Broadcasting and topology control in wireless ad hoc networks

Xiang-Yang Li ∗ Ivan Stojmenovic †

September 25, 2003

Abstract

Network wide broadcasting in Mobile Ad Hoc Networks(MANET) provides important control and route establish-ment functionality for a number of unicast and multicastprotocols. We present an overview of the recent progressof broadcasting and multicasting in wireless ad hoc net-works. We discuss two energy models that could be usedfor broadcast: one is non-adjustable power and one is ad-justable power. If the power consumed at each node isnot adjustable, minimizing the total power used by a re-liable broadcast tree is equivalent to the minimum con-nected dominating set problem (MCDS), i.e., minimize thenumber of nodes that relay the message, since all relayingnodes of a reliable broadcast form a connected dominat-ing set (CDS). If the power consumed at each node is ad-justable, we assume that the power consumed by a relaynode u is ‖uv‖β , where real number β ∈ [2, 5] depends ontransmission environment and v is the farthest neighborof u in the broadcast tree. For both models, we reviewedseveral centralized methods that compute broadcast treesconsuming the energy within a constant factor of the op-timum if the original communication graph is unit diskgraph. Since centralized methods are expensive to imple-ment, We further reviewed several localized methods thatcan approximate the minimum energy broadcast tree fornon-adjustable power case. For adjustable power case, nolocalized methods can approximate the minimum energybroadcast tree and thus review several currently best pos-sible heuristics. Several local improvement methods andactivity scheduling of nodes (active, idle, sleep) are alsodiscussed.

Keywords: Wireless networks, network optimiza-tion, power consumption, broadcasting, multicasting.

∗Department of Computer Science, Illinois Institute of Tech-nology, 10 W. 31st Street, Chicago, IL 60616, USA. Emailxli@cs.iit.edu, wangyu1@iit.edu.

†SITE, University of Ottawa, Ottawa, Ontario K1N 6N5,Canada. Email: ivan@site.uottawa.ca

1 Introduction

Wireless Ad Hoc Networks: Due to its poten-tial applications in various situations such as battle-field, emergency relief, environment monitoring, andso on, wireless ad hoc networks [1, 2, 3, 4] have re-cently emerged as a premier research topic. Wirelessnetworks consist of a set of wireless nodes which arespread over a geographical area. These nodes are ableto perform processing as well as capable of commu-nicating with each other by means of a wireless adhoc network. With coordination among these wire-less nodes, the network together will achieve a largertask both in urban environments and in inhospitableterrain. For example, the sheer numbers of wirelesssensors and the expected dynamics in these environ-ments present unique challenges in the design of wire-less sensor networks. Many excellent researches havebeen conducted to study problems in this new field[1, 2, 5, 3, 6, 4].

In this chapter, we consider a wireless ad hoc net-work consisting of a set V of n wireless nodes dis-tributed in a two-dimensional plane. Each wirelessnode has an omni-directional antenna. This is attrac-tive because a single transmission of a node can bereceived by many nodes within its vicinity which, weassume, is a disk centered at the node. We call theradius of this disk the transmission range of this wire-less node. In other words, node v can receive the signalfrom node u if node v is within the transmission rangeof the sender u. Otherwise, two nodes communicatethrough multi-hop wireless links by using intermediatenodes to relay the message. Consequently, each nodein the wireless network also acts as a router, forward-ing data packets for other nodes. By a proper scaling,we assume that all nodes have the maximum trans-mission range equal to one unit. These wireless nodesdefine a unit disk graph UDG(V ) in which there is anedge between two nodes if and only if their Euclideandistance is at most one.

In addition, we assume that each node has a low-power Global Position System (GPS) receiver, whichprovides the position information of the node itself. If

1

GPS is not available, the distance between neighbor-ing nodes can be estimated on the basis of incomingsignal strengths. Relative co-ordinates of neighboringnodes can be obtained by exchanging such informationbetween neighbors [7]. With the position information,we can apply computational geometry techniques tosolve some challenging questions in wireless networks.

