An enhanced approach to determine connected dominating sets for routing in mobile ad hoc networks. Chunchun Ni, Hui Liu, Anu G. Bourgeois and Yi Pan*

Transcription

1 Int. J Mobile Communications, Vol. X, No. Y, xxxx 1 An enhanced approach to determine connected dominating sets for routing in mobile ad hoc networks Chunchun Ni, Hui Liu, Anu G. Bourgeois and Yi Pan* Department of Computer Science, Georgia State University, Atlanta, GA 30303, USA nichunchun@hotmail.com hiu1@student.gsu.edu abourgeois@cs.gsu.edu E:mail: pan@cs.gsu.edu *Corresponding author Abstract: A mobile ad hoc network is a collection of wireless mobile nodes forming a temporary network without the support of any established infrastructure or centralised administration. Mobile ad hoc networks face a lot of challenges for designing a scalable routing protocol due to their natural characteristics. The IDea of virtual backbone routing has been proposed for efficient routing in mobile ad hoc networks because virtual backbone routing can reduce communication overhead and speed up the routing process compared with many existing routing protocols. Up to now, Minimum Connected Dominating Set (MCDS) is the main method used to form a virtual backbone. However, finding an MCDS is an NP-hard problem. A distributed protocol for calculating the connected dominating set was proposed by Wu and Li. In this paper, we propose a further extension to reduce the size of the dominating set as compared to their method. We conduct extensive simulations on these two related algorithms. These simulation results show that our approach can consistently outperform Wu and Li s method, particularly for a medium-density network. We discuss the tradeoff between cost and performance through theoretical analysis. Keywords: mobile ad hoc network; connected dominating set; routing. Reference to this paper should be made as follows: Ni, C., Liu, H., Bourgeois, A.G. and Pan, Y. (xxxx) An enhanced approach to determine connected dominating sets for routing in mobile ad hoc networks, Int. J. Mobile Communications, Vol. X, No. Y, pp Biographical notes: Chunchun Ni received her BS in Public Health from Zhejiang University, China, in 1995 and MS in Computer Sciences from Georgia State University in Currently, she is a research specialist at Emory University. Her working area includes wireless network, imaging processing, and data analysis in fmri. Hui Liu received her BEng and MEng degrees in Computer Science from Central South University, China in 1998 and 2001, respectively. Currently, she is a PhD student in the Department of Computer Science at Georgia State University. Her research interests include wireless networks, mobile computing, parallel algorithms, and optical networks. Copyright 2005 Inderscience Enterprises Ltd.

2 2 C. Ni, H. Liu, A.G. Bourgeois and Y. Pan Anu G. Bourgeois received her BS and PhD in Electrical and Computer Engineering from Louisiana State University in 1991 and 2000, respectively. Currently, she is an Assistant Professor in the Department of Computer Science at Georgia State University. Her research interests include dynamic reconfiguration, wireless networks, parallel algorithms, optical networks, and fault tolerance. Yi Pan received his BEng and MEng degrees in Computer Engineering from Tsinghua University, China in 1982 and 1984, respectively, and his PhD in Computer Science from the University of Pittsburgh, USA in Currently, he is a Professor in the Department of Computer Science at Georgia State University. Dr. Pan s research interests include parallel and distributed computing, optical networks, wireless networks, and bioinformatics. Dr. Pan has published more than 80 journal papers with 26 papers published in various IEEE journals. In addition, he has published over 90 papers in refereed conferences (including IPDPS, ICPP, ICDCS, INFOCOM, and GLOBECOM). He has also co-edited 13 books (including proceedings) and contributed several book chapters. His recent research has been supported by NSF, NIH, AFOSR, AFRL, JSPS, IISF and the states of Georgia and Ohio. Dr. Pan has served as an Editor-in-Chief or Editorial Board Member for eight journals including three IEEE Transactions and a guest editor for seven special issues. 1 Introduction Mobile ad hoc networks are widely deployed for many applications such as automated battlefield operations, wireless conferences, disaster rescues, connection to the internet in remote terrain, etc. A mobile ad hoc network is a particular type of wireless network in which an association of mobile nodes forms a temporary network, without any support of fixed infrastructure or central administration. Mobile nodes can control connections and disconnections by the distances between them and the willingness to collaborate during the formation of short-lived networks. That means a connection is achieved either through a single-hop radio transmission if two nodes are located within wireless transmission range of each other, or through relays by intermediate nodes that are willing to forward packets for them. Features of mobile ad hoc networks include dynamical topologies, high volatility of network information, multihop communication, and limited resources (i.e., lower bandwidth, lower battery, and limited CPU). Such essential characteristics have posed a lot of challenges in designing an efficient and scalable routing protocol in mobile ad hoc networks. Existing routing protocols are classified into two main classes. One is topology-based routing protocol, which is based on the information concerning links [1 5]. The other class is position-based routing protocol, in which mobile nodes know physical position information by geolocation techniques such as GPS [6 8]. Position-based routing protocols require that every mobile host is equipped with some kind of geolocation hardware device through which the mobile host can obtain its accurate position information. Thus, position-based routing protocols are expensive and not suitable in general occasions. Topology-based routing protocols are less expensive than position-based routing protocols. However, they exhibit a global flooding problem

