zone-based virtual backbone formation in wireless ad hoc networks

Available online at www.sciencedirect.com

Ad Hoc Networks 7 (2009) 183–200

www.elsevier.com/locate/adhoc

Zone-based virtual backbone formation in wirelessad hoc networks

Bo Han *

Department of Computer Science, University of Maryland, College Park, MD 20742, United States

Received 15 April 2007; received in revised form 25 December 2007; accepted 14 January 2008Available online 20 January 2008

Abstract

Efficient protocol for clustering and backbone formation is one of the most important issues in wireless ad hoc net-works. Connected dominating set (CDS) formation is a promising approach for constructing virtual backbone. However,finding the minimum CDS in an arbitrary graph is a NP-Hard problem. In this paper, we present a novel zone-based dis-tributed algorithm for CDS formation in wireless ad hoc networks. In this Zone algorithm, we combine the zone and level

concepts to sparsify the CDS constructed by previous well-known approaches. Therefore, this proposed algorithm can sig-nificantly reduce the CDS size. Particularly, we partition the wireless network into different zones, construct a dominating

tree for each zone and connect adjacent zones by inserting additional connectors into the final CDS (at the zone borders).Our comprehensive simulation study using a custom simulator shows that this zone-based algorithm is more effective thanprevious approaches. The number of nodes in the CDS formed by this Zone algorithm is up to around 66% less than thatconstructed by others. Moreover, we also compare the performance of Zone algorithm with some recently proposed CDSformation protocols in ns2 simulator.� 2008 Elsevier B.V. All rights reserved.

Keywords: Wireless ad hoc networks; Virtual backbone; Connected dominating set; Minimum connected dominating set; Distributedalgorithm

1. Introduction

Wireless ad hoc networks can be temporarily andspontaneously created by individual nodes withoutrequiring any infrastructure or central control. Inthis kind of networks, management tasks are typi-cally performed in a distributed manner. A wirelessnode can only directly communicate with othernodes that are within its transmission range. For

1570-8705/$ - see front matter � 2008 Elsevier B.V. All rights reserved

doi:10.1016/j.adhoc.2008.01.003

* Tel.: +1 301 405 2724; fax: +1 301 405 6707.E-mail address: [email protected]

nodes that are not within the transmission rangeof each other to communicate, some intermediatenodes must be utilized to relay packets for them.Generally, wireless ad hoc networks are deployedin emergent and temporary situations such as acci-dents or public gatherings.

The topology of wireless ad hoc networks canusually be modeled by a graph G = (V,E), wherethe vertex set V represents the wireless nodes andthere is an edge between two vertices if the corre-sponding nodes are within the transmission rangeof each other. If we assume all the mobile nodeshave the same maximum transmission range, the

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184 B. Han / Ad Hoc Networks 7 (2009) 183–200

topology of such wireless ad hoc networks can berepresented as a unit disk graph (UDG), a geomet-ric graph in which there is an edge between twonodes u and v if and only if their Euclidean distanceis at most one.

Although wireless ad hoc networks have no phys-ical infrastructure, it is natural to construct a virtualbackbone through connected dominating set forma-tion. A set is a dominating set (DS) if every node inthe network is either in the set or a neighbor of a nodein the set. When a DS is connected, where any twonodes in the DS can be connected through intermedi-ate nodes from the DS, it is denoted as a connected

dominating set. In practice, a core-extraction ad hocrouting algorithm is proposed in [1], where the coreis constructed by a dominating set. This core is alsoextended to support multicast in ad hoc networks[2]. Consequently, it is desirable to find a CDS formedby a small number of nodes. However, the minimumdominating set (MDS) and minimum connecteddominating set (MCDS) problems have been shownto be NP-Hard [3]. The problem of finding a MCDSin a unit disk graph is still NP-Hard [4].

Based on our previous work [5], we propose anovel zone-based algorithm for connected dominat-ing set formation in wireless ad hoc networks withconstant approximation ratio, linear time and mes-sage complexity. Generally, a tree-like CDS hassmaller size but the distributed leader election fordeciding the root of the tree generates a large num-ber of messages that consume the scarce resource(e.g., energy) in wireless networks. Whereas con-struction of a mesh-like CDS can reduce messageoverhead but the size of resulting CDS is relativelylarge. To achieve a tradeoff between the CDS sizeand message overhead, this Zone algorithm parti-tions the wireless network into different zones, usessome selection criteria to construct a dominatingtree for each zone and adjusts the zone borders tomake the adjacent zones connected (by insertingadditional nodes to the final CDS).

In contrast to tree-like CDS construction, theroot of dominating tree for each zone can be deter-mined automatically during the partition withoutextra communication overhead. For the dominatingtree construction, each node in this Zone algorithmis assigned a Level to indicate its logical distance tothe zone center. Compared with our previous algo-rithm [5], the contribution of this paper mainly liesin that we combine the Zone and Level conceptsto construct a dominating tree in each zone, whereasin the previous algorithm we form a mesh-like CDS

in each zone. Therefore, the approximation ratio ofCDS in a zone is further reduced from 44 to 7.8 andthe final CDS size is also significantly decreased.The results of our extensive simulation study showthat the number of nodes in the CDS formed by thisZone algorithm is up to around 66% less than thatconstructed by others. Meanwhile, we also keepthe communication overhead as low as possible.

The rest of the paper is organized as follows. Sec-tion 2 briefly reviews the related work. Section 3describes the network assumption and some prelim-inaries. In Section 4, we present our new zone-basedCDS formation algorithm. The performance analy-sis is given in Section 5. In Section 6, we discusssome implementation issues and the virtual back-bone maintenance. Section 7 presents the simulationresults using both custom and ns2 simulators. Wesummarize the major results and raise some futuredirections in Section 8.

2. Related work

Based on Guha and Khuller’s approximationalgorithm to calculate connected dominating set[6], Das et al. design a MCDS-based distributedrouting algorithm for wireless ad hoc networks [7].This distributed algorithm is not localized, since itutilizes central coordinators to direct the algorithmexecution. The algorithm proposed by Wu and Lifirst finds a connected dominating set and thenprunes some redundant nodes from the CDS usingtwo rules (Rule 1 and 2) [8]. In their marking pro-cess, each node is marked true (dominator) if ithas two unconnected neighbors. According to Rule1, a marked node can unmark itself if its neighborset is covered by another neighboring marked nodewith higher ID. According to Rule 2, a marked nodecan unmark itself if its neighborhood is covered bytwo other neighboring directly connected markednodes with higher IDs. The combination of Rule 1and 2 is fairly efficient to prune the redundantnodes. Recently, Dai and Wu propose a generalizedpruning rule, called Rule k [9]. According to Rule k,a marked node can unmark itself if its neighbor-hood is covered by a set of k neighbors with higherIDs whose induced sub-graph is strongly connected.Hereafter, the algorithms described in [8,9] arereferred to as Rule 1 and 2 and Rule k, respectively.