Power-Attenuation Model: Energy conservationis a critical issue in wireless network for the node andnetwork life, as the nodes are powered by batteriesonly. Each wireless node typically has a portable setwith transmission and reception processing capabili-ties. To transmit a signal from a node to the othernode, the power consumed by these two nodes con-sists of the following three parts. First, the sourcenode needs to consume some power to prepare the sig-nal. Second, in the most common power-attenuationmodel, the power needed to support a link uv is ‖uv‖β ,where ‖uv‖ is the Euclidean distance between u andv, β is a real constant between 2 and 5 dependent onthe transmission environment. This power consump-tion is typically called path loss. Finally, when a nodereceives the signal, it needs consume some power toreceive, store and then process that signal. For sim-plicity, this overhead cost can be integrated into onecost, which is almost the same for all nodes. Thus, wewill use c to denote such constant overhead. In mostresults surveyed here, it is assumed that c = 0, i.e., thepath loss is the major part of power consumption totransmit signals. The power cost p(e) of a link e = uvis then defined as the power consumed for transmittingsignal from u to node v, i.e., p(uv) = ‖uv‖β .

Broadcasting and Multicasting: Broadcasting isa communication paradigm that allows to send datapackets from a source to multiple receivers. In one-to-all model, transmission by each node can reach allnodes that are within radius distance from it, whilein the one-to-one model, each transmission is directedtoward only one neighbor (using, for instance, direc-tional antennas or separate frequencies for each node).The broadcasting in literature has been studied mainlyfor one-to-all model and we will use that model in thischapter. Broadcasting is also frequently referred to asflooding.

Broadcasting and multicasting in wireless ad hocnetworks are critical mechanisms in various applica-tions such as information diffusion, wireless networks,and also for maintaining consistent global network in-formation. Broadcasting is often necessary in MANETrouting protocols. For example, many unicast rout-

ing protocols such as Dynamic Source Routing (DSR),Ad Hoc On Demand Distance Vector (AODV), ZoneRouting Protocol (ZRP), and Location Aided Rout-ing (LAR) use broadcasting or a derivation of it toestablish routes. Currently, these protocols all relyon a simplistic form of broadcasting called Flooding,in which each node (or all nodes in a localized area)retransmits each received unique packet exactly onetime. The main problems with Flooding are that ittypically causes unproductive and often harmful band-width congestion, as well as inefficient use of node re-sources. Broadcasting is also more efficient than send-ing multiple copies the same packet through unicast.It is highly important to use power-efficient broadcastalgorithms for such networks since wireless devises areoften powered by batteries only.

Recently, a number of research groups have pro-posed more efficient broadcasting techniques [8, 9, 10,11, 12, 13, 14] with various goals such as minimiz-ing the number of retransmissions, minimizing the to-tal power used by all transmitting nodes, minimiz-ing the overall delay of the broadcasting, and so on.Williams and Camp [13] classified the broadcast pro-tocols into four categories: simple (blind) flooding,probability based, area based, and neighbor knowl-edge methods. Wu and Lou [15] classified broadcastingprotocols based on neighbor knowledge information:global, quasi-global, quasi-local, and local. The globalbroadcast protocol, centralized or distributed, is basedon global state information. In quasi-global broadcast-ing, a broadcast protocol is based on partial globalstate information. For example, the approximation al-gorithm in [16] is based on building a global spanningtree (a form of partial global state information) that isconstructed in a sequence of sequential propagations.In quasi-local broadcasting, a distributed broadcastprotocol is based on mainly local state information andoccasionally partial global state information. Clusternetworks are such examples: while clusters can be con-structed locally for most of the time, the chain reac-tion does occur occasionally. In local broadcasting, adistributed broadcast protocol is based on solely localstate information. All protocols that select forwardnodes locally (based on 1-hop or 2-hop neighbor set)belong to this category. It has been recognized thatscalability in wireless networks cannot be achieved byrelying on solutions where each node requires globalknowledge about the network. To achieve scalability,the concept of localized algorithms was proposed, asdistributed algorithms where simple local node behav-ior, based on local knowledge, achieves a desired globalobjective.

2

In this chapter, we categorize previously proposedbroadcasting protocols into several families: central-ized methods, distributed methods, localized meth-ods. Centralized methods calculate a tree used forbroadcasting with various optimization objectives ofthe tree. In localized methods, each node has to main-tain the the state of its local neighbors (within someconstant hops). After receiving a packets that neededto be relayed, the node decides whether to relay thepacket only based on its local neighborhood informa-tion. Majority of the protocols are in this family. Indistributed methods, a node may need some informa-tion more than a constant hop away to decide whetherto relay the message. For example, broadcasting basedon MST constructed in a distributed manner is a dis-tributed method, but not localized method since wecannot construct MST in a localized manner.