3 An enhanced approach to determine connected dominating sets 3 that may result in excessive redundancy, contention, and collision. It causes high protocol overhead and interference to ongoing traffic. As we know, wired networks have a fixed backbone. Although a mobile ad hoc network has no fixed backbone infrastructure, many topology-based routing protocols propose the promising IDea of virtual backbones such as cluster-based routing, backbone-based routing, and spine-based routing [9 12] to overcome the global flooding problem. The basic IDea behind these types of protocols is to divide a mobile ad hoc network into several small overlapping subnetworks, where each subnetwork is a clique (a complete subgraph). Each subnetwork has one or more virtual backbone hosts to connect to other parts in the network. These virtual backbone hosts form the core infrastructure of the ad hoc mobile network. The routing process is operated over the core. As a result, any broadcasting of control packets only happens in the core, and communications between core nodes and noncore nodes are all through unicast communications. Therefore, this can substantially reduce the protocol overhead caused by global flooding. The number of hosts forming the virtual backbone must be as small as possible in order to reduce the protocol overhead, to increase the convergence speed, and to simplify the connectivity management. Currently, Minimum Connected Dominating Set (MCDS) is the main method utilised to approximate the virtual backbone in a mobile ad hoc network. We assume that a mobile ad hoc network is deployed in a 2D space, and a mobile node is equipped with an omnidirectional antenna that has an equal maximum transmission range. Thus, the topology of such a mobile ad hoc network can be modelled as a unit-disk graph (UDG). A simple graph G (V, E) is used to represent a mobile ad hoc network, where V represents a set of mobile nodes and E represents a set of edges. An edge (u, v) in E indicates that nodes u and v are neighbours. For future reference, we now formally define dominating sets. Definition 1.1: A Dominating Set (DS) is defined by a subset of vertices of a graph where every vertex that is not in the subset is adjacent to at least one vertex in the subset. Definition 1.2: A Connected Dominating Set (CDS) is a dominating set that induces a connected subgraph. Definition 1.3: The Size, S(DS) of a Dominating Set is the number of nodes in DS. Finding an MCDS in a UDG is an NP-hard problem. Wu and Li [13] proposed a simple and distributed algorithm to calculate the CDS in a given connected graph that represents a mobile ad hoc network. In this paper, we modify Wu and Li s algorithm to calculate the CDS in mobile ad hoc networks. Our modifications reduce the size of the dominating sets by adding an enhanced rule. We then implement the two algorithms and compare the results. The simulation results show that the introduction of our enhanced rule can reduce the size of the DS by 10~20% in comparison with the existing distributed algorithm of Wu and Li, particularly in a medium-density space. However, we find that there are tradeoffs between computation complexity and the size of a DS. Specifically, as the size of the DS is smaller, the computational complexity is larger. In Wu and Li s original algorithm, the cost at each mobile node is O( 2 ), where is the maximum node degree of a given graph and the total amount of message exchanges is O( n), where n represents

