I Introduction
In recent years, the rapid advancement of vehicletovehicle (V2V) communications has enabled new applications for vehicle safety, traffic efficiency and autonomous driving. These applications could potentially improve the driving experience, reduce commute time and fuel consumption, enhance traffic safety, and contribute to other vehiclerelated areas in a significant way.
For many of these applications, a regulator vehicle is needed to regulate a group of vehicles, the size of the vehicle group is normally smmall (typically smaller than 40). Such leader vehicle plays an important role in many applications that aim to achieve a cooperative goal for all vehicles in the group. Viable applications include but not limit to Virtual Traffic Lights (VTL) [6, 18, 22]; intersection management/coordination [4, 16, 13]; onramp merging [1]; Cooperative Adaptive Cruise Control (CACC) system and platoon maintenance [21, 9], etc. These applications require vehicles to communicate through noisy V2V communication channel distributively and eventually come to a consensus on the selection of leader vehicle.
Therefore, a leader selection scheme for over vehicular adhoc network (VANET) is of great interest for connected vehicles (CV) and connected automated vehicles (CAV) related applications. While there are existing several results on leader selection in dynamic adhoc networks [7, 14, 19, 20, 3], these results are not particularly suitable for vehicular network. One of the crucial aspects is that vehicular network protocols such as SAE 2735[15] periodically broadcast messages to the surrounding vehicles, this special broadcast feature in vehicular network (which a general adhoc network normally doesn’t have), such feature could be helpful for leader selection within a small size of group (typically a size smaller than 40) and should be Incorporated into the leader selection process.
In this paper, a new proactive leader selection algorithm is introduced. A simple analysis is then carried out to evaluate the performance of the algorithm. Simulation results that evaluate the performance of the algorithm in realistic vehicular scenario are provided in the later part of the paper. Both analysis and simulations have shown that the algorithm has several desirable features:

The algorithm is extremely simple and only require a minimum amount of information sent. But very efficient in performance with fast convergence time and fast restabilization ability when erroneous events happen.

The algorithm can be highly bind with vehicular network protocols [10], the algorithm could be implemented by only add a customized field or header to the Basic Safety Message (BSM).

The algorithm is robust under highly dynamic environment. The algorithm does not require a static topology and reliable communication channels, in fact, the algorithm does not even require the whole network to be connected for all the time.
Ii Related Works
Previously, several algorithms have been proposed for leader selection in a distributed system where processes communicates over an unreliable channel. To give some examples,[7] gives a leader selection algorithm for a large group of nodes, assuming random link failure and random node crash, the algorithm is highly scalable, but with only a probabilistic guarantee; [19] gives two secure leader selection algorithm for an adhoc network, but it assumes static topology during the initialization phase, which could be unrealistic; several leader selection algorithms are proposed for dynamic adhoc networks [20, 3, 14], these algorithms propose to maintain a spanning tree or a directed acyclic graph (DAG), these algorithm needs the nodes to be constantly aware of network connectivity and needs to handle nodes crash and network joins explicitly, such algorithms are too complicated for a local cooperative application in vehicular network, and they are able to utilize the existing broadcast feature of vehicular network protocols. Another drawback of the aforementioned algorithms is that these algorithms are strict extremafinding algorithms that requires leader switch when there is the order of nodes changes, which is not a desirable feature for realistic applications. In realistic vehicular applications, it is not preferable to switch leaders often as it will create gap period during the process. As in most of the time a leader’s order drops from ’the first’ to ’the second’ will hardly affect the operation results in most of the applications, it is such leader switch is unnecessary and should be prevented.
Surprisingly, there are very few leader selection algorithms designed specifically in the context of vehicular networks applications. [17, 5] proposed leaderselection algorithm for a specific application, known as Virtual Traffic Lights (VTL), though the algorithm designed is in the context of vehicular network, the algorithm is still reactive and doesn’t take the aforementioned drawbacks into considerations.
The leader selection algorithm presented in this paper is a proactive algorithm that is highly bound with the vehicular network broadcast feature. Comparing with other algorithms, the algorithm introduced in this paper is specifically designed for VANET applications. The algorithm can be implemented with pure DSRC broadcasting, without any pointtopoint communication, which is a highly desirable feature for implementation in vehicular network. This algorithm is also an approximate extremafinding algorithm, that will select the best leader based on the order function, but will prevent unnecessary leader switch when the order of nodes changes, as discussed above this is a desirable feature for most applications.
Iii Problem Statement
Iiia Application Requirement
We consider a typical VANET scenario of small group (typically, 40 vehicles maximum) of moving vehicles, denote as the i’th vehicle. Based on different applications, the scenario could be different, it could be at the intersection, a merging ramp of the high way or a platoon of vehicles on highway. All vehicles considered are capable of V2V communications, but the communication channel could be noisy and unreliable, any packet could be lost. An order function is given, which gives a strict binary relationship for any two vehicles, it returns 1 or 0 based on the relation of the two vehicles. For convenience, in the rest of the paper, the term and are interchangeable, as well as, and . The order function can vary for different applications, as long as it satisfy