The solution proposed in [10] constructs a CDSrelying on all nodes having a common clock andrequires two-hop neighbor information. Stojmenov-ic et al. present a distributed construction of CDS in

B. Han / Ad Hoc Networks 7 (2009) 183–200 185

the context of clustering and broadcasting [11].Basically, the algorithm in [11] is an enhancementof Rule k, where more efficient pruning strategy isutilized and the degree, rather than node ID, is usedto sort the nodes. Dubhashi et al. propose a poly-logarithmic-time distributed algorithm that findsan approximation to the MCDS [12]. Cheng et al.design a polynomial time approximation schemefor the MCDS construction in ad hoc networks[13]. Kuhn and Wattenhofer propose a novel dis-tributed MDS approximation algorithm based onlinear programming relaxation techniques [14].Kuhn et al. compute a dominating set to initializenewly deployed ad hoc and sensor networks [15].Their model is very harsh, in which there is nounderlying MAC layer, asynchronous wake-up,scarce knowledge about the network topology.Wang et al. propose an efficient distributed methodto construct a low-cost weighted MCDS [16].

Several clustering algorithms choose cluster-heads according to some local condition [17–20].Baker and Ephremides [17] introduce a distributedcluster architecture and demonstrate its adaptabilityto network topological changes. The notion of clus-ter has been revisited by Gerla and Tsai [18] formultimedia communications with emphasis on theallocation of resources to support the multimediatraffic in ad hoc networks. The Distributed Cluster-ing Algorithm (DCA) [19] proposed by Basagniet al. is a common generalization of the algorithmsfor the clustering setup, such as the algorithms pre-sented in [17,18]. These previous approaches aregeneralized by allowing the choice of cluster-headsto be based on a generic weight associated with eachnode. Recently, Basagni et al. propose a clique clus-tering protocol to build and maintain a connectedbackbone for wireless sensor networks [20]. Mostof these clustering algorithms can also be utilizedto construct CDS in ad hoc networks. However,the CDS formed by cluster-heads and gateways(any node that can hear two or more cluster-heads[18]) in existing algorithms usually has a relativelylarge size. The proposed Zone algorithm is builton top of these previous clustering algorithms, butcan construct a CDS with much smaller size.

In the distributed clustering algorithms, it is notdesirable to have neighboring cluster-heads [21].For dominating set formation, it is also undesirableto have neighboring dominators. This observationleads to the well-known concept of maximal indepen-

dent set (MIS). An independent set of graphG = (V,E) is a subset S � V such that for any pair

of vertices in S, there is no edge between them. Obvi-ously, a MIS S is also an independent DS. The reasonis that for every node u that does not belong to S, itmust have at least one neighbor in S, otherwise ushould be added into S. Chlamtac and Farago presenta random clustering algorithm which essentially findsa small independent dominating set [22]. The two heu-ristic algorithms proposed by Alzoubi et al. [23,24]take advantage of the property of MIS in UDG, thuscan guarantee a constant approximation ratio of 8and 12, respectively. Although these two algorithmsare distributed, they are not localized because theyneed some global information. Moreover, both algo-rithms are implemented by first electing a leaderamong the nodes, which was going to be the root ofa spanning tree. However, distributed leader electionis expensive in practice, and exhibits a very low degreeof parallelism [25]. To address the problem of non-localized computation, based on their previousworks, Alzoubi et al. propose a message-optimallocalized algorithm with linear time and messagecomplexity [26]. The approximation ratio of this algo-rithm is bounded by 192. In the following, Alzoubi’salgorithm in [26] is referred to as AWF. Han andJia propose an area-based algorithm that constructsa mesh-like CDS in each area and uses additionalnodes to connect the CDS in adjacent areas [5]. Basa-gni et al. perform a comparative analysis for CDScomputation and propose another technique, calledDCA-S(i) to sparsify the mesh-like CDS by destroy-ing all small cycles with the maximum length i whilemaintaining connectivity [25]. DCA-S(i) is a naturalextension of the DCA protocol [19] and the proposedsparsification technique works on a ‘‘fat” CDS likethe one generated by [26]. More detailed literaturereviews of connected dominating set formation inad hoc networks can be found in [16,25].

Some variants of connected dominating set prob-lem have also been proposed by Wu and Stojmenov-ic to support power-aware computing in wireless adhoc networks [27,28]. The concept of dominating set

in undirected graphs is extended to cover directedgraphs by introducing another concept called absor-

bant set [29]. But for a directed graph, a node, say u,cannot directly know its absorbent neighborsbecause there is no edges from its absorbent neigh-bors to it, i.e., u cannot hear from its absorbentneighbors directly. This problem is partially solvedin [29] by means of k-hop broadcast. However, ina general graph, it is hard to guarantee that everynode will know all its absorbent neighbors unlessthe broadcast packet is forwarded throughout the


whole network. In this paper, we still concentrate onthe undirected graphs.

In fact, zone-based schemes have been exten-sively studied in wireless ad hoc networks [30–34].For instance, Wang and Olariu propose a novelhybrid routing protocol – the Two-Zone RoutingProtocol (TZRP) [30] – as a nontrivial extensionof ZRP proposed by Hass and Pearlman [31]. Deve-rapalli and Sidhu design a multicast protocol formobile ad hoc networks, called the Multicast rout-ing protocol based on Zone Routing (MZR) [32].Chen and Liestman propose a zonal algorithm forweakly connected dominating set construction inad hoc networks [33]. Liang and Hass present ahybrid Virtual Backbone Routing (VBR) frame-work for ad hoc networks, which combines localproactive and global reactive routing componentsover a variable-sized zone hierarchy [34].