4 4 C. Ni, H. Liu, A.G. Bourgeois and Y. Pan the total number of vertices in graph G. In our protocol, communication complexity remains the same, and computation complexity increases to O( 3 ) due to the addition of our enhanced rule. Since is typically a small number in ordinary networks, this is an acceptable increase in computation complexity as it provides the added benefits as described earlier. Almost concurrently with our research, Dai and Wu [14] proposed a rule K to reduce the number of dominating nodes in the initial CDS formed by Wu and Li s algorithm [13]. Their rule K is applied for unidirectional mobile ad hoc networks that can be visualised as directed graphs. In rule K, every vertex obtains all its marked neighbouring nodes with higher IDs than itself to form a set of vertices such as V +, and it decomposes V + into several strongly connected components. Then the vertex checks whether one of the strongly connected components can cover all its neighbouring nodes and itself. If it can be covered, the vertex is removed from CDS. There are some similarities between rule K and our approach. However, our algorithm is applicable for bidirectional mobile ad hoc networks that are induced into undirected graph. If one considers every undirected edge as two directed edges in order to apply rule K, then every vertex needs to check whether it can be dominated by the union of all its neighbouring nodes with higher IDs. It is very inefficient and time-consuming, especially in dense networks in which every vertex may have a lot of neighbours. Our algorithm does not have this type of problem. Furthermore through detailed simulation, we will determine what type of mobile ad hoc networks our algorithm is well suited for. This paper is organised as follows. Section 2 gives a brief review of related work. Section 3 provides preliminaries on dominating set-based routing protocols including Wu and Li s algorithm and its extension rules. We present our enhanced rule and performance analysis in Section 4. Section 5 describes our simulation results and Section 6 concludes the paper. 2 Related work Various virtual backbone based routing protocols have been proposed in recent years. Distributed approximation algorithms for MCDS in mobile ad hoc networks were first developed by Das et al. [9 10,15]. These algorithms provided distributed implementations of the two centralised algorithms given by Guha and Khuller [16]. In Das s algorithm, a CDS is found by growing a set U starting from a vertex with the maximum node degree. It then iteratively adds to U a node that is adjacent to the maximum number of nodes not yet in U until U forms a dominating set. Finally, it assigns each edge with a weight equal to the number of neighbours not in U, and then finds a minimum spanning tree T in the resulting weighted graph. All the nonleaf nodes form a CDS. This approach has two main improvements over previous protocols. First, only a few nodes need to keep global information that captures the topological structure changes of the whole network, and as long as network topological changes do not affect these MCDS nodes, there is no need to recapture global information. Thus it reduces the information access overhead and the update overhead. Second, each node only needs 2-distance neighbourhood information instead of information of the entire network topology. The main shortcoming of this algorithm is that the process of constructing a spanning tree is almost sequential, thus it needs a nonconstant number of rounds to

5 An enhanced approach to determine connected dominating sets 5 determine a CDS. Furthermore, the algorithm suffers from high implementation complexities and message complexity. Stojmenovic et al. [17] also proposed a distributed construction of CDS. Two types of nodes exist in the CDS: the cluster-heads and the border-nodes. The cluster-headers form a maximal independent set (MIS). The framework of constructing an MIS is described as follows: Each node uses a record key = (degree, x, y) as a unique rank parameter, where degree is the number of neighbours of the node and x and y are its two coordinates in the deployed network. The ranks are used to order all nodes. Each node with the lowest rank among all neighbours declares itself as a cluster-head by broadcasting. Whenever a node receives a message for the first time from a cluster-head, it gives up election as a cluster-head. Whenever a node has received the giving-up messages from all of its neighbours with lower ranks, it broadcasts a message to declare itself as a cluster-head. After a node collects the status of all neighbours, it joins the cluster centered at the neighbouring cluster-head with the lowest rank. Nodes adjacent to some node from a different cluster are selected as border-nodes. The implementation cost depends on the choice of rank. The algorithm is not very efficient because of the nonselective inclusion of all border-nodes. Also, the process for selecting cluster-heads may have to be serialised in some situations, such as in a linear network with monotonically increasing or decreasing ID distribution along the network. Recently, Alzoubi et al. [18] proposed a distributed solution with a constant approximation ratio for constructing a CDS. It also consists of two phases. One phase constructs an MIS and the other constructs a dominating tree. The first phase constructs a spanning tree rooted at a node v (selected through an election process). After such construction is finished, each node is IDentified according to a topological sorting order of the tree. Then, nodes are marked based on their tree levels in the order starting from root v. The root v is marked as black, and other nodes are all marked as white initially. Following the order, each node is marked black if it has no black neighbour. Let U be the set of black nodes, U forms an MIS. In the second phase, it constructs a tree spanning all the black nodes, and is referred to as a dominating tree. Let T be the dominating tree, where T is initially empty. The root v joins T at first. Then each black node (except v) selects a neighbour with the largest tree level but smaller than its own tree level and marks it as gray. Thus black and gray nodes form a CDS. Alzoubi et al. [18] prove that this algorithm has an approximation ratio of at most eight. Performance of this scheme is very good; however, a global infrastructure (spanning tree) is constructed before the node selection process. Also, two phases of the scheme are serialised. In addition, locality of maintenance is not realised in this approach, for a single change in network topology may destroy the spanning tree, thus causing the DS to be reconstructed. The performance comparison of these related algorithms is briefly summarised in Table 1. Since different assumptions and models are used in these proposed algorithms, it is difficult to make comparisons among them. The metrics used for comparison include computation and message complexities, the size of the generated CDS (this performance measurement is obtained either through theoretical analysis of the worst case in terms of

