# A Search Efficient Privacy-Preserving Location-Sharing Scheme in Mobile Online Social Networks
## 7 Simulation and Results
### Simulation Results  
Figure 8 shows the results under scenario-1. From Figure 7(a), it can be known that the number of effective users increases with \(f\) no matter which solution is used. The reason is that the increasing \(f\) leads to the increasing number of users' friends and the corresponding dummies. The number of effective users by using our scheme is smaller than other solutions since our SELS can efficiently filter out the issuing user's friends who are not in the issuing user's associated grids. However, other solutions have to provide all the issuing user's friends and the corresponding dummies to \(LS\), which inevitably increases the computational burden of \(LS\). From Figure 7(b), it can be seen that: (1) the query time increases with \(f\) in all schemes; (2) when \(50\leq f\leq 90\), the query time of our SELS is at least \(1/3\) of that of other schemes, i.e., the efficiency of our scheme is at least 3 times better than other solutions due to the same reason as above. In addition, it can be summarized that the effect of \(f\) on the query time is linear in all schemes. The reason is that when other parameters are fixed, the number of users (including friends and dummies) that need to be compared for distance with the issuing user is almost fixed, and the only factor that affects the query time is a linearly increasing number \(f\). Figure 9 shows the results under scenario-2. From Figure 8(a), it can be seen that when using our scheme, the number of effective users increases with \(l\), while the number of effective users is almost stable when using other solutions. The reason is that the number of the effective users increases  
Figure 8: The simulation results for scenario-1. with the query distance \(l\) which determines the number of associated grids in our scheme. However, for other solutions, the friends who are not in the issuing user's associated grids can not be filtered out, which leads to the result that all the friends have to be retrieved by \(LS\). Thus, it can be concluded that the efficiency of our SELS is better than other solutions since the number of effective users in our scheme is smaller than other solutions under different query distances. From Figure 8(b), it can be known that: (1) the query time increases with \(l\) in all schemes; (2) when the query distance \(l\) changes from 1 km to 5 km, the query time of our SELS is at least 1/3 of that of other schemes, i.e., the query efficiency is at least 3 times better than other solutions since a smaller number of effective users need less computational cost. In addition, we can also summarize that the effect of \(l\) on the query time is quadratic in all schemes. The reason is that when the query distance changes, the number of users (including friends and dummies) within the issuing user's query distance also changes, and the change of these users is quadratic. Furthermore, the reason for different curves in Figure 8(b) among our scheme and other solutions is as follows. When other parameters are fixed, different \(l\) leads to the change in the number of effective users and the number of users (including friends and dummies) within the issuing user's query distance in our scheme. While in other schemes, different \(l\) only leads to the change in the number of users (including friends and dummies) within the issuing user's query distance. Figure 10 shows the results under scenario-3. From Figure 9(a), it can be known that the number of effective users increases with \(k\) in all schemes. The reason is that increasing \(k\) leads to an increasing number of friends' dummies. The number of effective users by using our scheme is smaller than other solutions since our SELS can efficiently filter out the issuing user's friends who are not in the issuing user's associated grids. However, other solutions have to provide all the issuing user's friends and the corresponding dummies to \(LS\), which inevitably increase the computational burden of \(LS\).