Distributed or Localized Algorithms? Dis-tributed algorithms and architectures have been com-monly used terms for a long time in computer sci-ence. Unfortunately none of the already proposed ap-proaches are applicable to wireless ad-hoc networks.In order to address the needs of distributed computingin wireless ad hoc networks, one has to address howkey goals, such as power minimization, low latency,security and privacy, are affected by the algorithmsused. Some common denominators are almost alwayspresent, such as high relative cost of communicationto computation in wireless networks.

Due to the limited capability of processing power,storage, and energy supply, many conventional algo-rithms are too complicated to be implemented in wire-less ad hoc networks. Thus, the wireless ad hoc net-works require efficient distributed algorithms with lowcomputation complexity and low communication com-plexity. More importantly, we expect the distributedalgorithms for wireless ad hoc networks to be localized:each node running the algorithm only uses the infor-mation of nodes within a constant number of hops.However, localized algorithms are difficult to design orimpossible sometimes. For example, we cannot con-struct the minimum spanning tree (MST) locally. Re-cently, Li et al. successfully solved several challengingquestions [17, 18, 19, 20, 21, 24, 25] by giving effi-cient localized algorithms. In all these algorithms, weproved that the total communication cost of all nodesconstructing the structure together is O(n log n) bits.

MAC Specification: Collision avoidance is inher-ently difficult in MANETs; one often cited difficultyis overcoming the hidden node problem, where a node

cannot decide whether some of its neighbors are busyreceiving transmissions from an uncommon neighbor.The 802.11 MAC follows a Carrier Sense Multiple Ac-cess/Collision Avoidance (CSMA/CA) scheme. Forunicasting, it utilizes a Request To Send (RTS) / ClearTo Send (CTS) / Data / Acknowledgment (ACK)procedure to account for the hidden node problem.However, the RTS/CTS/Data/ACK procedure is toocumbersome to implement for broadcast packets as itwould be difficult to coordinate and bandwidth ex-pensive: a relay node has to perform RTS/CTS indi-vidually with all its neighbors that should receive thepackets. Thus, the only requirement made for broad-casting nodes is that they assess a clear channel beforebroadcasting. Unfortunately, clear channel assessmentdoes not prevent collisions from hidden nodes. Addi-tionally, no resource is provided for collision when twoneighbors assess a clear channel and transmit simulta-neously. Ramifications of this environment are subtlebut significant. Unless specific means are implementedat the network layer, a node has no way of knowingwhether a packet was successfully reached by its neigh-bors. In congested networks, a significant amount ofcollisions occur leading to many dropped packets. Themost effective broadcasting protocols try to limit theprobability of collisions by limiting the number of re-broadcasts in the network. Thus, it is often impera-tive the underlying structure for broadcasting is degreebounded and the links are at similar lengths. By us-ing a power adjustment at each node, the collision ofpackets and contention for channel will be alleviated.Notice that, if the underlying structure for broadcast-ing is degree bounded, we can either use RTS/CTSscheme to avoid hidden node problem, or we can re-broadcast the dropped packets (such rebroadcast willbe less since the number of intended receiving neigh-bors is bounded by a small constant).

Reliability: Reliability is the ability of a broadcastprotocol to reach all the nodes in the network. It canbe considered at the network or at the medium accesslayer. We will classify protocols according to their net-work layer performance. That is, assuming that MAClayer is ideal (every message sent by a node reachesall its neighbors), location update protocol providesaccurate desired information to all nodes about theirneighborhood, and the network is connected. Broad-cast protocols can be reliable or unreliable. In a re-liable protocol, every node in the network is reached,while in unreliable broadcast protocols, some nodesmay not receive the message at all.

3

Message Contents: The broadcast schemes mayrequire different neighborhood information, which isreflected in the contents of messages sent by nodeswhen they move, react to topological changes, changeactivity status, or simply send periodically updatemessages. For example, commonly seen hello messagemay contain (all or a subset of) the following infor-mation: its own ID, its position, one bit for dominat-ing set status (informing neighbors whether or not thenode itself is in dominating set), list of 1-hop neigh-bors, its degree. Other content is also possible, such aslist of 1-hop neighbors with their positions, or list of2-hop neighbors, or even global network information.

