SOLVING DYNAMIC LEADER ELECTION PROBLEM WITH PERFORMANCE-RELATED CHARACTERISTICS

Transcription

1 Journal of Engineering, National Chung Hsing University, Vol. 24, No. 3, pp SOLVING DYNAMIC LEADER ELECTION PROBLEM WITH PERFORMANCE-RELATED CHARACTERISTICS Che-Cheng Chang Ji-Chiang Tsai * ABSTRACT Leader election is a fundamental building block for many applications. Particularly, there have been many methods proposed for solving the leader election problem in the literature. Solving the leader election problem in static networks is easier than in dynamic networks because dynamic behavior of processes must be considered in the latter. A simple way to solve this problem in dynamic networks is attaching a synchronous clock to each process. But doing so violates the assumption of asynchrony. Moreover, a leader had better be a process with the best performance-related characteristic among all nodes within a connected component. In this paper, we present an efficient leader election algorithm with regard to performance-related characteristics for dynamic networks, without any synchronous clock assumption. Key words: leader election, dynamic networks, asynchronous systems, broadcast. 1. INTRODUCTION Leader election is a fundamental building block for many applications, such as group communication services [1], key distribution and management [2], routing coordination [3], sensor coordination [4], general control [5], and so on. Accordingly, there have been many methods proposed for solving the leader election problem in the literature. Obviously, solving the leader election problem in static networks is easier than in dynamic networks because we have to consider the dynamic behavior of processes in the latter networks. In the literature, a simple way to solve the leader election problem in dynamic networks is to attach a synchronous clock to each process in the system. For example, in [5], the authors used the concept of synchronous clock to indicate which node has the highest priority to be the current leader. More specifically, their algorithm directs any process that loses the connection with the current leader to start a negotiation with all other processes in the same connected component and then the process with the latest negotiation index will be elected as the leader until the topology changes again. Recently, in [6], the authors improved the algorithm proposed in [5] in a manner that the resultant convergence time of the new algorithm is less than the original algorithm. Moreover, the authors in [6] also modified the standard definition of the classic leader election problem for static networks. They assumed that every connected component will eventually have a unique leader, but not that eventually there is a unique leader in the system. Such a new definition is more proper for dynamic networks because dynamic networks may contain some disconnected parts. However, attaching a synchronous clock obviously violates the assumption of asynchronous systems. On the other hand, in [7], the authors considered that the elected leader should be the most-valued Department of Electrical Engineering, National Chung Hsing University, Taichung, Taiwan 402, R. O. C.; 國立中興大學電機工程學系 * Corresponding author, jichiangt@nchu.edu.tw DOI: /JENCHU

2 130 Solving Dynamic Leader Election Problem with Performance-Related Characteristics process among all the processes within the connected component it resides, where the value of a process is a performance-related characteristic. Particularly, such a performance-related characteristic may be remaining battery life, average distance to other nodes, etc. Hence, a most-valued process means that this process has the best performance-related characteristic among all processes within a connected component, e.g. the longest remaining battery life, the shortest average distance to other nodes, etc. Obviously, electing a leader with the best performance-related characteristic is more desirable than simply electing a random leader. For example, electing a leader with the longest remaining battery life can postpone the time for reelecting a new leader since the current leader will last longer before exhausting all its energy. In this paper, we present a new leader election algorithm with regard to performance-related characteristics for dynamic networks. In other words, the leader that the algorithm will choose is the process currently with the best performance-related characteristic among all processes within a connected component. Likewise, we do not assume any synchronous clock. Moreover, our algorithm can elect a new leader more quickly than the algorithm proposed in [7]. This paper is structured as follows. Section II describes the preliminaries of the leader election problem. In Section III, we first introduce the concept of our algorithm for static networks for the sake of clear presentation, and then in Section IV, we apply the previous concept to dynamic networks, by only adding some extra