6 6 C. Ni, H. Liu, A.G. Bourgeois and Y. Pan approximation ratio to MCDS or through simulation on average case), whether the method supports locality of maintenance, and whether it requires global information. Table 1 Performance comparison of the algorithms in [9 10,13,15,17 18] and that proposed in this paper [9 10,15] [17] [18] [13] New Message O(n 2 ) O(n 2 ) O(n log n) O(n ) O(n ) Time O(n 2 ) O(n 2 ) O(n ) O( 2 ) O( 3 ) Size (2ln +3)opt Sim 8opt+1 Sim Sim Maintenance locality no no no yes yes Information Global Partial global Partial global Local Local Notes: opt is the size of MCDS n is the number of nodes is the maximum degree Sim means that the size of the generated CDS is measured through simulation no means it does not support locality of maintenance yes represents it supports locality of maintenance the last column labelled by new corresponds to our algorithm We see our algorithm is superior over those algorithms in [9 10,15] and [17]. The algorithm in [18] has good performance and approximation ratio to MCDS, but it needs to establish a global spanning tree, which makes local maintenance and propagation of information impossible. It is difficult for [13,17], and our algorithm to obtain a theoretical approximation ratio of the generated CDS size to that of MCSD. Thus, we conducted extensive simulations to compare the sizes of the generated CDS in [13] and our algorithm on average cases. Although the algorithm in [13] takes less time than our algorithm, simulation results demonstrate that our algorithm reduces the size of the CDS by 10~20% compared to that of [13], while the algorithm in [13] has better performance than those in [9 10,15]. As we know, the most important objective in designing a routing protocol is to reduce broadcast redundancy to save limited resources and to avoid the broadcast storm problem. This is achieved by generating a smaller CDS. Therefore, our algorithm does contribute in this aspect and provide benefits, though there is an acceptable increase in computation complexity. 3 Preliminary existing algorithm Wu and Li [13] proposed a simple and efficient distributed algorithm that can quickly find a DS in a mobile ad hoc network. Each node is marked as white initially. Let N(v) be the open neighbour set of vertex v, which means N(v) includes all the neighbours of vertex v. Let N[v] be the closed neighbour set of vertex v, the set of all neighbours and itself. By assumption, each node has a unique ID number. This algorithm runs in two phases. In the first phase, each node broadcasts its neighbour set N(v) to all its neighbours, and after collecting all adjacency information from all neighbours every node marks itself as black if there exist two unconnected neighbours. All black nodes form the initial CDS.