The broadcast message sent by the source, orretransmitted, may contain broadcast message only.In addition, it may contain a various of informationneeded for proper functioning of broadcast protocol,such as the same type of information already listedfor hello messages, some constant bits of the systemrequirements (such as the maximum broadcast delay),or list of forwarding neighbors of current relaying node,informing them whether or not to retransmit the mes-sage.

Jitter and RAD: Suppose a source node originatesa broadcast packet. Given that radio waves propa-gate at the speed of light, all neighbors will receive thetransmission almost simultaneously. Assuming similarhardware and system loads, the neighbors will pro-cess the packet and rebroadcast at the same time. Toovercome this problem, broadcast protocols jitter thescheduling of broadcast packets from the network layerto the MAC layer by some uniform random amount oftime. This (small) offset allows one neighbor to obtainthe channel first, while other neighbors detect that thechannel is busy (clear channel assessment fails) andthus delay their transmissions to avoid collision. Sincethe node has to backup all received broadcast packetswithin RAD, the RAD cannot be too large also. Onthe other hand, if RAD is small, this node may re-peatively broadcast a same packet, and thus causingthe infinity loop of rebroadcast.

Many of the broadcasting protocols require a nodeto keep track of redundant packets received over ashort time interval in order to determine whether torebroadcast. That time interval, which were termedRandom Assessment Delay (RAD) [13], is randomlychosen from a uniform distribution between 0 andTmax seconds, where Tmax is the highest possible de-lay interval. This delay in transmission accomplishestwo things. First it allows nodes sufficient time toreceive redundant packets and assess whether to re-

broadcast. Second, the randomized scheduling pre-vents the collisions of transmission.

Performance Measurement: The performance ofbroadcast protocols can be measured by variety of met-rics. A commonly used metric is the number of mes-sage retransmissions with respect to the number ofnodes. In case of broadcasting with adjusted transmis-sion power (thus adjusted disk that the message canreach), the total power can be used as performancemetrics. The next important metric is reachability,or the ratio of nodes connected to the source that re-ceived the broadcast message. Time delay or latencyis sometimes used, which is the time needed for thelast node to receive broadcast message initiated at thesource. Note that retransmissions at MAC layer arenormally deferred, to avoid message collisions. Someauthors consider alternative more restricted indicator,whether or not the path from source to any node isalways following a shortest path. This measure maybe important if used as part of routing scheme, sinceroute paths are created during the broadcast process.

Organization The rest of the chapter is organizedas follows. In Section 2, we discuss in detail severalcentralized broadcasting methods. Note that bothminimizing the number of retransmissions and min-imizing the total power used by transmitting nodesare NP-complete problems even for unit disk graphs.We thus discuss in detail several methods that canachieve constant approximation ratio in polynomialtime. Since centralized methods are expensive to im-plement for wireless ad hoc networks, in Section 3, wethen review several protocols that use only localized in-formation. In Section 4, we discuss how to judiciouslyassign operation model to each node thus saving over-all energy consumptions in the network. In Section5, we conclude this chapter with discussion of somepossible future works.

2 Centralized Methods

We assume that two energy models could be used forbroadcast: one is non-adjustable power and one is ad-justable power. If the power consumed at each nodeis not adjustable, minimizing the total power used bya reliable broadcast tree is equivalent to the minimumconnected dominating set problem (MCDS), i.e., min-imize the number of nodes that relay the message,since all relaying nodes of a reliable broadcast forma connected dominating set (CDS). If the power con-sumed at each node is adjustable, we assume that the

4

power consumed by a relay node u is ‖uv‖β , wherereal number β ∈ [2, 5] depends on transmission en-vironment and v is the farthest neighbor of u in thebroadcast tree. In the rest of the section, for these twoenergy models respectively, we reviewed several cen-tralized methods that can build some broadcast treewhose energy consumption is within a constant factorof the optimum if the original communication graph ismodelled by unit disk graph.