information on messages. At last, simulation results are demonstrated in Section V for showing the efficiency of our algorithm, and we make a conclusion for our work in Section VI. 2. PRELIMINARIES 2.1 System Model A finite set of processes {p 1, p 2,, p n } with n 1 is considered in the text. Unlike the usual model of traditional fixed networks, the processes in are not necessarily aware of each other initially. This assumption reveals the self-organization nature of the considered unstructured peer-to-peer network. Processes in communicate with other processes by exchanging messages through a reliable channel. Namely, there is no message creation, corruption and duplication. Moreover, process p i can send a message to another process p j only if p j is within the transmission range of p i, and vice versa. We also assume that all processes in have the same transmission range. This means that the communication is symmetric. Particularly, since the transmission range is finite, some processes in may disconnect with others. Hence, we use the term connected component to represent a subset of processes, in which any process can communicate with all other processes by routing messages. Furthermore, there is no bound neither on the transmission delay of messages nor on relative speeds of processes. Thus we face an asynchronous system. On the other hand, a crashed process is considered as a leaving process since any process cannot communicate with a crashed process anymore. Similarly, we also consider a recovered process as a new joining one because any process can communicate with the recovered process from now on. In order to facilitate explaining the ideas proposed in this paper, we describe a few technical terms first. Definition 1: In a network, a process is called a neighbor of p i if and only if it is within the transmission range of p i. In the text, we denote the set of neighbors of p i as neighbors i. In particular, a process can employ the techniques of probe and reply proposed in [7] to find which neighbor has left its transmission range or which node has entered its transmission range. Definition 2: In a network, a process is called a participant of p i if and only if it and p i are in the same connected component of the network. Namely, a process can route a message to any of its participants since there exists at least one path from the process to any participant in a connected component of the network. In the text, we denote the set of participants of p i as participants i. Note that the two sets neighbors i and participants i vary with time since the topology of a dynamic network is not fixed. 2.2 The Specification of Leader Election In the classical leader election problem for static networks, every process p i in the network eventually elects the same leader [8]. Later, the authors in [5] adapted the classical definition for dynamic networks. Moreover, in [7], the elected leader must be the mostvalued node because electing a leader with the best performance-related characteristic is more desirable than

3 Journal of Engineering, National Chung Hsing University, Vol. 24, No simply electing a leader randomly. Definition 3: The specification of the leader election problem for dynamic networks is: (1) For every connected component, there is a unique eventual leader. (2) The eventual leader of a connect component is the highest priority node among all the nodes within the component, where the priority of a node is based on the considered performance-related characteristic, using the node ID to break ties among nodes with the same value. 3. THE LEADER ELECTION ALGORITHM FOR STATIC NETWORKS In this section, we start to present our algorithm for solving the leader election problem. Foremost, for the sake of clear presentation, we employ the static fixed networks to introduce the basic concept of our algorithm. Then we adapt the algorithm for static fixed networks to the dynamic version in the next section. In the following, we present the core property of our algorithm. Theorem 1: In a connected component, if every process broadcasts its neighbor set and any received neighbor set from another process, every process can finally know the existence of all other processes in the same component. Proof: After a process p i broadcasts its neighbor set, any neighbor of p i, say p j, first receives the message and then learns the neighbor set of p i. Subsequently, the neighbor set of p i is also broadcast to any neighbor of p j, say p k, via p j such that p k learns the neighbor set of p i, and so on. Since there is a path between any process and p i in a connected component, we know that every process in the same connected component as p i will eventually learns the neighbor set of p i. This means that the neighbor set of every process can be learned by any other processes in the same component. Hence, every process can finally know the existence of all other processes in the same component. According to theorem 1, a