Mobile Networks and Applications 9, 141–149, 2004 2004 Kluwer Academic Publishers. Manufactured in The Netherlands.
Distributed Construction of Connected Dominating Set in Wireless Ad Hoc Networks
PENG-JUN WAN Department of Computer Science, Illinois Institute of Technology, Chicago, IL 60616, USA
KHALED M. ALZOUBI Saint Xavier University, Computer Science Department, 3700 West 103rd street, Chicago, IL 60655, USA
OPHIR FRIEDER Department of Computer Science, Illinois Institute of Technology, Chicago, IL 60616, USA
Abstract. Connected dominating set (CDS) has been proposed as virtual backbone or spine of wireless ad hoc networks. Three distributed approximation algorithms have been proposed in the literature for minimum CDS. In this paper, we first reinvestigate their performances. None of these algorithms have constant approximation factors. Thus these algorithms cannot guarantee to generate a CDS of small size. Their message complexities can be as high as O(n2), and their time complexities may also be as large as O(n2) and O(n3). We then present our own distributed algorithm that outperforms the existing algorithms. This algorithm has an approximation factor of at most 8, O(n) time complexity and O(nlog n) message complexity. By establishing the (nlog n) lower bound on the message complexity of any distributed algorithm for nontrivial CDS, our algorithm is thus message-optimal. Keywords: wireless ad hoc networks, distributed algorithm, connected dominating set, independent set, leader election, spanning tree
1. Introduction it is desirable to find a minimum connected dominating set (MCDS) of a given set of nodes. However, finding an MCDS Wireless ad hoc networks can be flexibly and quickly de- in unit-disk graphs is NP-hard [6], and thus only distributed ployed for many applications such as automated battlefield, approximation algorithms in polynomial time are practical for search and rescue, and disaster relief. Unlike wired net- wireless ad hoc networks. In this paper, we further show that works or cellular networks, no physical backbone infrastruc- any distributed algorithm for nontrivial CDS requires at least ture is installed in wireless ad hoc networks. A communi- O(nlog n) messages, where n is the number of nodes and the cation session is achieved either through a single-hop radio message size is a constant multiple of the number of bits rep- transmission if the communication parties are close enough, resenting the node IDs (a CDS is said to be trivial if it consists or through relaying by intermediate nodes otherwise. In this of all nodes). paper, we assume that all nodes in a wireless ad hoc network Since the networking nodes in wireless ad hoc networks are distributed in a two-dimensional plane and have an equal are very limited in resources, a virtual backbone should not maximum transmission range of one unit. The topology of only be “thinner”, but should also be constructed with low such wireless ad hoc network can be modeled as a unit-disk communication and computation costs. In addition, the com- graph [6], a geometric graph in which there is an edge be- munication and computation costs should be scalable as the tween two nodes if and only if their distance is at most one (see figure 1). Although a wireless ad hoc network has no physical back- bone infrastructure, a virtual backbone can be formed by nodes in a connected dominating set of the corresponding unit-disk graph [1,7,10]. Such virtual backbone, also referred to as spine, plays a very important role in routing, broad- casting and connectivity management in wireless ad hoc net- works [1]. In general, a dominating set (DS) of a graph G = (V, E) is a subset V ⊂ V such that each node in V −V is adjacent to some node in V ,andaconnected dominating set (CDS) is a dominating set which also induces a connected subgraph. A (connected) dominating set of a wireless ad hoc network is a (connected) dominating set of the corresponding Figure 1. Model the topology of wireless ad hoc networks by unit-disk unit-disk graph. To simplify the connectivity management, graphs. 