3. Network assumptions and preliminaries

In this paper, we assume that an ad hoc networkcomprises a group of wireless nodes communicatingthrough a common broadcast channel using omni-directional antennas with the same transmissionrange (the corresponding topology graph is UDG).Scheduling of transmission is the responsibility ofthe MAC layer. That is, like many existingapproaches, we do not deal with the issues on howthe messages use a shared wireless channel to avoidcollisions. Each node has a unique ID (e.g., IPaddress) and also knows the ID and node degree ofits one-hop neighbors. We also assume that eachnode will be in a relative-static state in a reasonableperiod of time.

We call the nodes in the dominating set domina-

tors, the nodes not in the dominating set dominatees,and the nodes that connect two dominators connec-

tors. Especially, we call the connectors that connecttwo or three hops away dominators as one-hop con-

nectors and two-hop connectors, respectively. Next,we give some well-known preliminaries which willbe used later.

Preliminary 1. By building a dominating set throughMIS construction, the number of dominators insidethe disk centered at every node with radius k-units isbounded by a constant lk.

Proof. Alzoubi et al. gave a proof through calcula-tion that lk < (2k + 1)2 � 1 [26]. When k = 2, 3, wehave lk = 23, 47. Let D be the maximum density

of the packing of n non-overlapping equal small cir-cles with radius r in a large circle with radius R.Thus, D = npr2/pR2 = nr2/R2 [35]. The upper boundof D is also given in [36] as

D 6n

1�ffiffi3p

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi34þ 2

ffiffi3p

p ðn� 1Þq� �2

: ð1Þ

For a given k, R = kr, D = nr2/R2 = n/k2. Fromabove inequality 1, we can calculate the value lkwhich is equal to n. For example, for k = 2, wecan get a smaller l2 = 21. Similarly, we can getl3 = 42. h

Preliminary 2. Let G be a UDG and OPT be thesize of a minimum CDS for G, then the size ofany MIS for G is at most 3.8 � OPT + 1.

The proof of this preliminary bounds the size ofany MIS in G and can be found in [37].

Preliminary 3. In a DS, the maximum distance toanother closest dominator from any dominator is 3.

Proof. By contradiction. Assume that the maxi-mum distance from a dominator u to the closestdominator v is 4, and the shortest path between u

and v is {u,x,y,z,v}. According to the definitionof dominating set, node y must have a dominator,say w, which is one hop closer (three hops) to u thanv. This contradicts the assumption that v is the clos-est dominator to u. h

4. Zone-based CDS formation algorithm

4.1. Overview

The Zone algorithm is motivated by the follow-ing observations. After the network is divided intoseveral zones through the common MIS construc-tion approaches, in each zone, the maximum dis-tance to another closest dominator from anydominator is exactly 2. Therefore, we do not needto connect every pair of three-hop away neighboringdominators which will introduce two more nodesinto the final CDS. Connecting every dominatorto only one specific two-hop away neighboringdominator is enough to guarantee the connectivityin each zone.

The main objective of this Zone algorithm is toreduce the size of CDS and meanwhile keep thecommunication cost as low as possible. As we dis-


cussed above, the two algorithms in [23,24] con-struct dominating tree for the whole topologygraph. This dominating tree has a small size butleads to a high message overhead. The AWF algo-rithm [26] reduces the communication cost by con-necting all two or three hops away dominatorpairs and thus forms a large mesh-like CDS. TheZone algorithm achieves a tradeoff between thesealgorithms by first constructing a dominating treein each zone and then building a mesh-like con-nected overlay network of these zones.

The Zone algorithm follows the general cluster-ing approaches [17–19]. At first, a DS is constructedthrough MIS formation, and then we insert addi-tional nodes into this DS to make it connected.Compared with existing approaches, the novelty ofthis algorithm is that during the DS formation,the wireless network is divided into several zonesnaturally and the root of the dominating tree foreach zone is determined automatically. Therefore,we do not need to use distributed leader electionto determine the root of the dominating tree. More-over, the root of each zone does not need to controlthe zone size and adjust the zone borders.

4.2. Degree based algorithm

In this paper, we define two kinds of most-valued-nodes, the nodes with the minimum ID and thenodes with the maximum degree among all the can-didates of dominators or connectors. The resultingZone algorithm is named Min ID and Max Degree,respectively. For simplicity, in the following descrip-tion of the Zone algorithm, we will use node degreeas the selection metric. We stress that our algorithmcan be easily extended to support other selection cri-teria, such as available energy, node mobility, orsome combinations of them. That is, we can alsointegrate the node weight defined in [19] into ouralgorithm to extend the dominator selection criteria.More such important criteria for wireless ad hocnetworks can be found in [19].

Define the rank of node u to be an ordered pairof (du, idu) where du is the node degree and idu isthe node ID. We say that a node u with rank (du, idu)has a higher order than a node v with rank (dv, idv) ifdu > dv, or du = dv and idu < idv. A node stays in oneof the four states: unmarked, dominatee, dominator

and connector. Initially, each node is in unmarkedstate and enters into dominatee or dominator statelater. The connector state can only be entered fromthe dominatee state. As mentioned above, this Zone

algorithm can be divided into three phases: (a) zonepartition, (b) dominating tree construction in eachzone and (c) adjustment at the zone borders.

4.2.1. Zone partition

In the first phase, we partition the network intodifferent zones and each zone is supposed to havea unique zone ID. Each node knows its own zoneID that indicates to which zone it belongs (detailsabout how nodes obtain their zone IDs will be dis-cussed later). First, a node u with the highest rankamong its one-hop neighbors becomes a dominatorand broadcasts a DOMINATOR message to all itsneighbors. Note that such node does exist in thebeginning. After receiving a DOMINATOR mes-sage, a node, say v, becomes a dominatee if its cur-rent state is unmarked. If it is the first time that v

receives a DOMINATOR message, v will broadcasta DOMINATEE message to all its neighbors. Thesame procedure is repeated among the remainingnodes, until each node becomes either a dominatoror a dominatee. In a word, a node u becomes a dom-inator if and only if

(1) node u has the highest rank among all its one-hop neighbors or

(2) node u has neighbors with higher ranks thanitself, but these neighbors already becomedominatees.

We call a dominator that satisfies condition (1) asseed dominator and a dominator satisfying condition(2) as non-seed dominator. Obviously, the conditionthat a node becomes a non-seed dominator is indi-rectly affected by some seed dominator.