transitivity property: and , then .

Antisymmetry: and , then
As the vehicles concerned are unique, in the following discussion, we also use ’’ and ’’ symbol when the two items for comparison are obviously not the same object. Notice some literature prefer height function in extremafinding to order function, but since this paper is applicationoriented, we choose order function to describe the problem as height functions is not as straightforward as order functions in some applications. Notice it is always simple to convert height function to order function by directly compare the height of the two nodes.
With the scenario setting above, the following functionalities desirable:

The vehicles select a leader based on order function , all the vehicles should come to the consensus and be aware of the same leader. The order function will give a strict order to all the vehicles, desirably, the best vehicle should be the leader.

The leader status needs to be maintained: When the leader disappears, the new leader selection will start; when new vehicles join the group, they will come to the same consensus.

After selecting the leader, the leader status should be persistent, even if the order of nodes changes, or a better vehicle joins, as leader switches will create a time gap which is undesirable for applications.

The leader should be able to broadcast uniform information to all vehicles in the group. We denote this message as leader messages
Iv Algorithm
Iva The basic proactive leader selection algorithm
The leader selection algorithm introduced in this paper is based on proactive broadcasting. All vehicles will broadcast leader message periodically with a period of , to maintain the current leader status. The leader message contains information of the current leader, it will be only generated by the leader vehicle itself, all other vehicles will only relay the leader message. The algorithm introduced here is very simple that leader messages are the only messages sent. The leader messages function as information carrier in leader selection process, as well as a heartbeat indicator to show that the leader still exists after convergence to one leader. Figure 1 gives the flowchart of the leader selection process.
The leader message is defined to contain the following field:

leader: the leader message specifies the id of the current leader.

sequence number: the unique identifier of each message, so that each message will only be relayed once from each vehicle.