142 P.J. WAN ET AL. wireless ad hoc networks are typically deployed with large Proof. Let A be any distributed algorithm for spanning tree network size. In this paper, we first reinvestigate the per- in wireless ad hoc networks whose unit-disk graph is a ring. formance of the three known distributed approximation algo- Note that any spanning tree of a ring consists of all edges in rithms for MCDS, proposed by Das et al. in [1,7,10], by Wu the ring except one. Thus it has exactly two leaves which are and Li in [12], and by Stojmenovic et al. in [11], respectively. also neighbors. Thus after an spanning tree is completed, the While the first one has a (log n) approximation factor, the two leaves can exchange a message to select the leader be- other two both have (n) approximation factors. Thus none tween them according to some symmetry-breaking criterion, of them can guarantee to generate a CDS of small size. The al- for example by their IDs. After the leader is identified, it then gorithms also have very high implementation cost in terms of notifies all other nodes in linear number of message. Thus message complexity and/or time complexity. We thus present from algorithm A, we can derive a distributed algorithm for our own distributed algorithm that always outputs a nontrivial leader election whose message complexity is (n) more than CDS. This algorithm has an approximation factor of at most 8, the number of messages sent by A. From theorem 2, the mes- O(n) time complexity and O(nlog n) message complexity. sage complexity of A is at least (nlog n). As (nlog n) is a lower bound on the message complexity of any distributed algorithm for nontrivial CDS, our algorithm A distributed algorithm for leader election in wireless ad is thus a message-optimal distributed algorithm for nontrivial hoc networks has been proposed in [5]. This algorithm has CDS. message complexity O(nlog n) and therefore is message- The remaining of this paper is organized as follows. In sec- efficient. Its actual implementation also constructs a spanning tion 2, we establish a (nlog n) lower bound on the message tree rooted at the leader. complexity of any distributed algorithm for nontrivial CDS. In sections 3–5, we analyze the performances of the three Theorem 4. In asynchronous wireless ad hoc networks existing algorithms by Das et al. in [1,7,10], by Wu and Li whose unit-disk graph is a ring, any distributed algorithm for in [12], and by Stojmenovic et al. in [11], respectively. In sec- nontrivial CDS sends at least (nlog n) messages. tion 6, we present a better distributed algorithm and analyze its performance. Finally, we conclude this paper in section 7. Proof. Let A be any distributed algorithm for CDS in wire- less ad hoc networks whose unit-disk graph is a ring. Note that for any nontrivial CDS of a ring consists of all nodes ex- 2. Lower bound on message complexity cept either a unique node or two neighboring nodes. So after an nontrivial CDS is completed, the leader can be elected as In this section, we establish the (nlog n) lower bound on the follows. A dominatee declares itself as the leader if both its message complexity for distributed algorithms for leader elec- neighbors are dominators, or one of its neighbor is a domina- tion, spanning tree and nontrivial CDS in wireless ad hoc net- tee but has larger ID. The leader then notifies all other nodes works. The reduction is made from the following well-known in linear number of message. Thus from algorithm A, we can bound on the message complexity of distributed leader elec- derive a distributed algorithm for leader election whose mes- tion in asynchronous ring networks with point-to-point trans- sage complexity is (n) more than the number of messages mission. A leader election is a process to elect a unique node sent by A. From theorem 2, the message complexity of A is as the leader by all nodes. at least (nlog n). Theorem 1 [2]. In asynchronous rings with point-to-point transmission, any distributed algorithm for leader election 3. Das et al.’s algorithm sends at least (nlog n) messages. The centralized version of the distributed algorithm proposed Theorem 2. In asynchronous wireless ad hoc networks by Das et al. consists of three stages. The first stage finds an whose unit-disk graph is a ring, any distributed algorithm for approximation to Minimum Dominating Set, which is essen- leader election sends at least (nlog n) messages. tially the well-studied Set Cover problem. Not surprisingly, the heuristic proposed by Das et al. in [1,7,10] is a transla- Proof. Let A be any distributed algorithm for leader election tion