To partition the network, the node ID of a seeddominator automatically becomes the ID of the cor-responding zone centered at this seed dominator.We add an item, Zone ID, into the DOMINATORmessage to indicate the zone that the dominatorbelongs to. When an unmarked node receives thefirst DOMINATOR message, it becomes a domina-tee of the zone indicated in this message. Eachdominatee also inserts its zone ID into the DOMIN-ATEE message that it broadcasts to all itsneighbors. From its neighboring dominatees, everynon-seed dominator knows to which zone itbelongs. If neighboring dominatees have differentzone IDs, the non-seed dominator can arbitrarilyselect one zone to join. The nodes with the samezone ID form a zone eventually and the networkis divided into different zones through a sweep of

Fig. 1. Border nodes in the zone borders.


the network spreading outwards from the seed dom-inators. Moreover, at this time, all the dominatorsalso form a MIS of the topology graph.

4.2.2. Dominating tree formation

In the second phase, a tree can be constructedrooted at the seed dominator of each zone. Givena tree T, the level of its root is 0 and the level of anode is the hop count of the shortest path in T

between itself and the root of T. We add anotheritem, Level, into the DOMINATOR message toindicate the level of a dominator. The level of theseed dominator (i.e., the root of dominating tree)of each zone is assigned to be 0. When an unmarkednode, say node v, receives the first DOMINATORmessage, it can get its own level by increasing thelevel in this message by 1. Then v inserts its obtainedlevel into the DOMINATEE message and broad-casts it to all its neighbors. Note that, each non-seeddominator has at least one two-hop away neighbor-ing dominator (who dominates its neighbor withhigher rank than itself) in the same zone if there isany other dominator existing. Thus, from its neigh-boring dominatees, every non-seed dominator canalso know its own level by increasing the minimumlevel of its known neighboring dominatees (in thesame zone) by 1.

To connect these two-hop away dominators andform a dominating tree in each zone, every domina-tee broadcasts a ONE-HOP-DOMINATOR mes-sage which contains the node IDs, zone IDs andlevels of all its one-hop away dominators. Afterthat, every node can know its two-hop away neigh-boring dominators. Then each non-seed dominatoronly needs to connect to one of its two-hop awaydominators with the same zone ID and the mini-mum level, by notifying the related dominatee tobecome a one-hop connector using the ONE-HOP-CONNECTOR message. If there are multiplepaths between these two dominators, the pathincluding the dominatee with the highest rank ischosen to break the tie. We will use an example toillustrate this procedure later.

4.2.3. Adjustment along the zone borders

Combining the dominating tree in each zone maynot necessarily generate a CDS for the whole graph.To form the final CDS, we need to choose somedominatees to become connectors at the zone bor-ders. For this purpose, we need to know whichnodes are on the zone borders. A border dominatee

is a dominatee that has a neighbor with different

zone ID. For example, in Fig. 1, nodes p, q, x andz are all border dominatees. In fact, each dominateecan know whether it is on the zone border after thezone partition. Similarly, a node is a border domina-tor if it has a two or three-hop away neighboringdominator with different zone ID. To make thedominators know their three-hop away neighboringdominators, border dominatees need to broadcast aTWO-HOP-DOMINATOR message that includesthe node IDs and zone IDs of all their knowntwo-hop away neighboring dominators. Then, theborder dominator can select some neighbors tobecome connectors from its own local view.

For clarity, we still use Fig. 1 to illustrate this pro-cess. Take nodes u and v as examples. From node u’slocal view, to connect zones A1 and A2, A1 and A3,nodes x and z should be selected as connectors.From node v’s local view, to connect A2 and A3, bothnodes p and q should be selected as connectors. Toavoid collision and reduce the message overhead,in a pair of two or three hops away neighboringdominators, only the dominator with lower ID needsto adjust the zone border. Thus, border dominator w

will take no action. Finally, dominator u will notifynodes x and z to become one-hop connectors usingBORDER-ONE-HOP messages and dominator v

will notify q to become a two-hop connector usinga BORDER-TWO-HOP message. After receivingthe BORDER-TWO-HOP message from v, q willnotify p to become a two-hop connector using aBORDER-TWO-HOP message.

Note that, all the decisions for the adjustmentsalong the zone borders are made locally. If everyborder dominator also sends its own local view tothe seed dominator in its own zone, after collectingthe complete information about its neighboringzones, each seed dominator can use this quasi-glo-bal information to construct only one path to


connect to a given neighboring zone. Although thisapproach can reduce the number of nodes that areselected as connectors at the zone borders, and thusreduce the size of the final CDS, it will also signifi-cantly increase the message complexity because ofthe quasi-global information collection. Moreover,if every zone knows some global information aboutthe underlying topology graph which is messageexpensive in a distributed environment, even a span-ning tree among these zones can be build to furtherreduce the final CDS size.

4.2.4. Example

To illustrate the whole algorithm more clearly,Fig. 2 gives an example of CDS formation usingMax Degree algorithm. In this figure, the IDs ofnodes are labeled beside the nodes. Black nodes rep-resent the dominators, black nodes with outer circlerepresent the seed dominators and gray nodes repre-sent the connectors. A possible execution scenario isshown in Fig. 2b–d and explained below:

1. Initially all nodes are unmarked (Fig. 2a).2. Nodes 7 and 14 declare themselves as domina-

tors, since they have the highest ranks among

(a) Initial topology

(c) More dominators selected

Fig. 2. CDS construction by Max Degree algorithm. (a) Initial topolog(d) final CDS constructed.

their unmarked one-hop neighbors. They are alsoseed dominators at level 0. After receiving aDOMINATOR message, nodes 4, 5, 8, 15, 16,19, 20, 21 and 23 declare themselves as domina-tees at level 1 and broadcast DOMINATEE mes-sages (Fig. 2b).

3. After receiving some DOMINATEE messagesfrom their neighbors, nodes 1, 6, 9, 10, 11 and18 declare themselves as dominators at level 2.The reason is that all their neighbors with higherranks became dominatees, thus their ranksbecome the highest among their unmarked neigh-bors. At this time, all the dominators form a MISand these nodes are divided into two zones, sep-arated by the dash line (Fig. 2c).