information of leader: This field contains information of the leader in this leader message. This field is used for vehicles to compare leader candidate. It should contain all information needed for order function . Most commonly, this field contains GPS information of the vehicle and its unique id to break ties.
Initially, vehicles assign themselves as leader and all of them will issue leader messages and broadcast them. Periodically, the vehicle compare the current leader with all the leader messages in the message buffer with the order function . If any message contains the better leader than the current leader, the vehicle will swap the current leader with the better leader, and relay that better leader message. Notice only the vehicles that consider themselves as leaders issue leader messages, other vehicles only relay leader messages. In this way, after a certain period of time, all vehicles will come to the consensus of one true leader.
The leader status is maintained by the leader which keeps broadcast leader messages, such leader messages are treated as ’heart beat’ that indicates the leader is still functional. Each vehicle will check periodically if the heartbeat of the current leader still exists (by checking if any leader selection message is received within ), if the heartbeat still exists, it means the leader is still alive (functional), otherwise, it means the leader disappeared, and the vehicle will reset its leader status by assigning itself as leader and issue leader messages again.
Notice the vehicle will only start to assign the leadership to itself (reset the leader status) when no leader messages are received. As long as leader messages are received, even if the vehicle itself becomes the better leader candidate, it will not override leader status. This is a desirable design in vehicular network leader selection, as the vehicles are dynamically moving and the GPS signal can be noisy, two vehicle might be always racing. This mechanism is designed to stabilize the leader selection process and quickly converge to a stable leader, and to prevent frequent and unnecessary toggling. Algorithm 1 shows a pseudo code of the algorithm.
While the algorithm described in Algorithm 1
is functional, it can be further optimized to have better performance. In the remaining of this section, further optimizations of the algorithm are introduced in terms of reducing bandwidth, increasing packet receiving probability and a precautionary mechanism.
IvB Preventing broadcast storm
While the aforementioned algorithm is functional, we can further optimize the algorithm in many aspects, one of these aspects is to reduce redundant broadcasting to save communication bandwidth.
Like all proactive algorithms in adhoc networks, the introduced algorithm will generate large amount of the packets and will possibly jam the limited bandwidth in some situation (i.e., the precious bandwidth of DSRC). This is a known drawback of all proactive algorithm. Therefore, in this subsection we introduce several mechanism to address this issue.
IvB1 Avoid unnecessary relay
Since the broadcast messages are In the vehicular network scenario, all vehicles will broadcast Basic Safety Message (BSM) periodically, vehicles will be able to sense their neighbors and maintain a list of all their neighbors, this list will be attached as payload of the leader selection message. Therefore, the vehicle will check the neighbor list of the received leader selection message(s), if all of this vehicle’s neighbors are already in the neighbor list of the received leader selection message(s), the vehicle will not broadcast this message. This will reduce the messages within a platoon, especially when most of the vehicles are connected to each other.
IvB2 Reduce the broadcast frequency when already reach consensus
The leader message plays different roles before and after reaching consensus. Before reaching an consensus, the leader messages are the information carriers, the broadcast frequency of the leader messages directly determine the convergence time. Highfrequency leader messages broadcast will make vehicles come to consensus quickly. After reaching consensus, the leader messages are used as heartbeat to indicate that the leader still exists, it is not critical to remain high broadcast frequency. Therefore, for each vehicle, as long as it doesn’t receive the conflict leader message, it will broadcast (or relay) leader message at a lower rate. This mechanism will make highfrequency broadcasting only happens at the the leader selection process, or whenever a new vehicle joins. Notice that though that the method introduced is based on periodical broadcasting, and will consume some bandwidth, it is the optimal way to maintain the leader status at the intersection, with the minimum bandwidth.
It is worth mentioning that since leader messages are also used for the leader to broadcast application information to other vehicles, if the application information require a high refreshing frequency, the leader shouldn’t reduce frequency after reaching the consensus.
IvC Prevent probability vanishing
In the basic proactive leader selection algorithm, the leader messages are only forwarded when a node receives one, this is desired as we want to prevent the case when leader disappear. However, such design will affect the performance in a network topology when there is a long chain. The probability of successfully delivery of a leader message will be where is the link probability and is the number of hops the message needs to go through. Such a probability vanishes exponentially as the number of hops increase, which is undesirable. Therefore, a modification is introduced to address this issue.
At each period, if the node does not receive the leader message from the current leader, and none of the other leader messages received gives a better candidate, the node will still broadcast the current leader in a special leader message, denote as dummy leader message. Dummy leader messages work the same as normal leader messages, except that they will not reset timer for the detection of heartbeats.
IvD Precautionary mechanism
There is a slight chance that the arbitrary order function that’s based on vehicles’ geoinformation might cause trouble as the geoinformation of vehicles constantly changes. In very slight probability, the vehicles might not be able to break tie because of this. Therefore, precautionary mechanism is introduced that when there is an unresolved leader (conflict leader) for a certain amount of iterations, the vehicle will use a geoinformation free order function (such as comparison of id) to decide leader.
V Analysis
