## 4 Data Acquisition via V2U Communications  
### Data Distribution at Roadside Units Based on Stable Matching of Bipartite Graph  
Stable matching is a specific approach in bipartite graph matching theory. It requires that each vertex on one side has a strict preference ranking over the vertices of the other side. A matching between the left vertices and the right vertices is stable if there is no left-right pair who both strictly prefer each other to their current partners. The concept of stability captures fairness conditions for market participants and has had enormous influence on the design of real world matching markets [28]. In stable matching theory, take matching students and colleges as an example. The goal is to design a self-reinforcing admissions process given a set of preferences among colleges and students. Student \(x\) and college \(y\) are unstable if \(x\) prefers \(y\) to its assigned college, and meanwhile \(y\) prefers \(x\) to one of its admitted students. The unstable pair indicates a way to improve the matching, such as assigning \(x\) to \(y\) in the instance. An assignment with no unstable pairs is called a stable assignment. Moreover, in order to handle the possible weak ranking orders in stable matching, [PERSON] and [PERSON] design a polynomial-time stable improvement cycles algorithm for the computation of a student-optimal school choice when some ranking tie exists by discovering and removing the stable improvement cycles [29]. When formulating the content distribution problem as a two-sided stable matching problem, we take the content \(\mathcal{C}\) as the left vertices \(L\) and the storage cells of roadside units \(\mathcal{U}\) as the right vertices \(R\), and initialize the edge set \(E\) with \(\emptyset\). The bipartite graph is \(BG=<L,R,E>\). Since the bipartite graph matching is one-to-one mapping, for those contents having more than one replica carried by different roadside units, a preliminary algorithm is required, which will be discussed later. Then, set the ranking rules to calculate the left vertices' preference profile and the right vertices' priority rankings. Thereafter, solve the stable matching problem using Gale-Shapley algorithm [30] and the stable improvement cycles algorithm. We explain modelling the preference ranking of content and the priority ranking of a storage cell below. In order to achieve a high access ratio and a short access latency, a compromise solution is designed, in which the preferences and the priorities are calculated based on the corresponding prior knowledge. A preference profile is a vector of linear orders as \(P=(p_{k})_{k\in\mathbb{C}}\), where \(p_{k}\) denotes the preference ranking of content \(c_{k}\) over all the storage cells. Take an arbitrary left vertex \(L(k)\), which represents content \(c_{k}\), as an example. The preference of \(L(k)\) over an arbitrary right vertex \(R(j,z)\), which representsthe \(z\)th storage cell in a roadside unit \(u_{j}\), is denoted as \(p(c_{k},u_{j})\). To calculate the preference, we set a response factor matrix as  
\[M=F^{T}\times G. \tag{3}\]  
In \(M=(m_{kj})_{k\in\mathcal{C},j\in U}\), \(m_{kj}=\sum_{i\in V}f_{ik}g_{ij}\), which indicates the number of satisfied requests by a replica of \(c_{k}\) carried by \(u_{j}\). In this article, we set  
\[p(c_{k},u_{j})=m_{kj}=\sum_{i\in V}f_{ik}g_{ij}. \tag{4}\]  
Thus, the preference implies the access ratio of a content replica. It is observed that the larger \(p(c_{k},u_{j})\) is, the more requests will be satisfied because of the replica of \(c_{k}\) at \(u_{j}\). Therefore, in the data acquisition model, content \(c_{k}\) prefers to be held by a roadside unit \(u_{j}\) having a larger \(p(c_{k},u_{j})\). To achieve this, sort the roadside units in a descending order of \(p(c_{k},u_{j})\), which indicates the preference ranking of the left vertex \(L(k)\) over all the storage cells of roadside units. Especially noteworthy is that the different storage cells of a roadside unit have the same preference.