4. After each dominatee broadcasts ONE-HOP-DOMINATOR message, every dominatorknows its two-hop away neighboring domina-tors. Nodes 4, 8, 16 and 21 are selected as theconnectors by dominators 1, 6, 9, 10, 11 and 18at level 2 to connect seed dominators 7 and 14at level 0. Nodes 2, 6, 10, 17 and 20 know thatthey are at the zone borders after receiving theDOMINATOR, DOMINATEE or ONE-HOP-DOMINATOR messages. Node 14 knows it is

(b) Seed Dominators selected

(d) Final CDS constructed

y, (b) seed dominators selected, (c) more dominators selected, and


at the zone border after receiving a TWO-HOP-DOMINATOR message from node 20. To adjustthe zone borders, node 17 is selected as a connec-tor by dominator 6 to connect dominator 10.Obviously, all the black and gray nodes form aconnected dominating set of the graph and theinduced sub-graph is indicated by the thick blacklines (Fig. 2d).

Note that dominator pairs {1,11} and {1,18},which are all at level 2 and separated by threehops, need not to connect to each other. The rea-son is that to guarantee the connectivity they onlyneed to connect the seed dominator 7 at level 0.Border dominator 10 needs not to connect borderdominator 14 in another zone. Because dominator10 knows that it has connected with dominator 6 inthe same zone as dominator 14. From the aboveexample, we can see that the benefit of using theZone and Level concepts is that dominators canselectively connect to their two or three hops away

Fig. 3. Connected dominating sets constructed by different algorithmsRule 1 and 2, (c) Rule k, (d) AWF, (e) Min ID, and (f) Max Degree.

neighboring dominators, and thus reduce the sizeof the final CDS.

Fig. 3 shows a comparison of Rule 1 and 2, Rulek, AWF, Min ID and Max Degree through a samplenetwork with 140 nodes. The original topology ofthe network is depicted in Fig. 3a. Fig. 3b–f showthe CDS generated by Rule 1 and 2 (66), Rule k

(60), AWF (59), Min ID (50) and Max Degree(43), respectively. The number in the above paren-theses depicts the size of the CDS constructed bythese five algorithms. In these five figures, only nodesin the CDS and the induced graph by the CDS areshown. We can see that the CDS constructed bythe Zone (Max Degree, followed by Min ID) algo-rithm contains the least number of nodes.

5. Performance analysis

In this section, we analyze the time and messagecomplexity, show the correctness, and give theapproximation ratio of this Zone algorithm.

on the same sample network. (a) Original connected network, (b)


Theorem 1. Both message and time complexity of this

Zone algorithm are O(n).

Proof. Since each node sends out a constant num-ber of messages, the total number of messages isO(n). The time complexity of this algorithm isbounded by MIS construction which has theworst-case time complexity O(n). The worst caseoccurs when all nodes are distributed in a line andin either ascending or descending order of theirranks. Both the second and the third phases can ter-minate within two steps, broadcasting ONE-HOP-DOMINATOR or TWO-HOP-DOMINATORmessages and selecting one-hop connectors or two-hop connectors. Thus, both the message and timecomplexity of this algorithm are O(n). h

Our proposed algorithm has relatively slowerconvergence time than purely localized algorithmswhich take only constant number of steps to termi-nate. However, the disadvantage of purely localizedalgorithms is that they usually cannot guaranteeconstant approximation ratio [9]. Generally, theconvergence time of the Zone algorithm is deter-mined by the largest distance, in terms of hop count,between a border dominatee and its seed dominator(in the same area). This distance is usually muchsmaller than n, the number of nodes in the topologygraph.

Theorem 2. The dominators and connectors selected

by this Zone algorithm construct a CDS.

Proof. In this Zone algorithm, each dominator hasat least one two-hop away neighboring dominatorin the zone it belongs to, if there is any other dom-inator existing in the same zone. Since each non-seed dominator connects to one two-hop awayneighboring dominator with the same zone ID andthe minimum level, we can guarantee the connectiv-ity inside these zones. We also connect each pair ofadjacent zones by at least one path, thus, this theo-rem is proved. h

Theorem 3. Let G be a unit disk graph and OPT be

the size of a minimum CDS for G, then the size of

CDS constructed by this Zone algorithm is within a

constant approximation ratio of OPT.

Proof. From Preliminary 1, each dominator has atmost 42 three-hop away neighboring dominatorsand each pair of at most three-hop away neighbor-ing dominators introduces at most two nodes to the

final CDS. Therefore, the size of the final CDS is42 � 2/2 + 1 times as large as the size of the MISwhich consists of all the dominators. Also fromPreliminary 2, we know that the size of a MIS isat most 3.8 � OPT + 1. Therefore, the number ofnodes in the CDS is at most (42 � 2/2+1) � (3.8 �OPT+1) = 163.4 � OPT + 43. h

6. Discussions

6.1. Approximation ratio and network model

Note that, Theorem 3 only gives a worst-caseupper bound of the approximation ratio for the pro-posed Zone algorithm. The worse case happenswhen there are no two-hop away neighboring dom-inators, i.e., every zone has only one seed domina-tor. Moreover, these dominators in different zonesare all separated by exactly three hops. Therefore,to connect these zones, each pair of three-hop awayneighboring seed dominators will add two morenodes into the final CDS.

However, if each zone has more than one domi-nators, after each non-seed dominator has con-nected to one of its two-hop away neighboringdominators in the same zone through a one-hopconnector, the dominators and connectors in thesame zone form a dominating tree of the inducedsub-graph of each zone, G0. Let OPT0 be the sizeof a minimum CDS for G0. Through similar analysisof Theorem 3, the size of a dominating tree for azone is at most 2 � 3.8 � OPT0 + 1 = 7.6 �OPT0 + 1. In our previous algorithm [5], the sizeof CDS in each zone is at most 44 � OPT0 + 12.Therefore, this new algorithm is more powerful thanthe previous one. Moreover, since we just add onemore item, Level, into DOMINATOR andDOMINATEE messages, the total number of con-sumed messages is not increased.

As mentioned above, the corresponding topologygraph is assumed to be a UDG and the achievedapproximation ratio is based on this assumption.However, the Zone algorithm is not limited onlyto UDG. It also works for other network modelswhere different nodes have different transmissionranges, although the approximation ratio may nothold for the non-UDG network models. Note that,we do assume that there is no unidirectional link inthe network which is also a common assumption formost of the existing literature for CDS constructionalgorithms.