We model the VANET as an incomplete graph with the vertex set and edge set . Some nodes can’t connect to other nodes due to some physical obstacles. The nodes pair connected can communicate with each other with a probability of successful packet transmission. The edges between vehicles blocked by the obstacles are removed. Figure 2 shows one example of it. In this figure, a building blocked some of the communication between the vehicles on different approach, we remove the edges between these vehicles.
For any pair of vehicles , if there exists an edge, we denote the probability of being connected to be . Notice that, it is realistic to assume a lower bound , since we can directly remove those edges with small probability to be connected. The benefit of directly remove edges instead of assigning a small connectivity probability for them is that here we can still assume the connectivity between different pairs to be independent (but for shadowing scenarios, the connectivity are not independent as vehicles in the same area are likely to be disconnected at the same time).
As the topology of the network can variate a lot, we can’t give exact calculation of expected time to converge, but we can give an upper bound:
Theorem 1.
In the incomplete graph described above, if the diameter of the incomplete graph is , for any node in the graph, denote the expected time that this node converges to the actual leader as , is bounded by:
(1) 
Proof.
For any pair of neighbor nodes if sends a leader message, the expected time that this message propagates to is .
For any node in the graph, we first find the shortest path toward the leader node : , the length of this path is . Denote the expected time steps that the leader message propagate through this path . Since there there might exist more than 1 path toward the leader, the expected convergence time for this node is smaller than . Therefore, we have:
(2) 
∎
Theorem 1 gives an ideal upper bound for the convergence time, especially for vehicular network situations, since in a realistic VANET, the graph topology is typically of low diameter. For example, the scenario in figure 2 has a diameter of 3. It is also worth mentioning in a realistic use case of vehicular network, is usually large (close to 1), for example or . For the case in figure 2, the expected time of convergence is . If the broadcast period is 100ms, the expected convergence time is 330 ms.
Vi Simulation Evaluation
Via Simulation scenario
We perform several simulations to evaluate the above algorithm in different aspects. To generate trustful realistic results, we developed a hybrid simulation that simulates both the mobility of vehicles and probabilistic DSRC channel. The mobility is simulated using SUMO, a popular opensource mobility simulator
[12].As for the communication channel modeling, to yield realistic results, proper probability model for packet reception rate (PRR) is required. Previous researches has proposed the Nakagamim distribution for DSRC channel modeling [11], the PRR can be obtained by the following equation:
(3) 
In the equation, is the fading parameter of the signal. It has different values due to the weather, the congestion of the network or the number of buildings. This parameter varies from 1 to 3, corresponds to a harsh communication condition, and corresponds to a good communication condition. Parameter corresponds to the distance between the two vehicles. corresponds to the intended communication range by the radio [2]. This parameter is determined by the radio transmission power. As 802.11p specifies five power levels, can take five different values: 100, 200, 300, 400, 500 [11]. For DSRC OnBoard Units (OBU), is normally used.
Figure 3 shows the packet reception rate on different distance for both and . In the simulation the PRR is truncated at 100 meters, when the distance between the two vehicles is larger than 100, the PRR is set to be 0.
We perform the simulations on a typical onelane intersection. The length of each approach is 100 meters. A traffic light is placed at the intersection that performs periodical phase changes. Only the vehicles before passing the intersection select leader. Therefore, when the leader vehicle passes the intersection, the leader will disappear and other vehicles need to detect this event and reelect the leader. Vehicles arrive at the intersection according to a Poisson process, for convenience, the arrival rate of 4 approaches are set to be the same.
ViB Qualitative results
We first observe the leader status in each SUMO simulation visually to obtain a qualitative understanding of the algorithm. Because that a traffic light is located at the intersection that performs periodically phase switch, the leader will stop at the intersection for a while and leave the intersection. We observe that at the moment that the leader vehicle leaves the intersection, all vehicles that are still at the intersection detect this event automatically and start electing a new leader, this new leader selection period is very short (within 1 second). In terms of leader status maintenance, the performance of the
basic leader selection algorithm is almost identical to the performance of the optimized leader selection algorithm (there is a distinct difference between the two algorithm in terms of messages exchange, more details can be found in section VIC).Figure 4 shows the leader status of one typical simulation in terms of time. The status ’leader selected’ is the status that leader selection algorithm successfully selected one leader at the intersection, the status ’selecting leader’ is the status that the leader selection algorithm is not yet converged and in the process of selecting leader. Because that leader vehicle will periodically leave the intersection, the leader selection algorithm will need to reselect leader periodically, the reselecting process can be clearly observed and located by looking at the spikes in the figure. These spikes are the very short period of leader selection process. In the figure 4, the width of the spikes can’t be observed clearly because they are very short, in VIC, quantitative measurement of these spikes are given, they are less than 1 second. Figure 4 and the simulation observation results qualitatively show that the leader selection algorithm works as expected in a realistic vehicular simulation setting.
ViC Quantitative results
In this subsection, we illustrate quantitative measurements which show the performance of the algorithm. The following metrics are chosen to illustrate the performance, these statistical metrics are collected from 100 simulations, each of which lasts 3 minutes:

