## 4 Proposed Method
### Basic Strategy to Solve Design Challenges  
(i-a) Stimulating mobile hosts at suitable positions to volunteer to serve as REPs  
Each mobile host volunteers to serve as a REP if neither its one-hop neighbors nor itself has been a REP within time \(T^{\prime}{}_{q}\) that is shorter than \(T_{q}\). \(T^{\prime}{}_{q}\) is a configuration parameter set at a value less than or equal to \(T_{q}\). Thus, one mobile host volunteers to serve as a REP for its one-hop neighbors at an interval shorter than or equal to \(T_{q}\). Then, the REP sends a Bid Invitations Query message to the server. To deal with changes in one-hop neighbors due to their movement, a REP finishes its role as REP after sending a Bids message to the server, and another mobile host volunteers to serve as a new REP.  
Later, we will introduce another type of REP to mitigate the effects of misbehaving REPs that fail to fulfill the role of REP, and we need to distinguish them from the normal REPs described here. We denote a normal REP as a primary REP (P-REP) hereafter and the REP that will be introduced later as a secondary REP (S-REP). However, in the following, we just use the term REP when there is no need to distinguish between P-REP and S-REP.  
(i-b) Selection of REPs to which the server sends a bid invitation for a data request  
The server estimates abstracted locations of mobile hosts and selects a subset of REPs to which it sends a bid invitation for a data request in a similar way to the case with the existing method using a neighbor existence area (NEA). An NEA abstracts information on the locations of a REP and its one-hop neighbors with a constant data size. An NEA consists of a small circle (center O, radius \(r\)) that covers the area where the REP and its one-hop neighbors exist and the number of mobile hosts \(N\) (Figure 9). The server receives NEA information from each REP with a Bid Invitations Query message. For each data  
Figure 8: Example of case where \(\cup_{i\in S_{\text{r}}}G_{i}\
eq S_{\text{T}}\). request, the server calculates the minimum value of \(R\), the radius of the circle \(C\) centered on the POI for data request \(q\), so that the expected number of mobile hosts in circle \(C\) can be greater than or equal to the configuration parameter \(\theta_{n}\) (Figure 10). Mobile hosts in the NEA are considered to be in circle \(C\) only when circle \(C\) covers the NEA. The server determines that a P-REP or one-hop neighbors of the P-REP are likely to obtain data for data request \(q\) if the NEA reported by the P-REP overlaps with circle \(C\). Then, the server sends a bid invitation for data request \(q\) to the P-REP (Figure 10). Thus, \(S_{p}(q)\supseteq S_{e}(q)\) will be satisfied. Next, we explain how the above strategy can make \(S_{p}(q)\) include a mobile host that is not in \(S_{e}(q)\) and how it can also make \(S_{p}(q)\) exclude a mobile host in \(S_{e}(q)\). With this strategy, the server sends a bid invitation for data request \(q\) to a P-REP and its one-hop neighbors if the NEA reported by the P-REP overlaps with circle \(C\). Because the NEA includes areas where neither the P-REP nor its one-hop neighbors exist, the P-REP and its one-hop neighbors can receive bid invitations for more data requests than in the existing method. Therefore \(S_{p}(q)\) can include a mobile host that is not in \(S_{e}(q)\) (Figure 6). Moreover, a mobile host can be a one-hop neighbor of different REPs. Thus, each mobile host may be counted multiple times in the number of mobile hosts \(N\) of different NEAs (e.g., Figure 11). In this case, the radius \(R\) of circle \(C\) can be estimated to be smaller than its actual value because the number of mobile hosts in circle \(C\) is estimated to be greater than the actual value. If the estimated value of \(R\) becomes less than its actual value, mobile hosts that actually need to receive a bid invitation for a data request may not receive the bid invitation. Therefore, \(S_{p}(q)\) may not include a mobile host that is in \(S_{e}(q)\).