## 2 Related Works
### Probabilistic Data Dissemination Algorithms for VANETs Based on Distance  
Many probabilistic data dissemination schemes that use the relative Euclidean distance among vehicles in VANETs can be found in the literature [18; 19; 23; 28]. In ref. [18], the authors study the performance of p-persistence scheme for VANET scenarios, which determines lineally the retransmission probability with respect to the relative Euclidean distance between two vehicles and the vehicle's wireless communication range, which is assumed to be equal for all the vehicles in the VANET. The p-persistence scheme employs the following expression:  
\[\mathbf{p}=\frac{\mathbf{d_{ik}}}{\mathbf{R}}\ \mathbf{0}<\mathbf{d_{ik}}<\mathbf{1} \tag{1}\]  
where \(\mathbf{p}\) is the retransmission probability of the receiver to retransmit an incoming message, \(\mathbf{R}\) is the vehicles' wireless communication range and \(\mathbf{d_{ik}}\) is the relative Euclidean distance between two vehicles \(\mathbf{i}\) and \(\mathbf{k}\). A simple evolution of p-persistence scheme is the polynomial scheme [22], which uses an exponent \(\mathbf{b}\) to calculate the retransmission probability according to the following equation:  
\[\mathbf{p}=\left(\frac{\mathbf{d_{ik}}}{\mathbf{R}}\right)^{\mathbf{b}} \tag{2}\]  
Moreover, in [28] the authors propose irresponsible scheme, a probabilistic scheme based on the Cumulative Distribution Function (CDF) of the Euclidean distance between two vehicles. The main idea of irresponsible forwarding is that a vehicle should not retransmit an incoming message if it is probable that there is another node located farther away from the sender. The following expression is used by irresponsible scheme to determine the retransmission probability:  
\[\mathbf{p}=\left(\mathbf{1}-F_{x_{ij}}(\mathbf{R}-\mathbf{d_{ik}})\right)^{\mathbf{1}/p} \tag{3}\]\(F_{\mathbf{x}_{ij}}\) is the CDF of the Euclidean distance between two vehicles \(i\) and \(\mathbf{k}\), and \(\mathbf{v}\) is a tuning parameter to adjust the retransmission probability. The value of \(F_{\mathbf{x}_{ij}}\) depends on the spatial distribution of the vehicles in the VANET. In ref. [18], the authors demonstrate that exponential and lognormal distributions are likely to be found in VANET scenarios. In the case of the exponential distribution, the retransmission probability can be determined as follows:  
\[\mathbf{p}=\mathbf{\exp}\left(-\frac{\mathbf{\rho}_{\mathbf{s}}(\mathbf{R}-\mathbf{d}_{\mathbf{ik}})}\mathbf{{v}}\right) \tag{4}\]  
where \(\mathbf{\rho}_{\mathbf{s}}\) is the spatial density of vehicles in the VANET (vehicles/meter) and \(\mathbf{v}\) the mentioned adjusting parameter. Other data dissemination schemes based on the irresponsible forwarding scheme can be found in [19; 23]. In ref. [19], the authors modified the expression (4) to reflect that vehicles may have different wireless communication ranges. The resulting expression is:  
\[\mathbf{p}=\mathbf{\exp}\left(-\frac{\mathbf{\rho}_{\mathbf{s}}(\mathbf{t}-\mathbf{d}_{\mathbf{ik}})}{ \mathbf{v}}\frac{\mathbf{R}}{\mathbf{R}_{i}}\right) \tag{5}\]  
where \(\mathbf{R}_{i}\) is the wireless communication range of the \(i\)th receiving vehicle and the ratio \(\mathbf{R}/\mathbf{R}_{i}\) represents the differences in the vehicles' wireless communication areas. In [23], the authors improve the silencing mechanism proposed and analyzed in [22]. The idea is to use the relative Euclidean distance among vehicles to enhance the silence mechanism. In ref. [24], the authors demonstrate that dissimilarity metrics can also be suitable parameters to improve the performance of the silence mechanism. All reviewed data dissemination scheme for VANETs based on the Euclidean present the same problem; all of them rely on the hypothesis that the wireless communication range of vehicles can be known. However, in real testbeds, it has been shown that the actual wireless communication range of a wireless node can be very different from the nominal range [29].