6.2. Implementation issues

We assume each node knows the node ID anddegree of all its one-hop neighbors. This can beachieved through requiring each node to broadcastits node ID initially, using a HELLO message. Aftereach node knows all its one-hop neighbors, it canbroadcast its degree. Certainly, nodes are requiredto broadcast this information periodically in amobile scenario. In this Zone algorithm, after eachdominatee has received either a DOMINATOR ora DOMINATEE message from all its one-hopneighbors, it will broadcast a ONE-HOP-DOMI-NATOR message. To guarantee each node cancheck whether it has received some given messagefrom all its one-hop neighbors, each node will alsomaintain a data structure to record the receptionsof DOMINATOR, DOMINATEE, ONE-HOP-DOMINATOR and TWO-HOP-DOMINATORmessages. In these messages (e.g., HELLO, DOMI-NATOR and DOMINATEE), the length of nodeID and zone ID is 2 bytes and the length of Levelis 1 byte. Table 1 shows the fields for a TWO-HOP-DOMINATOR message (this message hasthe most complicated structure). In this table,DOM stands for DOMINATOR. Suppose this mes-sage is generated by node u that has k two-hop awaydominators. For the limited space, we will not pres-ent the detailed structures of other control messageshere.

6.3. Backbone maintenance

Ad hoc networks often need to face dynamictopology changes. Therefore, backbone mainte-

Table 1Detailed structure of TWO-HOP-DOMINATOR message

Field name Description

Message Type The type of message, e.g., HELLO orDOM (1 bytes)

Node ID Node ID (2 bytes), u

Number of Two-hopDOM

The number of two-hop awaydominators (2 bytes)

DOM-0 Node ID Node ID of two-hop away dominator 0DOM-0 Zone ID Zone ID of two-hop away dominator 0Connector Node ID for

DOM-0Node ID of neighboring connector fordominator 0

� � � � � �DOM-k Node ID Node ID of two-hop away dominator k

DOM-k Zone ID Zone ID of two-hop away dominator k

Connector Node ID forDOM-k

Node ID of neighboring connector fordominator k

nance is also an important issue. For the CDS main-tenance, dynamic topology change can be handledby the methods proposed in [8,26]. Generally, thereare two kinds of mechanisms to handle the topologychange: periodical reconstruction and on-demandupdate. Each method has its own pros and cons.In the former scheme, the period of time elapsedbefore the reconstruction is critical to the systemperformance. If it is too short, higher communica-tion overhead will be introduced and if it is too long,maybe the old CDS cannot guarantee the domi-nance property or connectivity under the new topol-ogy. On-demand update is efficient in slightlytopology change and will lose its effectiveness whenfacing major topology change. We can use tech-niques similar as [26] to maintain the CDS duringthe network lifetime. Note that, how to preservethe approximation quality in mobile ad hoc net-works is still an open problem [16].

7. Experimental results

We evaluate the performance of Zone algorithmusing two groups of simulation. The first group isconducted using a custom simulator in ideal net-works without packet losses, and thus, retransmis-sions. This simulator focuses on some graphproperties of the CDS, such as the size of CDSand the average hop stretch and the message over-head in the ideal environment. In the second group,the Max Degree version of our Zone algorithm iscompared with several recently proposed algo-rithms, in terms of message overhead, energy con-sumption and protocol duration, in more realisticnetworks using the network simulator ns2. The cus-tom simulator, due to its simplicity, runs much fas-ter than the ns2 simulator which makes it possible tosimulate networks with large number of nodes.

7.1. Simulation in ideal networks

We first show the effectiveness of this Zone algo-rithm through extensive simulation study in idealnetworks without channel contention, packet colli-sion and retransmission. Especially, we comparethe performance of our two Zone algorithms, MinID and Max Degree, with other representative algo-rithms, Rule 1 and 2 [8], Rule k [9] and AWF [26].We use a custom event-driven simulator written inC language to simulate wireless ad hoc networks.To measure the performance of the algorithms, fourparameters are set up for the evaluations. They are


the size of CDS, the average node degree in theinduced sub-graph of the constructed CDS, theaverage hop stretch and the average message over-head (in bytes).

7.1.1. Size of CDS

First parameter is the size of CDS as we aim dur-ing the construction. The constructed virtual back-bone can be used for routing protocol design inwireless ad hoc networks. In this kind of routingalgorithm, only the nodes in CDS will relay datafor others and all the dominatees just send theirdata to their dominators. Therefore, we prefer smal-ler values of this parameter and perform the com-parison under two scenarios. In the first scenario,a given number of nodes (ranging from 60 to 200with increment step of 20 and from 200 to 1000 withincrement step of 100, respectively) were randomlyand uniformly distributed in a square simulationarea of size 100 by 100 units. Each node has a fixedtransmission range r (r = 15, 30 units, respectively).

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Fig. 4. The number of nodes in CDS when r is 15 units. (a) Number o(n 2 [200,1000]).

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Fig. 5. The number of nodes in CDS when r is 30 units. (a) Number o(n 2 [200,1000]).

All the simulation results presented in this sectionwere obtained by running these algorithms on 300connected graphs. This allows us to test these algo-rithms with increasing density of network fromn = 60 and r = 15 (sparse network) to n = 1000and r = 30 (very dense network).

Fig. 4a and b shows the simulation results whenthe node’s transmission range is 15 units and thenumber of nodes in the network ranges from 60 to200 and from 200 to 1000, respectively. From thefigures, we notice that the number of nodes in theCDS increases when more nodes join the network.Among these algorithms, the Zone algorithm out-performs the others. Fig. 5a and b shows the resultswhen the node’s transmission range is set as 30 unitsand number of nodes in the networks ranges from60 to 200 and from 200 to 1000, respectively. Com-paring Fig. 4a and b with Fig. 5a and b, we find thatincreasing the node’s transmission range canincrease the coverage area of each node and, there-fore, increase the density of the network, which

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leads to a smaller size of the CDS. Among thesealgorithms, Max Degree outperforms the otherfour, followed by Min ID. When the number ofnodes in the network reaches 1000, the number ofnodes in the CDS constructed by Max Degree isonly about 34% of that constructed by Rule 1 and 2.

In the second scenario, a fixed number of nodes(n = 200 and 1000, respectively) were randomly dis-tributed in the same simulation area. The networkdensity is determined by the node’s transmissionrange r. For each fixed number of nodes, we run dif-ferent experiments where the value of r changesfrom 5 to 60 units with increment step of 5 units.Fig. 6a and b shows the performance observations.When the transmission range r increases, the num-ber of nodes in the CDS decreases because theincrease of r results in the higher density of the net-work. Thus, fewer nodes are needed to cover thesimulation area. We also found that these five algo-rithms perform closely in the extremely dense net-work. For example, when n = 1000 and r = 45, thenumber of nodes in the CDS is only about 10. In

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Fig. 6. The number of nodes in CDS when r is from 5 units to 45 units.