process can know all its participants by all processes broadcasting their neighbor sets as well as any received neighbor sets. Initially, any process p i only knows its neighbor set. Consequently, it can only elect the current leader from the set. Later, upon receiving a broadcast message from its neighbors, p i learns more of its participants and then chooses the process with the best performance-related characteristic among its participants known so far as its new leader. Finally, p i will know all processes in the same connected component and then choose the one with the best performance-related characteristic in the connected component as its eventual leader. Obviously, the values of the considered performancerelated characteristic of any two nodes may not be distinct. In other words, it is possible to have some nodes in a connected component with the same value. For example, in Figure 1, the value of the considered performance-related characteristic of each process is shown in the bracket in the corresponding node. We can see that p 4 and p 7 in Figure 1 have the same value. If such a scenario happens, we use the node ID of these nodes to break the tie. Particularly, the node with the smallest ID will be considered. For instance, in Figure 1, we will choose p 4 as the highest priority node in this component. Doing so can help the processes in a connected component elect a unique node as the eventual leader. Based on theorem 1, our leader election algorithm for static networks is presented as algorithm 1 in Table 1. In this algorithm, every process p i takes its identification and the value of the considered performance-related characteristic as the input (lines 1 ~ 2). Subsequently, it initiates the algorithm by executing the init phase. First, since using the technique of probe and reply proposed in [7] can help a process know all its neighbors, we assume that every process can obtain the whole set of its neighbors. Therefore, p i adds these nodes within its transmission range and itself to its neighbors i set (lines 7 ~ 8). Furthermore, it sets the participants i set to neighbors i to indicate that its current knowledge about participants is only the set of neighbors (line 9). Then, p i initializes the received i set as an empty set to prepare for recording the information to be received (line 10). Figure 1. A topology contains two connected two components.

4 132 Solving Dynamic Leader Election Problem with Performance-Related Characteristics Table 1. The leader election algorithm for static networks Moreover, it elects the node with the highest priority in its current participants i set as the current leader (line 11). Finally, p i broadcasts neighbors i to all its neighbors such that they can learn the neighbor set of p i (line 12). On the other hand, upon receiving neighbors j, if p j is not in the received i, it means that p i may increase its knowledge about participants through neighbors j (line 13). Accordingly, p i updates its participants i (line 14). Furthermore, p i elects the node with the highest priority in its participants i set as its current leader (line 15). Then p i broadcasts neighbors j to all its neighbors (line 16). More specifically, the condition in line 13 can avoid processes in the static network from broadcasting the same information forever, by merely broadcasting information that it has not yet forwarded. For example, in Figure 1, if p 3 broadcasts neighbors 9 to p 4, p 4 will receive it and then broadcast it to p 3 again. Such a condition can let p 3 ignore this message from p 4 and will not broadcast it again. Otherwise, neighbors 9 would be broadcast within the connected component forever. In the following, we demonstrate that the algorithm shown in Table 1 is sufficient for solving the leader election problem in static networks. Theorem 2: The algorithm shown in Table 1 is sufficient for solving the leader election problem in static networks. Proof: We need to prove that the algorithm can meet the specifications defined in definition 3. According to theorem 1, each node in a connected component can finally know all other nodes in the same component. Namely, each node in the component finally has the same participant set as all other nodes in the same component. Hence, since all nodes choose the node with the highest priority among all nodes within the component as their leader, the leader that they choose will be eventually unique because the node with the highest priority in the component is unique. 4. THE LEADER ELECTION ALGORITHM FOR DYNAMIC NETWORKS In [7], the authors used several distinct cases to explain the behavior of dynamic networks. However, these cases cannot cover all possible behavior of dynamic networks due to rapid node mobility. Furthermore, the algorithm in [7] adopted the concept of growing and shrinking to discover all processes in the same component. Then any process employs the collective information to elect the leader and broadcasts its decision to all nodes in the same component. Obviously, such a concept requires that messages traverse a component three times for an election, i.e. growing, shrinking and broadcasting the decision about the leader. Yet, our algorithm merely employs the concept of increasing the knowledge of a