of Chvatal’s greedy algorithm [4] for Set Cover, and thus in wireless ad hoc networks whose unit-disk graph is a ring. guarantees an approximation factor of H( ),where is the Let A∗ be the algorithm by replacing each wireless transmis- maximum degree and H is the harmonic function. Let U de- sion by two point-to-point transmissions. Then A∗ is a dis- note the dominating set output in this stage. The second stage tributed algorithm for leader election in asynchronous rings constructs a spanning forest F . Each tree component in F with point-to-point transmission. Note that the algorithm A∗ is a union of stars centered at the nodes in U.Thestarsare sends twice messages of that sent by A. Thus from theorem 1, generated by letting each dominatee node pick up an arbitrary A must also send at least (nlog n) messages. neighbor in U. The third stage expands the spanning forest F to a spanning tree T . All internal nodes in T form a CDS. It Theorem 3. In asynchronous wireless ad hoc networks is easy to show that the CDS generated in this way contains whose unit-disk graph is a ring, any distributed algorithm for at most 3|U| nodes, and therefore is a 3H( )-approximation spanning tree sends at least (nlog n) messages. of MCDS. DISTRIBUTED CONSTRUCTION OF CONNECTED DOMINATING SET 143
as the CDS at the end of the second stage. This can be proven by induction as follows. Initially, the degree of node vi is − 2 · 2i 1 + (k − 1) + 2 = 2i + k + 1;
the degrees of the node u1 and u2 are both k − 2i 1 + k + 1 = 2k + k; i=1 and the degree of any other node is k − 2i 1 − 1 + 1 + 1 = 2k. i=1
So vk is the first node to be selected. Now we assume that the nodes vk,vk−1,...,vj have been added to the dominating set. For any node vi with i
(log )/2 − 1/2and3H( ). Both its message complexity and time complexity are O(n2).
4. Wu and Li’s algorithm
While the algorithm proposed by Das et al. first finds a DS and then grow this DS into a CDS, the algorithm proposed by Wu and Li in [12] takes an opposite approach. The algorithm in [12] first finds a CDS and then prune certain redundant nodes from the CDS. The initial CDS U consists of all nodes (a) (b) which have at least two non-adjacent neighbors. A node u in U is considered as locally redundant if it has either a neigh- Figure 3. Instance for which the CDS output by Wu and Li’s algorithm con- bor in U with larger ID which dominates all other neighbors sists of all nodes but the MCDS consists of only two nodes. of u, or two adjacent neighbors with larger IDs which together The distributed implementation of the above algorithm dominate all other neighbors of u. The algorithm then re- given in [12] runs in two phases. In the first phase, each moves all locally redundant nodes from U. This algorithm ap- node first broadcasts to its neighbors the entire set of its neigh- plies only to wireless ad hoc networks whose unit-disk graph bors, and after receiving this adjacency information from all is not a complete graph. As indicated in [12], the approxima- neighbors it declares itself as dominator if and only if it has tion factor of this algorithm remains unspecified. Obviously, two nonadjacent neighbors. These dominators form the ini- the MCDS of any wireless ad hoc network whose unit-disk tial CDS. In the second phase, a dominator declares itself as graph is not complete graph consists of at least two nodes. a dominatee if it is locally redundant. Note a dominator can On the other hand, any CDS contains at most n nodes. Thus, find whether it is locally redundant from the adjacency infor- the approximation factor of the above algorithm is at most mation of all its neighbors. It is claimed in [12] that the to- n/2wheren is the number of nodes. Next, we show that the tal message complexity is O(n ) and the time complexity at approximation factor of the above algorithm is exactly n/2. each node is O( 2). A more accurate message complexity is This means that the above algorithm does perform extremely (m) where m is the number of edges in the unit-disk graph, poorly over certain instances. as each edge contributes two messages in the first phase. The When n is even, we consider the instance illustrated in fig- O( 2) time complexity, however, is not correct. In fact, in ure 3(a). These nodes are evenly distributed over the two hori- order to decide whether it is locally redundant