7 An enhanced approach to determine connected dominating sets 7 Wu and Li introduced the concept of a gateway node. A node is a gateway node if it belongs to the DS. However, considering only the first phase, there are too many gateway nodes. So in the second phase, the algorithm executes two extensional rules to eliminate local redundancy. Extensional rule 1 is as follows: Consider any two nodes u and v belonging to the DS. If N[v] N[u] and ID(v) < ID(u), then change v s colour to white. That means if all neighbours of v and itself are covered by u, and v is connected to u and has lower ID, v can be removed from the DS. Rule 2 is described as follows: Consider any three nodes u, v, and w belonging to a DS, such that u and w are two black neighbours of v. If N(v) N(u) N(w) and v has the smallest ID of three nodes, then v s colour is changed to white. In other words, if each neighbour of v is covered by u and w together, where u and w are both connected neighbours of v, then v can be eliminated from the list of gateway nodes. Thus, the second phase removes some nodes from the original DS and the size of a DS is further reduced. Wu and Li provided simulation results to show that the cardinality of a DS is largely reduced. Figure 1 shows the results of Wu and Li s algorithm using two phases and the two extensional rules. There are ten nodes randomly located in 2D space. Some nodes are connected if they are within the transmission range. Nodes are marked as black if the node has any two neighbouring nodes that are not connected. For example, node 0 has neighbour nodes 6 and 9 that they are not connected; therefore node 0 is marked black by the basic rule. In the figure, only node 3 is not marked because all of its neighbours are connected. (The neighbours of node 3 are node 1, node 4, and node 9; there are lines between 1 and 4, 4 and 9, 1 and 9.) Applying extension rule 1, nodes 0, 6, 4, 1, 5 are unmarked as shown by a black line. For example: N[0] = {0, 2, 6, 9}, N[2] = {0, 2, 6, 8, 9}, N[0] N[2] and ID(0) < ID(2); therefore, node 0 can be unmarked by extensional rule 1. N[6] = {0,2,6,7,8}, N[8] = {0,2,6,7,8}, N[6] N[8] and ID(6) < ID(8), therefore, node 6 also can be unmarked by extensional rule 1. Applying extensional rule 2, nodes 0, 1, 4, 5 are unmarked as shown by black cross. For example: N(0) = {2,6,9} and N(2) = {0, 6, 8, 9}, N(9) = {0, 2, 7, 1, 3, 4, 1}. N(0) N(2) N(9) and ID(0) = min{id(0), ID(2), ID(9)}. Node 0 can be unmarked by extensional rule 2. There is an overlap between extensional rules 1 and 2. Many nodes are unmarked by extensional rule 1, but they can also be unmarked by extensional rule 2. For example, node 0 can be unmarked by both extensional rules 1 and 2. In this algorithm, each node only needs to know local information and 2-hop neighbourhood information. On the other hand it is proven in [13] that the black vertices form a DS of unit disk graph G. At the same time, G', induced from the DS is connected and includes all the intermediate nodes of any shortest path between two vertices in G. The paper claims that the cost of computation at each vertex is O( 2 ), where represents the maximum node degree of graph G. Also, the total amount of message exchange is O(n ), where n is the cardinality of the vertex set of graph G. This algorithm needs a fixed number of rounds. By introducing dominating sets, the routing process in a mobile ad hoc network can be simplified.

8 8 C. Ni, H. Liu, A.G. Bourgeois and Y. Pan Figure 1 Examples of Wu and Li s algorithm 1. Connected graph 2. Marked by basic rule 3. Marked by extensional rule 1 4. Marked by extensional rule 2 4 Proposed extensions As we know, nodes in the DS maintain up-to-date information about their domain and exchange such information between each other to keep global information about the whole network. In order to minimise the relative accesses and update overhead, it is highly desirable to reduce the size of the CDS. Wu and Li s paper [13] states that rule 2 can be extended to a more general case in which the open neighbour set of vertex v is covered by the union of open neighbour sets of more than two neighbours of v in G'. We will address this particular problem. In some cases, the open neighbour set of a node cannot be covered by any other two marked neighbours, but its open neighbour set can be covered by three marked neighbours. In order to prevent illegal simultaneous removal of vertices in G', we consider if its ID is the smallest one among these four, then it can be removed from a DS. Thus, we propose our enhanced rule for determining the DS.