5 Journal of Engineering, National Chung Hsing University, Vol. 24, No node. This concept only needs that any message from each node traverses a component one time for electing the leader. In particular, upon receiving messages from all other processes in the same component, a process can immediately find the process with the best performancerelated characteristic among all its participants. Unlike static networks, the knowledge of a node in dynamic networks may be decreasing. Hence, we have to deal with such a critical situation. Instead of adopting the assumption of synchronous clocks, we use the following concept of routing history. Definition 4: The routing history of a message contains identities of all processes by which the message is traversed. Trivially, while a process p i receives a message, p i can learn identities of all processes involved in the path traversed by the message via the routing history carried on the message. For example, in Figure 1, when p 1 starts to broadcast a message created by itself, the routing history of the message contains the source, i.e. p 1, initially. After another process p 9 receives the message broadcast by p 1, p 9 adds its ID to the routing history of the message prior to broadcasting it. Thus upon receiving such a message, p 3 knows that the routing path of the message so far is p 1 p 9 p 3. The purpose of the routing history is to prevent a node from sending the backflow message defined below. Definition 5: A backflow message is a message that one process p i transmits to another process p j and p j transmits back to p i later. Obviously, the information piggybacked on a backflow message may be obsolete and thus makes the problem more sophisticated and more difficult to solve. For instance, consider the network topology shown in Figure 1. Since there are two disconnected components in it, two leaders will be elected in each component, i.e. p 4 and p 7. Now assume that some nodes move, and then the foregoing topology becomes the topology depicted in Figure 2. Particularly, because p 4 moves out of the transmission range of p 3, p 4 and p 3 cannot communicate with each other from now on. Simultaneously, p 2 enters the transmission range of p 9 such that p 2 and p 9 can communicate mutually. By continuously broadcasting message, p 3 can tell p 9 the fact that p 4 has left. Without the routing history piggybacked on messages, after p 7 transmits a backflow message to p 9 that had been originally created by p 3 before p 4 left, p 9 will consider that p 4 is in the component again if the backflow message Figure 2. The resultant topology after some nodes in Figure 1 move. arrives after p 3 has told p 9 that p 4 left. If the aforementioned scenario happens again and again, the participant set of p 9 will not be stable forever. Accordingly, if a backflow message is transmitted in the component, even after the topology is stable, some processes in it may never have a stable participant set. This means that processes in the same component may not be able to elect the same process as their leader eventually. With the routing history, the foregoing scenario can be avoided because a node can never receive a message originally sent from itself by checking the routing history carried on received messages. Consider the above example again. All backflow messages broadcast by p 7 back to p 9 were sent by p 9 earlier. Therefore, p 9 is contained in the routing histories of these messages. By checking the routing history, p 9 can discard those backflow messages. Now we begin to describe our leader election algorithm for dynamic networks presented in Table 2. In the beginning, every process p i takes its identification and the value of the performance-related characteristic as the input (lines 1 ~ 2). Subsequently, it initiates the algorithm by executing the init phase. First, p i adds those nodes within its transmission range and itself to neighbors i (lines 7 ~ 8). Similarly, it sets participants i to neighbors i to indicate that its initial knowledge about other processes only includes its neighbors (line 9). Then it elects the node with the highest priority in its current participants i set as the current leader (line 10). Finally, p i broadcasts neighbors i for informing the existence of its neighbors to all other nodes in the same component (line 11). On the other hand, upon receiving a message carrying neighbors j, if p i is not in the routing history of the message, it means that p i can update its knowledge through neighbors j (line 12). Such a condition can prevent p i from receiving a backflow message as well as broadcasting forever, even after they already know all nodes in the same component.