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Fig. 7. The average node degree in CDS when n is from 100 to 1000. (a)units.

this kind of network, a few nodes (less than 1%)are enough to dominate all the other nodes.

7.1.2. Node degree in the CDS

Second parameter is the average node degree inthe induced sub-graph of the constructed CDS, G0.The interference aspect is often maintained bydevelopers of topology control algorithms in adhoc networks and solved by sparseness or low nodedegree of the resulting topology graph. Low nodedegree may cause less interference, thus, smalleraverage node degree is preferred.

Fig. 7a and b plots the average node degree in theinduced sub-graph by the CDS as a function of thenumber of nodes in the network (ranging from 100to 1000 with increment step of 100) when the trans-mission range r is 15 and 30 units, respectively. Allthese curves have a rising trend as the number ofnodes in the network increases. When the transmis-sion range r is 15 units, the average node degree inthe CDS constructed by Rule k is the smallest, fol-lowed by Min ID and Max Degree. But when the

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transmission range r is 30 units, the average nodedegree in the CDS constructed by Min ID becomesthe smallest.

7.1.3. Hop stretch

Third parameter is the average hop stretch and itis defined as follows: Let DG(p, q) be the hop countof the shortest path between p and q in G, DG0(p,q)be the hop count of the shortest path between p andq in the sub-graph G0 induced by the CDS, i.e., allthe intermediate nodes along the path belong tothe CDS. Then the hop stretch is expressed asHSðp; qÞ ¼ DGðp; qÞ=DG0 ðp; qÞ. The sub-graph G0

dilates the network so that the hop counts betweendifferent pairs of nodes in G0 may be larger than thecorresponding hop counts in the original network.We do not wish the dilation to be too large, so smal-ler average hop stretch is preferred.

Fig. 8a and b plots the average hop stretch of allpairs of nodes as a function of the number of nodesin the network (ranging from 100 to 1000 with incre-ment step of 100) when the transmission range r is15 and 30 units, respectively. There is a trade offbetween the size of CDS and the average hopstretch. The smaller size of the CDS that we con-struct, the larger the hop stretch will be. If we selectall the nodes in the original network into the CDS,the hop stretch will be 1, which is the smallest value.Among these algorithms, AWF performs best interm of hop stretch because it constructs the largestCDS. Comparing Figs. 4 and 5 with Fig. 8, we haveobserved that the CDS constructed by Max Degreehas both smaller size and hop stretch than the one

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Fig. 8. The average hop stretch when n is from 100 to 1000. (a) Transm

constructed by Rule k. In this simulation scenario,all the average hop stretches are less than 2.

7.1.4. Message overhead

One important aspect to be considered in theCDS formation algorithm is the quality of the infor-mation needed by the algorithm. As we mentionedabove, there is a trade off between information qual-ity and message consumption: the more accurate isthe required information, the better performanceof the algorithm can be achieved. If every nodehas the global topology information, the algorithmcan obtain the optimal result. However, the priceto be paid, in terms of messages to be exchanged,to obtain high quality information must be carefullyconsidered. In this sense, we prefer small messageoverhead.

Fig. 9a and b relates the message overhead to thenumber of nodes in the network (ranging from 100to 1000 with increment step of 100) when the trans-mission range r is 15 and 30 units, respectively. Inboth cases, the y-axis denotes the average numberof bytes of messages transmitted by the nodes. SinceRule k has the same message complexity with Rule 1and 2, we only plot the curve for Rule 1 and 2.Among these algorithms, Rule 1 and 2 consumesthe most number of bytes of messages. The reasonis that, in this algorithm, each node needs to knowits two-hop neighbor information which is messageexpensive in large scale dense ad hoc networks.When n = 1000 and r = 15, this kind of informationexchange accounts for about 96% of the total

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message overhead and demands significant time andenergy consumption.

AWF, Min ID and Max Degree show similarperformance because in these algorithms each nodeonly requires the knowledge of its one-hop neigh-bors and a constant number of two-hop and three-hop neighbors. Compared with AWF, Min IDand Max Degree introduce slightly more messageoverhead because two extra items, Zone ID andLevel, are associated with each node in the messageexchange. When n = 1000 and r = 30, the numberof bytes of messages consumed by Rule 1 and 2 is439808, nearly 12 times more than that of the otherthree. To show the difference of AWF, Min ID andMax Degree clearly, in Fig. 9b we omit the curve ofRule 1 and 2.

7.2. Simulation in realistic networks

We also implement the Zone algorithm in ns2.29to study its performance in more realistic networks.This group of simulation is based on the clusteringframework implemented in [38]. We mainly com-pare the performance of the Max Degree versionof Zone algorithm, named Zone hereafter, with avariant of the recently proposed DCA-S(4) proto-cols, where cluster-heads (i.e., dominators) areselected based on the node degree [38]. This variantis called DCA-D-S(4) protocol. These two protocolsbelong to the same class according to the classifica-tion in [38], i.e., protocols with a lower degree oflocalization. They all first compute a dominatingset through MIS construction, and then make itconnected by adding more nodes. As the baseline,we also include the localized algorithm proposedin [11], named Stojmenovic hereafter, into the com-

parison, which is proved to be very efficient andexhibit a high degree of localization [38].

7.2.1. Simulation setupThe simulation setup of ns2 is very similar with

that in [38] where parameters have been modifiedto take sensor nodes characteristics into account.The major differences are that we disable the RTS/CTS exchange at the MAC layer and reduce theMAC layer packet header overhead. In the simula-tion, a number of static wireless nodes with a max-imum transmission radius of 30 meters arerandomly and uniformly deployed in a square areaof size 200 � 200 m. The initial energy of every nodeis 1 J. The power consumptions in transmitting,receiving, and asleep modes are 24, 14.4, and0.015 mW, respectively. The number of nodesranges from 50 to 300 with increment step of 50,which allows us to evaluate these protocols onincreasingly dense networks with average nodedegree ranging from 3.51 to 18.59. The simulationresults are average values based on 300 connectedgraphs.