9 An enhanced approach to determine connected dominating sets Enhanced Rule Consider any four nodes u, v, w, x dominating set, such that u, w, and x are three marked neighbours of v. If N(v) N(u) N(w) N(x) and ID(v) = min {ID(v), ID(u), ID(w), ID(x)}, change the colour of v to white as unmarked. The enhanced rule together with Wu and Li s basic rule and extensional rules will generate a CDS for all graphs that are not complete graphs. Note that there is no need for a CDS when a network is complete, since every node in the network forms a CDS Theorem 1 If the given graph is not a complete graph, the set of nodes selected by Wu and Li s basic marking rule, and extensional rules together with our enhanced rule forms a CDS Proof It has been shown in [13] that nodes derived from the basic rule and the two extensional rules form a CDS. We only need to show that whenever a smallest ID node v is removed based on our enhanced rule, the resultant nodes (G' {v}) still form a CDS. Removing v only affects the neighbour nodes of v and itself. Because u, v, w are three marked neighbours of v, v is also adjacent to a vertex in V'. Nodes in the open neighbour set of v are all adjacent to at least one vertex (u, x, or w) in V' since v s open neighbour set is covered by the union the of open neighbour sets of its three marked neighbours. Thus G' {v} is still a dominating set. Since N(v) N(u) N(w) N(x) implies u, w, and x are connected, G' {v} is still a connected dominating set. Figure 2 An example of the proposed enhanced rule If the density of a network is high and there are many links between the nodes, this enhanced rule should unmark some dominating set nodes and decrease the size of the DS. For example in Figure 2, node v and its open neighbour set cannot be covered by any other two marked neighbours, but this set can be covered by three marked neighbours (u, w, and x). Assuming v s ID is the smallest, it can be removed from the list of gateway nodes by our enhanced rule. Our simulation results (to be discussed in the following section) confirm this observation. Now we measure the performance of our new algorithm by computation complexity within each node and communication complexity between nodes. Communication complexity can be measured by the total amount of message exchanges throughout the procedure. In our approach, the enhanced rule needs to check every three neighbours of a

10 10 C. Ni, H. Liu, A.G. Bourgeois and Y. Pan vertex, so the cost at each vertex is O( 3 ), where is the maximum node degree of the graph induced by a mobile ad hoc network. The total amount of message exchange is O(n ), where n is the cardinality of G s vertices set. Since is only a small number in ordinary networks, our strategy is slightly more complex than Wu and Li s algorithm, but less complex than Das s algorithm in all measurements. Our approach also only needs a constant number of rounds. This is very important because the topology of a mobile ad hoc network changes very frequently due to the movement of nodes and thus, the dominating set will have to be updated and recalculated frequently. Through simulation, we will see that another important measurement, the size of the DS generated, is reduced further than that of Wu and Li s algorithm. 5 Simulation and results We conducted a simulation study to measure the size of the CDS derived from our enhanced rule and compare it with the one generated by Wu and Li s algorithm. We have simulated three algorithms: Wu and Li s algorithm without applying any extensional rules, their algorithm with their extensional rules, and our proposed algorithm with the enhanced rule. In our simulation, random graphs are generated in a square units of a 2-D simulation area, by randomly including a certain number of mobile nodes. We assume that each mobile node has the same transmission range r, thus the generated graph is undirected. If the distance between any two nodes is less than radius r, then there is a connection link between the two nodes. Basic, Wu s, and New are three parameters used to represent the size of the CDS calculated by Wu and Li s basic rule, their algorithm with the two extensional rules, and our new algorithm respectively. We performed two groups of simulations. In the first group, we set the transmission radius of mobile nodes r to 125, 150, 175, 200, 225, and 250 units. In this way, we can control the density of the generated graphs because the density increases when r increases. For each transmission range r, the number of nodes n is varied from 80 to 150. For each n, the number of simulations is 500 times. For each case, we calculate Basic, Wu s, and New, and then record the average number. We compare the results in terms of the size of the CDS generated. It is well known that the lower the number of dominating nodes, the better the result. Thus, our goal is to generate a small CDS to facilitate a fast routing process and reduce access and update overhead. Figures 3a to 3f show the average number of dominating set nodes versus the number of nodes in the network for the increasing order of transmission radius r. Notice that the performance of the basic rule without extensional rules is very poor and the ratio of nodes in the DS to all nodes in the network is almost one, specifically, Moreover, when the transmission range is very large, almost every node belongs to a DS. Because the basic rule is loose, when the density of generated graphs increases as the transmission range increases, it is very easy to find two neighbours of a node that are not connected. After applying the two extensional rules of Wu and Li s algorithm, the size of the DS is largely reduced. The ratio of the number of dominating nodes over the total number of nodes in the network changes from 0.6 to 0.22 as the transmission range increases.