6 134 Solving Dynamic Leader Election Problem with Performance-Related Characteristics Table 2. The leader election algorithm for dynamic networks Then p i adds itself to the routing history of the message carrying neighbors j for noting that it has received such a message and accordingly has updated its participants i set (lines 13 ~ 14). Also, p i elects the node with the highest priority in its current participants i set as the current leader (line 15). Subsequently, p i broadcasts neighbors j to other processes in the same component (line 16). There are two more affairs required to be dealt with. First, upon detecting neighbor changing, p i updates both its neighbors i and participants i sets (lines 17 ~ 18). Hence, it must re-elect the node with the highest priority in its current participants i set as the new leader (line 19). Subsequently, p i broadcasts the new neighbors i set for informing the existence of nodes in neighbors i to other nodes in the same component (line 20). Particularly, p i also broadcasts the reelection i message to ask other nodes in the same component to broadcast their neighbor sets and elect their leaders again (line 21). On the other hand, upon receiving the reelection j message, if p i is not in the routing history of reelection j, p i adds itself to the routing history of reelection j for noting that it has received such a message (lines 22 ~ 23). Then it also broadcasts its neighbors i set again and routes the reelection j message to its neighbors (lines 24 ~ 25). More importantly, the routing history can also help a process tolerate network partition by executing the update operation. In particular, the update operation in algorithm 2 is more than just the union operation in algorithm 1. Since the participants of a process p i in dynamic networks may be decreasing instead of increasing, the update operation may remove some processes from its participants i set while p i detects network partition or receives a message about network partition. Now reconsider the situation where the topology shown in Figure 1 is changed to the one in Figure 2. We can see that p 4 knows that a neighbor, i.e. p 3, has left its transmission range or crashed. Then p 4 must discard all information obtained via p 3. Here, p 4 checks all information to find the part related to routing paths that have traversed p 3 and then deletes the records. In other words, neighbor 3, neighbor 9, neighbor 1 and neighbor 7 will be removed because all these messages have traversed p 3. Hence,

7 Journal of Engineering, National Chung Hsing University, Vol. 24, No there remains only one node, i.e. p 4, in both the two sets neighbors 4 and participants 4. This means that p 4 will elect itself as the current leader since only itself is in the component. In the following, we demonstrate that the algorithm presented in Table 2 is sufficient for solving the leader election problem in dynamic networks. Theorem 3: The algorithm presented in Table 2 is sufficient for solving the leader election problem in dynamic networks. Proof: Since the algorithm shown in Table 2 is the version adjusted for dynamic networks, such an algorithm is also sufficient for solving the leader election problem in static networks according to theorem 2. Therefore, to prove its sufficiency for dynamic networks, we only need to consider how it deals with the change of network topology. Particularly, upon detecting topology change, p i broadcasts reelection i to find nodes that are in the same component as it now. By theorem 1, such a message will be received by each node in the same connected component. Namely, each node in the same component will broadcast its neighbor set again. Hence, we can regard such a scenario as a new leader election procedure. After the topology eventually becomes stable, the leader election procedure will be completed since each node in the component will know all other nodes in the same component and then elects the node with the highest priority as its leader according to theorem SIMULATION RESULTS In this section, we use the PARSEC language to perform our simulation experiments [9]. In the beginning, we describe the settings of our simulation below. (1) The first experiment: (a) The simulation environment is an area of 1,000 m 1,000 m. (b) The transmission range is set to 250 m, 300 m and 350 m, respectively. (c) There are 100 nodes randomly deployed in the considered environment. (2) The second experiment: (a) The simulation environment is an area of 1,000 m 1,000 m. (b) The transmission range is set to 300 m. (c) There are 80, 100 and 120 nodes randomly deployed in the considered environment, respectively. (3) The third experiment: (a) The simulation environment is an area of 1,000 m 1,000 m. (b) The transmission range is set to 250 m, 300 m and 350 m, respectively. (c) There are 100 nodes randomly deployed in the considered environment. (d) Nodes may move randomly within some an unknown but finite time. More specifically, each scenario is executed 1,000 times, and the average execution is shown in Figure 3. According to the simulation results of Figure 3, we can see that our algorithm is more efficient than the algorithm proposed in [7]. We explain this observation with the following statements. First, by utilizing our algorithm, a process can increase its knowledge about participants through exchanging messages with its neighbors. Therefore, if the considered topology is more strongly connected, each process in the considered topology can know all its participants more quickly. In other words, the more strongly connected the topology Figure 3. (a) Results of the first experiment; (b) Results of the second experiment; (c) Results of the third experiment.

