location in \(C\), we compute how many base stations that the driving location can cover under the given latency constraint. For example, if a driving location \(d\) covers 2 base stations in \(P_{1}\), 3 base stations in \(P_{2}\), and 4 base stations in \(P_{4}\), we consider that \(d\) covers 9(=2 + 3 + 4) base stations. Furthermore, when doing this counting, we only count the consecutive base stations from each \"end\" of the paths in \(P\). In other words, even though a driving location covers 10 base stations in the \"middle\" of a path, we consider that the driving location covers 0 base stations because it does not cover any base station at the two ends of the path. The reason is because we do not want to break the sequence of base stations into many pieces. For example, consider a path with 100 base stations.