in the second zontal sides of a unit-square. Each node has exactly m neigh- phase, a node u in the initial CDS may have to examine as bors, one in the opposite horizontal side and the rest in the many as O( 2) pairs of neighbors, and for each pair of neigh- same horizontal side. The two nodes at the left two corners bors, as much as O( ) time may be taken to find out whether of the unit-square form an MCDS. However, the CDS out- such pair of neighbors together dominates all other neighbors put by the algorithm in [12] consists of all nodes. Indeed, for of u. Therefore, the time complexity at each node may be as each node u, the unique neighbor lying in the opposite hori- high as O( 3), instead of O( 2). Note that m and can zontal side is not adjacent to all other neighbors of u. Thus, be as many as O(n2) and O(n), respectively. Thus, the mes- the initial CDS U consists of all nodes. In addition, no single sage complexity and the time complexity of the distributed neighbor of a node u can dominate all other neighbors of u. algorithm in [12] are O(n2) and O(n3), respectively. The in- Furthermore, if a pair of neighbors of u are adjacent, they stances shown in figure 3 do require such complexities. must lie in the same horizontal side as u, and therefore nei- In summary, we have the following performance results of ther of them is adjacent to the unique neighbor of u lying in the distributed algorithm in [12]. the opposite horizontal side. So no node is locally redundant. Consequently the output CDS still consists of all nodes. Theorem 6. The approximation factor of the distributed al- When n is odd, we consider the instance illustrated in fig- ∗ gorithm proposed by Wu and Li in [12] is exactly n/2. ure 3(b). The node with the largest ID, denoted by u ,isthe Its message complexity is (m) and its time complexity is center of the left vertical side of a unit-square, and all other O( 3). n − 1 nodes are evenly distributed over the two horizontal sides of the unit-square. The two nodes at the left two cor- ners of the unit-square form an MCDS. On the other hand, the 5. Stojmenovic et al.’s algorithm CDS output by the algorithm in [12] also consists of all nodes. In fact, following the same argument as in the even case, all In the context of clustering and broadcasting, Stojmenovic nodes other than u∗ are in the initial CDS U. The node u∗ is et al. [11] presented three synchronized distributed construc- also in the initial CDS U as well. Since u∗ is not adjacent to tions of CDS. In each of the three constructions, the CDS con- the two nodes at the right corners of the unit-square, all nodes sists of two types of nodes: the cluster-heads and the border- other than u∗ are not locally redundant. The u∗ itself is also nodes. The cluster-heads form a maximal independent set not locally redundant as it has the maximum ID. Therefore, (MIS), i.e., a dominating set in which any pair nodes are non- the output CDS still consists of all nodes. adjacent. A node is a border-node if it is not a cluster-head DISTRIBUTED CONSTRUCTION OF CONNECTED DOMINATING SET 145
6. A better distributed algorithm
Our distributed algorithm for CDS consists of two phases. These two phases construct a maximal independent set (MIS), and a dominating tree, respectively. They are described and analyzed in the next three subsections.
6.1. MIS construction
By definition, any pair of nodes in an MIS are separated by at least two hops. However, a subset of nodes in an MIS Figure 4. Instance for which the CDS output by Stojmenovic et al.’s algo- rithm consists of all nodes but the MCDS consists of only one node. may be three hops away from the subset of the rest nodes in this MIS. The MIS constructed in this section guarantees that the distance between any pair of its complementary subsets is exactly two hops. Our construction uses a carefully chosen and there are at least two cluster-heads within its 2-hop neigh- rank definition. The ranking is induced by an arbitrary rooted borhood. The set of cluster-heads is induced by a ranking of spanning tree T , which can be constructed by the distributed nodes which give rise to a total ordering of all nodes. Three leader-election algorithm in [5] with O(n) time complexity rankings are used: the ID only [8,9], an ordered pair of de- and O(nlog n) message complexity. Given a rooted spanning gree and