8 136 Solving Dynamic Leader Election Problem with Performance-Related Characteristics is, the better the performance is. Particularly, since each process connects to any other processes in the complete connected topology, in such a topology, a process can know all its participants through only exchanging messages with its neighbors once. On the other hand, the algorithm proposed in [7] merely directs several processes to start a computation for electing a new leader, and only one computation will eventually exist. This means that these processes must deal with the competitions among different computations. Hence, while being applied to a more strongly connected topology, the more competitions need to be dealt with. Besides, such an algorithm requires that messages traverse the whole topology three times for an election, i.e., growing, shrinking and broadcasting the decision about the leader. However, our algorithm only needs that any message from each node traverses a component one time for an election. Additionally, even in dynamic networks, our algorithm also has its superiority on the convergence time. After the network topology is stable, the system will work like that in static network topology. Hence, just like the above discussions, our algorithm still has better performance than that proposed in [7]. 6. CONCLUSIONS In this paper, we have explored the problem of leader election in dynamic networks, where not only the topology will change but also processes may crash. We employ the routing history of messages to deal with the dynamic behavior of networks instead of adopting the synchronous clock assumption. Doing so makes the leader election problem easier to solve in practice. More importantly, our algorithm can guarantee that the elected leader is the one with the best performance-related characteristic among all nodes within the same connected component. REFERENCES 1. Roman, G.-C., Huang, Q. and Hazemi, A., Consistent Group Membership in Ad Hoc Networks, Proceeding of the 23rd International Conference on Software Engineering, Toronto, pp (2001). 2. Asokan, N. and Ginzboorg, P., Key Agreement in Ad Hoc Networks, Computer Communications, Vol. 23, No. 17, pp (2000). 3. Amis, A.D., Prakash, R., Vuong, T.H.P. and Huynh, D.T., Maxmin D-Cluster Formation in Wireless Ad Hoc Networks, Proceeding of the IEEE International Conference on Computer Communications, Tel Aviv, pp (2000). 4. Heinzelman, W.R., Chandrakasan, A. and Balakrishnan, H., Energy-Efficient Communication Protocol for Wireless Microsensor Networks, Proceedings of the 33rd Hawaii International Conference on System Sciences, Maui, pp (2000). 5. Malpani, N., Welch, J.L. and Waidya, N., Leader Election Algorithms for Mobile Ad Hoc Networks, Proceedings of the 4th International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, Boston, pp (2000). 6. Derhab, A. and Badache, N., A Self- Stabilizing Leader Election Algorithm in Highly Dynamic Ad Hoc Mobile Networks, Proceeding of the IEEE Transactions on Parallel and Distributed Systems, Vol. 19, No. 7, pp (2008). 7. Vasudevan, S., Kurose, J. and Towsley, D., Design and Analysis of a Leader Election Algorithm for Mobile Ad Hoc Networks, Proceedings of the 12th IEEE International Conference on Network Protocols, Berlin, pp (2004). 8. Lynch, N.A., Distributed Algorithms, Morgan Kaufmann, San Francisco (1996). 9. Zeng, X., Bagrodia, R. and Gerla, M., GloMoSim: A Library for Parallel Simulation of Large-Scale Wireless Networks, Proceedings of the 12th Workshop on Parallel and Distributed Simulations, Alberta, pp (1998). Manuscript Received: Apr. 21, 2013 Revision Received: Jun. 25, 2013 and Accepted: Jun. 25, 2013