11 An enhanced approach to determine connected dominating sets 11 Figure 3 Average number of dominating set nodes relative to the number of nodes, n (a) Number of nodes (b) Number of nodes (c) Number of nodes (d) Number of nodes (e) Number of nodes (f) Number of nodes

12 12 C. Ni, H. Liu, A.G. Bourgeois and Y. Pan The simulation also shows that our algorithm outperforms Wu and Li s algorithm with the two extensional rules. By using our new algorithm (enhanced rule), the size of the DS is further decreased; it is less than the number derived from their algorithm, especially in the medium-density space. Since, in a low-density space, every node has few neighbours, it is possible that a node has less than three neighbours, thus this possibility prevents it from even applying our enhanced rule. In the other extreme condition of high-density networks, the size of the DS has already been reduced greatly with the two extensional rules. For example, when the transmission range equals 250, and the number of nodes equals 150, the size of the DS is 29. In this case, the size of the DS divided by the number of nodes in whole network is 0.22, which is very low. In other words, most of nodes are already unmarked by extensional rule 1 or 2. There are very few nodes remaining that can be decreased by using our enhanced rule. Therefore, our enhanced rule is very efficient and useful in medium-density networks. The ratio of the dominating set size induced by our algorithm over the number of nodes changes from 0.59 to 0.19 as the transmission range increases. In order to confirm the observation that our algorithm has better performance in the medium-density mobile ad hoc networks, we consider a fixed number of nodes, 150, and vary the transmission range from 100 to 250 units as an example. In Figure 4, the x-axis represents the increase of the transmission range from 100 to 250 units, and the y-axis IDentifies the improvement percentage of our algorithm as compared with Wu and Li s method with two extensional rules. The figure clearly shows that our algorithm performs better in medium-density space than low and high-density space. Figure 4 Improvement percentage of our algorithm versus density of networks over a network of 150 nodes

13 An enhanced approach to determine connected dominating sets 13 Figure 5 Average number of dominating nodes relative to radium r (a) Transmission range (b) Transmission range (c) Transmission range (d) Transmission range (e) Transmission range (f) Transmission range (g) Transmission range (h) Transmission range

14 14 C. Ni, H. Liu, A.G. Bourgeois and Y. Pan In order to fully understand this relative performance between Wu and Li s algorithm and ours, we conducted a second group of simulations. Figures 5a to 5h show the number of dominating set nodes with respect to radius, r, for the increasing order of the number of nodes, n. The number of nodes in the DS decreased smoothly as the transmission range increased by applying their algorithm and our new algorithm. From the simulation results, we have drawn the following conclusions: The simulation results verify Wu and Li s paper in that the algorithm using only the basic rule is poor and generates a large dominating set. We also verify that their algorithm using extensional rules 1 and 2 is efficient. Our approach by applying the enhanced rule consistently outperforms Wu and Li s algorithm with extensional rules, especially in medium-density networks. It is suggested that the enhanced rule can be used in medium-density networks to reduce the size of the dominating set. 6 Conclusion In this paper, we proposed an enhanced rule to reduce the size of the Connected Dominating Set (CDS) based on Wu and Li s algorithm for calculating the CDS. The enhancement is achieved with a little increase in computation complexity. Our approach calculates the CDS in O( 3 ) time with 2-hop neighbourhood information, where is the maximum node degree of the graph. In addition, the algorithm also uses constant rounds of message exchanges, and the amount of exchanged messages is the same as in their algorithm. The gateway nodes selected by our new algorithm form a CDS, then a reduced graph can be generated from the CDS and the searching space for a routing process can be reduced to this graph. The effectiveness of our enhancement is confirmed through a simulation study on both sparse and dense networks. Our simulation results show that the algorithm using only the basic rule of Wu and Li is poor and generates a large dominating set. It also shows that Wu and Li s algorithm using extensional rules 1 and 2 is efficient to reduce the size of the DS. Finally, we see that our algorithm with the enhanced rule consistently outperforms their algorithm by further reducing the size of the dominating set, especially in the medium-density space. Acknowledgment This research was supported in part by NSF under Grants ECS and ECS , and by NSFC under Grant No ( two bases project). The authors would also like to thank the two anonymous reviewers for their constructive comments.

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