ID [3], and an order pair of degree and location [11]. tree T , the (tree) level of a node is the number of hops in T The selection of the cluster-heads is given by a synchronized between itself and the root of T . (Thus the level of the root distributed algorithm, which can be generalized to the fol- is 0.) The rank of a node is then given by the ordered pair of lowing framework. Initially all nodes are colored white. In its level and its ID. Such ranking gives rise to a total ordering each stage of the synchronized distributed algorithm, all white of the nodes in the lexicographic order. The following distrib- nodes which have the lowest rank among all white neighbors uted process enables each node to calculate its own rank and are colored black; then all white nodes adjacent to the these the number of lower-ranked neighbors. black nodes are colored gray; finally the ranks of the remain- Each node maintains two local metering variables x1 ing white nodes are updated. The algorithm ends when all and x2.Thevariablex1 counts the number of neighbors whose nodes are colored either black or gray. All black nodes then levels have not yet been identified and is thus initialized to the form the cluster-heads. number of neighbors. The variable x2 counts the number of Regardless of the choice of the ranking, the algorithms children who have not yet reported the completion and is thus in [11] have a (n) approximation factor. Such inefficiency initialized to the number of children. Each node also main- stems from the non-selective inclusion of all border-nodes. tains a levelList that records the levels of its neighbors and is In fact, if the rank is ID only, figure 4 shows a family of in- initially empty, and a local variable y which stores the num- stances which would imply the approximation factor to be ex- ber of lower-ranked neighbors. After the rooted spanning tree actly n, the worst possible. In these instances, the node with T is constructed, the root announces its level 0 by broadcast- the largest ID is located at the center of a unit-disk and all ing a LEVEL message. Upon receiving a LEVEL message, other nodes are evenly distributed in the boundary of the unit- a node appends an entry consisting of the sender’s ID and disk. After the cluster-heads are selected, all other nodes be- level to levelList and then decrements x1 by 1. If the sender come border-nodes. Thus the CDS would consist of all nodes. is its parent in T , it sets its own level to one plus the sender’s On the other hand, the node at the center dominates all other level, and then announce this level by broadcasting a LEVEL nodes. If the rank is an ordered pair of degree and ID or an message. If x1 = 0, it sets y to the number of lower-ranked order pair of degree and location, the instances shown in fig- neighbors which can be calculated from levelList.Ifitisa = ure 3 imply that their approximation factors are at least n/2. leaf in T (i.e. x2 0 initially) and its own level has been All algorithms in [11] have O(n2) message complexity determined, it transmits a LEVEL-COMPLETE message to and (n) time complexity. This can be illustrated in the fol- its parent. Upon receiving a LEVEL-COMPLETE message = lowing instance: All n nodes are evenly distributed in an in- towards itself, a node decrements x2 by 1; if x2 0after terval of length n − 1 with two nodes being the endpoints of the update and it is not the root, a node transmits a LEVEL- the interval. The ith node from the left endpoint of interval COMPLETE message to its parent and then resets x2 to the = has ID i (i.e., the IDs increase from left to right). number of children. When the local variable x2 0atthe In summary, we have the following performance results of root, the root simply resets x2 to the number of children. By this time, all nodes know the ranks of its own and all its neigh- the distributed algorithm in [11]. bors and thus the root will move on the construction of the MIS by a color-marking process. Theorem 7. The distributed algorithms proposed by Stoj- All nodes are initially marked with white color and will menovic et al. in [11] have an approximation factor of n/2 be marked with either gray or black eventually. Each node 2 or n, O(n ) message complexity, and (n) time complexity. also maintains a blackList which records the IDs of its black 146 P.J. WAN ET AL.