These algorithms are implemented in a backbonelayer placed between the MAC and link layers inns2. In our implementation, we use broadcast trans-mission only for DOMINATOR, DOMINATEEand ONE-HOP-DOMINAOTR messages and uni-cast transmission for all other messages. Thesechoices can reduce the possible packet collisionsand save node energy. To guarantee packet delivery,timers are launched for all the intended receivers ofbroadcast messages and the destination of unicastmessages, and a backbone layer CONFIRM mes-sage is utilized for acknowledgement. After timeoutoccurs, the related message will be retransmitted


using unicast transmission. Note that, the domina-tees only need to send ONE-HOP-DOMINATORmessages to its neighboring dominators andtherefore these messages will be ignored by otherdominatees. We also slightly modify the TWO-HOP-DOMINATOR message to optimize ourimplementation. The new TWO-HOP-DOMINATORmessage only includes the ID of neighboringzones and is only sent to the dominator of a given

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Fig. 10. The simulation results for realistic networks. (a) Average peroverhead at backbone layer, (d) message overhead at MAC layer, (e) e

dominatee. This modification is based on the factthat a border dominator only needs to make itselfconnected to its neighboring zones and does notcare about to which two or three-hop awayneighboring dominators it connects.

To compare the performance of these protocols,the metrics considered are the size of CDS (i.e.,backbone), backbone robustness, the messageoverhead per node (in bytes) at backbone and

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centage of nodes in CDS, (b) backbone robustness, (c) messagenergy consumption (J), and (f) protocol duration (s).


MAC layers, average node energy consumption andprotocol duration. Backbone robustness is definedas the number of backbone nodes whose disappear-ance will make the backbone disconnected or someof the dominatee nodes uncovered [38]. The differ-ence of message overheads at backbone and MAClayers is that the MAC layer overhead includes boththe overhead at the backbone layer and the over-head for the ACK and retransmission for unicastpackets (at the MAC layer).

7.2.2. Performance evaluation in ns2The simulation results are presented in Fig. 10.

The average percentage of nodes in the CDS isshown in Fig. 10a. As we can see, Stojmenovic onlyoutperforms Zone and DCA-D-S(4) for small net-works. The sparsification techniques of Zone andDCA-D-S(4) are proved to be very effective whenthe network becomes dense. However, due to its rel-ative large size, the robustness of the CDS generatedby Stojmenovic is higher than those of Zone andDCA-D-S(4) which is depicted in Fig. 10b.

Fig. 10c and d shows the per node message over-head (in bytes) at backbone layer and MAC layer,respectively. These figures clearly show that Zonealgorithm requires less bytes than DCA-D-S(4).The reason is that the technique of find cycles evenwith small size can consume more messages. Zonerequires more backbone layer message overheadthan Stojmenovic since it needs to confirm most ofthe (broadcast and unicast) messages at this layer.Due to its simplicity and localization, Stojmenoviccan save lots of messages by avoiding the CON-FIRM messages. However, as mentioned above,the message size of Stojmenovic protocol isunbounded. Since every node needs to exchangeits neighbor list with its own neighbor, this messagesize will become very large in dense networks. More-over, in practice, the loss rate of large packets willbe much higher than that of small packets [39]. Notethat, the sizes of all the messages used by Zone algo-rithm are bounded by some constants. ComparingFig. 10c with Fig. 10d, we find that the differenceof MAC layer message overhead for Zone and Sto-jmenovic protocols is more significant than that ofbackbone layer. The reason is that Zone heavilyuses unicast packets which cause lots of ACKsand retransmissions at MAC layer. This is also con-firmed by the per node average energy consumptionshown in Fig. 10e.

Fig. 10f shows the protocol duration for thesethree protocols which is measured as the time used

by them to finish backbone formation. From thisfigure, we find that Zone and Stojmenovic take lesstime than DCA-D-S(4). For small networks, Zoneeven requires less time than Stojmenovic. But Sto-jmenovic outperforms Zone for dense networks,again due to its high degree of localization. Notethat, for Zone and DCA-D-S(4), most of the timeis used to select the dominators, through MIS for-mation, which shows a lower degree of localization.

8. Conclusion and future work

In this paper, we propose an efficient zone-baseddistributed algorithm for connected dominating setformation in ad hoc networks. Both time and mes-sage complexity of this algorithm are O(n). In thisZone algorithm, we partition the network into dif-ferent zones, construct a dominating tree for eachzone and adjust the zone borders by inserting addi-tional nodes into the final CDS. Our comprehensivesimulation study on both sparse and dense networksverifies the effectiveness of this algorithm. From thesimulation study using a custom simulator, we haveobserved that this Zone algorithm always outper-forms the other three existing well-known algo-rithms regardless of the size and density of thenetworks, in terms of the size of CDS. Moreover,we also compare the performance of the Zone algo-rithm with some recently proposed CDS protocolsin the ns2 simulator.

Currently, we are implementing the proposedalgorithm on the Emulab fixed wireless test-bed[40] to evaluate its performance in practice. As men-tioned above, our future work includes investigatingthe pros and cons of using other selection criterioninstead of node ID and node degree. Moreover,constructing virtual backbones in multihop wirelessnetworks composed of selfish nodes [41] will beanother interesting future work.

Acknowledgements

Most of the work was done when the author wasa graduate student at City University of Hong Kongand was supported by Graduate Scholarship fromthe University Grants Committee of Hong Kong,Research Tuition Scholarship and Research Activi-ties Funds from City University of Hong Kong.The author is grateful to Dr. Weijia Jia andDr. Peng-Jun Wan for discussions and suggestions.The author would also like to thank Dr. StefanoBasagni and the anonymous reviewers for valuable


comments which improve the quality of the paper ina number of different ways.

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Bo Han is a Ph.D. student in Departmentof Computer Science at University ofMaryland, College Park. He received hisB.E. in Computer Science and Technol-ogy from Tsinghua University in 2000and M.Phil. in Computer Science fromCity University of Hong Kong in 2006.In 2003, he worked as a research assis-tant in City University of Hong Kong.He was a visiting scholar at Departmentof Computer Science, University of

Maryland, College Park from December 2005 to July 2006. Heworked as research intern at AT&T Labs Research for summer
2007. His research interests include Wireless Communication,Distributed Algorithms and Internet Computing.
http://www.pp.bme.hu/ci/index.html

zone-based virtual backbone formation in wireless ad hoc networks

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