Stable percentage: Percentage of time when there is a unique leader at the intersection, it’s the average value over all simulations.

Average convergence time: Time needed to converge to a unique leader, it’s the average over all simulation trials.

Maximum convergence time: Time needed to converge to a unique leader, it’s the maximum of all simulation trials.

Number of messages sent in each simulation. we use average number of messages sent over all simulation trials. This will give the performance of the algorithm in the channel usage aspect.
Algorithm  Basic  Optimized  

traffic volume  medium  dense  medium  dense 
Stable percentage  97%  98%  97%  98% 
Average convergence time (s)  0.66  0.6  0.51  0.39 
Maximum convergence time (s)  0.88  0.83  0.91  0.64 
Number of messages  13474  54743  5080  8829 
Table I shows the results obtained from the simulation. From the table, we observe several interesting facts. The stable percentage, the percentage of the time that all vehicles are having the same leader, is in general very high (97%  98%). the remaining 23% are the duration of leader selection process. This can be justified from the leader status figure in Figure 4.
As for convergence time, the average convergence time is roughly half a second, and the maximum convergence time is less than 1 second. This is an ideal performance as most of the vehicular applications need the leader selection process to be converging fast. An average of half a second and maximum less than a second is sufficient for most of vehicular applications.
Another interesting observation is that the traffic volume does not affect stable percentage and convergence time in a major way, this is counterintuitive at first glimpse, as when vehicles number increases, more vehicles need to come to the same agreement. The reason behind this is that when there are more vehicles, the leader message will be broadcast more times by each vehicle, hence increasing the probability for each vehicle to receive the correct leader message.
Meanwhile, we notice that, compare to the basic algorithm, the optimized algorithm will reduce the amount of messages in a significant way. It reduces 60% of messages in medium traffic and 85% in dense traffic. It is desirable that optimized algorithm reduce more percentage of the messages in dense traffic, because the goal of the optimized algorithm is to reduce broadcast storm, especially under the dense traffic condition.
Vii Discussion
The algorithm described in this paper is based on pure broadcasting, only one message type, the leader message, is needed, and broadcast periodically. Notice this is a highly desirable feature for VANET. Such algorithm can be very easily implemented with SAE2735 protocol for vehicular network. While in the paper, we state the leader message as an independent message, it can be sent with other messages in the form of a header or a customized field. For example, the leader message information can be embedded into the BSM as customized fields, in that case, the algorithm does not even require a special type of the message broadcast. The information required by the leader message is very short, the size of the BSM is not going to be affect in a noticeable way.
One of the future works is to carry out a more detailed mathematical analysis of the performance as well as a strict proof of the correctness of the algorithm. A more comprehensive scheme that specifies leader switches as well as the messages from all member nodes to the leader is also to be researched in the future.
Viii Conclusion
In this paper, a new leader selection algorithm is introduced for VANET. The algorithm is based on proactive broadcasting, which is able to tightly integrated into the SAE 2735 protocol for vehicular uses.
Simulation results have shown that the algorithm converges to a unique leader in very short time within 1 second. When using the algorithm at a intersection, 98% of the time has 1 leader at the intersection. The remaining 2% is the time of leader switching, each switching is less than 1 second.
To make it realistic for vehicular usage, the algorithm is optimized to prevent broadcast storm under heavy traffic flow, as well as the design of other precautionary cases. The optimized algorithm will yield same leader selection results with only 15%40% (depend on different traffic condition) of message exchanges of the original basic algorithm. The message reduction is most efficient when the traffic volume is high, which is highly desirable.
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