Figure 5. An example of the MIS construction ((a)–(g)) and dominating tree construction ((h)–(k)). neighbors. Note that the blackList can contain at most five ing a MARK-COMPLETE message towards itself, a node black nodes. The root first marks itself black and broad- decrements x2 by 1; if x2 = 0 after the update and it is casts a BLACK message. Upon receiving a BLACK mes- not the root, a node transmits a MARK-COMPLETE mes- sage, a node adds the sender’s ID to blackList, and if its color sage to its parent. By the time when the local variable x2 = 0 is still white, it marks itself gray and broadcasts a GRAY at the root, all nodes have been marked with either gray of message which contains its level. Upon receiving a GRAY black and thus the root will move on the construction of the message, if the rank of the sender is lower than its own, a CDS. white node decrements y by 1; if y = 0 after the update, it Figure 5 illustrates the algorithm for color marking in this marks itself black and broadcasts a BLACK message. When phase. In the graph, the IDs of the nodes are labelled be- a leaf node is marked with either gray or black, it transmits side the nodes, and node 0 is the leader elected in the first a MARK-COMPLETE message to its parent. Upon receiv- phase. The solid lines represent the edges in the rooted span- DISTRIBUTED CONSTRUCTION OF CONNECTED DOMINATING SET 147 ning tree T , and the dashed lines represents other edges in the mits a JOIN message towards the sender, and then broad- unit-disk graph. The ordering of the nodes by rank is given casts an INVITE1 message. by 0, 4, 12, 2, 5, 8, 10, 3, 6, 9, 11, 1, 7. A possible execution • Upon receiving an INVITE1 message, a gray node with scenario is shown in figures 5(a)–(g). The nodes 0, 5, 3, and z = 0setsz = 1andparent to the ID of the sender, trans- 7 are the black nodes and form a CDS. mits a JOIN message towards the sender, and then broad- The construction of the CDS in the next phase relies on the casts an INVITE2 message. following property of the black nodes. • Upon receiving a JOIN message towards itself, a node adds the ID of the sender to childrenList. Theorem 8. All black nodes form an MIS and any pair of complementary black subsets are separate by exactly two Theorem 8 guarantees that whenever there is any black hops. node outside the current T ∗, at least one black node would join T ∗. Thus eventually all black nodes will join T ∗. Conse- ∗ Proof. Let U ={ui:1 i k} where ui is the ith node quently, all gray nodes will join T eventually. The internal ∗ which is marked black. From the construction, any pair of nodes of T are then output as the CDS. black nodes are not adjacent to each other and thus U is an Figures 5(h)–(k) illustrate a possible scenario of the domi- MIS. For any 1 j k,letHj be the graph over {ui:1 nating tree construction. The thick links are edges in the dom- i j} in which a pair of nodes is connected by an edge if inating tree. The internal nodes 12, 0, 5, 7, 2, 3 form a CDS. and only if their graph distance in G is two. We prove by induction on j that Hj is connected. Since H1 consists of 6.3. Performance analysis a single vertex, it is connected trivially. Assume that Hj−1 is connected for some j 2. When the node uj is marked We first analyze the message complexity and time complexity black, its parent in T must be already marked gray. Thus, of our distributed algorithm. After the rooted spanning tree T there is some node ui with 1 i Table 1 Performance comparison. [1,7,10] [12] [11] This paper n Approx. factor (log n) n 2 , n 8 Msg. complexity O(n2) (m) O(n2)O(nlog n) Time complexity O(n2)O( 3) (n) O(n) Nontrivial Yes No No Yes approximation algorithms for MCDS. After that we presented our own algorithm. The performance comparison of these Figure 6. Ui lie in a sector of at most 240 degree within the coverage range four algorithms is listed in table 1. From this table, we can of node v . i conclude that our algorithm outperforms the existing algo- Lemma 9 and its proof implies the following upper-bound rithms. on the size of the CDS generated by our algorithm. Lemma 10. The number of internal nodes in T ∗ is at most References − 8opt 2. [1] V. Bharghavan and B. Das, Routing in ad hoc networks using minimum connected dominating sets, in: Proc. of International Conference on Proof. If there is a black node in OPT, then following the Communications’97, Montreal, Canada (June 1997) pp. 376–380. similar proof to lemma 9 we can show that the total number [2] J. Burns, A formal model for message passing systems, Technical Re- of black nodes is at most port TR-91, Computer Science Department, Indiana University (May 1980). 1 + 4(opt − 1) = 4opt − 3. [3] G. Chen and I. Stojmenovic, Clustering and routing in wireless ad hoc networks, Technical Report TR-99-05, Computer Science, SITE, Uni- ∗ Since each internal gray node in T has at least one black versity of Ottawa (June 1999). child, the total number of internal gray nodes in T ∗ is no more [4] V. Chvátal, A greedy heuristic for the set-covering problem, Mathemat- than the number of black nodes. Thus the total number of ics of Operation Research 4(3) (1979) 233–235. ∗ [5] I. Cidon and O. Mokryn, Propagation and leader election in multihop internal nodes in T is at most broadcast environment, in: Proc. of 12th International Symposium on 2(4opt − 3) = 8opt − 6. DIStributed Computing (DISC98), Greece (September 1998) pp. 104– 119. Now we assume that no black node is in OPT.Letk be the [6] B.N. Clark, C.J. Colbourn and D.S. Johnson, Unit disk graphs, Discrete T ∗ k Mathematics 86 (1990) 165–177. number of black nodes adjacent to the root of .Then 5, [7] B. Das, R. Sivakumar and V. Bharghavan, Routing in ad-hoc networks and following the similar proof to lemma 9 we can show that using a spine, in: Proc. of International Conference on Computers and the total number of black nodes is at most k + 4(opt − 1). Communications Networks ’97, Las Vegas, NV (September 1997). Note that the root of T ∗ has exactly k black children and any [8] M. Gerla and J. Tsai, Multicluster, mobile, multimedia radio network, internal gray node other than the root of T ∗ has at least one ACM–Baltzer Journal of Wireless Networks 1(3) (1995) 255–265. black child. Thus the total number of internal gray nodes in [9] C.R. Lin and M. Gerla, Adaptive clustering for mobile wireless net- works, IEEE Journal on Selected Areas in Communications 15(7) T ∗ other than the root of T ∗ is at most 4(opt − 1).Sothe ∗ (1997) 1265–1275. number of internal nodes in T is at most [10] R. Sivakumar, B. Das and V. Bharghavan, An improved spine-based − + + + − = − + infrastructure for routing in ad hoc networks, in: Proc. of IEEE Sym- 4(opt 1) k 1 4(opt 1) 8opt 7 k posium on Computers and Communications ’98, Athens, Greece (June 8opt − 7 + 5 1998). = − [11] I. Stojmenovic, M. Seddigh and J. Zunic, Dominating sets and neigh- 8opt 2. bor elimination based broadcasting algorithms in wireless networks, in: Proc. of IEEE Hawaii Int. Conf. on System Sciences (January 2001). Thus the lemma is true in either case. [12] J. Wu and H.L. Li, On calculating connected dominating set for ef- ficient routing in ad hoc wireless networks, in: Proc. of Proceedings In summary, we have the following performance results of of the 3rd ACM International Workshop on Discrete Algorithms and our distributed algorithm. Methods for Mobile Computing and Communications (1999) pp. 7–14. Theorem 11. Our distributed algorithm has an approxima- tion factor of at most 8, O(n)time complexity, and O(nlog n) Peng-Jun Wan is an Associate Professor in Com- message complexity. puter Science at Illinois Institute of Technology. He received his Ph.D. in computer science from Univer- sity of Minnesota in 1997, MS in operations research and control theory from Chinese Academy of Sci- 7. Conclusion ence in 1993, and BS in applied mathematics from Qsinghua University in 1990. His research interests In this paper, we have established a (nlog n) lower bound include wireless networks and optical networks. on message complexity of any distributed algorithm for non- E-mail: [email protected] trivial CDS. We then reinvestigated three known distributed DISTRIBUTED CONSTRUCTION OF CONNECTED DOMINATING SET 149 Khaled M. Alzoubi is an Assistant Professor in Ophir Frieder is the IITRI Chair Professor of Com- Computer Science at Saint Xavier University. He re- puter Science at the Illinois Institute of Technology. ceived his Ph.D. in computer science from Illinois In- His research focus spans the area of scalable infor- stitute of Technology in 2002, and BS in mathemat- mation systems including search and retrieval and ics and computer science from University of Illinois communications issues. He is on the editorial boards at Chicago in 1987. His research interests include of the Journal of the American Society for Informa- distributed algorithms for routing in wireless ad hoc tion Science and Technology, Information Retrieval, networks. the Journal of Supercomputing, and Optical Net- E-mail: [email protected] works Magazine. He is a Member of the ACM and a Fellow of the IEEE. E-mail: [email protected]