2.1 Assumptions

We first study the adjustable power model. Minimum-energy broadcast/multicast routing in a simple ad hocnetworking environment has been addressed by thepioneering work in [26, 27, 28, 29]. To assess thecomplexities one at a time, the nodes in the net-work are assumed to be randomly distributed in atwo-dimensional plane and there is no mobility. Nev-ertheless, as argued in [29], the impact of mobilitycan be incorporated into this static model becausethe transmitting power can be adjusted to accom-modate the new locations of the nodes as necessary.In other words, the capability to adjust the trans-mission power provides considerable “elasticity” tothe topological connectivity, and hence may reducethe need for hand-offs and tracking. In addition, asassumed in [29], there are sufficient bandwidth andtransceiver resources. Under these assumptions, cen-tralized (as opposed to distributed) algorithms werepresented by [29, 30, 31, 32] for minimum-energybroadcast/multicast routing. These centralized algo-rithms, in this simple networking environment, are ex-pected to serve as the basis for further studies on dis-tributed algorithms in a more practical network envi-ronment, with limited bandwidth and transceiver re-sources, as well as the node mobility.

2.2 Based on MST and Variations

Some centralized methods are based on optimiza-tion. The scheme proposed in [33] is built uponan alternate search based paradigm in which theminimum-cost broadcast/multicast tree is constructedby a search process. Two procedures are devisedto check the viability of a solution in the searchspace. Preliminary experimental results show thatthis method renders better solutions than BIP, thoughat a higher computational cost. Liang [30] showedthat the minimum-energy broadcast tree problem isNP-complete, and proposed an approximate algorithmto provide a bounded performance guarantee for theproblem in the general setting. Essentially they reduce

the minimum-energy broadcast tree problem to an op-timization problem on an auxiliary weighted graph andsolve the optimization problem so as to give an ap-proximate solution for the original problem. They alsoproposed another algorithm that yields better perfor-mance under a special case. Das et al. [31] proposed anevolutionary approach using genetic algorithms. Thesame authors also presented in [34] three different in-teger programming models which can be used to findthe solutions to the minimum-energy broadcast/ mul-ticast problem. The major drawback of optimizationbased schemes are, however, that they are centralizedand require the availability of global topological infor-mation.

Some centralized methods are based on greedyheuristics. Three greedy heuristics were proposed in[29] for the minimum-energy broadcast routing prob-lem: MST (minimum spanning tree), SPT (shortest-path tree), and BIP (broadcasting incremental power).The MST heuristic first applies the Prim’s algorithmto obtain a MST, and then orient it as an arborescencerooted at the source node. The SPT heuristic appliesthe Dijkstra’s algorithm to obtain a SPT rooted at thesource node. The BIP heuristic is the node version ofDijkstra’s algorithm for SPT. It maintains, through-out its execution, a single arborescence rooted at thesource node. The arborescence starts from the sourcenode, and new nodes are added to the arborescenceone at a time on the minimum incremental cost basisuntil all nodes are included in the arborescence. Theincremental cost of adding a new node to the arbores-cence is the minimum additional power increased bysome node in the current arborescence to reach thisnew node. The implementation of BIP is based on thestandard Dijkstra’s algorithm, with one fundamentaldifference on the operation whenever a new node q isadded. Whereas the Dijkstra’s algorithm updates thenode weights (representing the current knowing dis-tances to the source node), BIP updates the cost ofeach link (representing the incremental power to reachthe head node of the directed link). This update isperformed by subtracting the cost of the added linkpq from the cost of every link qr that starts from qto a node r not in the new arborescence. They havebeen evaluated through simulations in [29], but littleis known about their analytical performances in termsof the approximation ratio. Here, the approximationratio of a heuristic is the maximum ratio of the energyneeded to broadcast a message based on the arbores-cence generated by this heuristic to the least necessaryenergy by any arborescence for any set of points.

For a pure illustration purpose, another slight

5

variation of BIP was discussed in detail in [35]. Thisgreedy heuristic is similar to the Chvatal’s algorithm[36] for the set cover problem and is a variation ofBIP. Like BIP, an arborescence, which starts with thesource node, is maintained throughout the execution ofthe algorithm. However, unlike BIP, many new nodescan be added one at a time. Similar to the Chvatal’salgorithm [36], the new nodes added are chosen tohave the minimal average incremental cost, which isdefined as the ratio of the minimum additional powerincreased by some node in the current arborescenceto reach these new nodes to the number of these newnodes. They called this heuristic as the Broadcast Av-erage Incremental Power (BAIP). In contrast to the1 + log m approximation ratio of the Chvatal’s algo-rithm [36], where m is the largest set size in the SetCover Problem, they showed that the approximationratio of BAIP is at least 4n

ln n − o (1), where n is thenumber of receiving nodes.

Wan et al. [35] showed that the approximationratios of MST and BIP are between 6 and 12 and be-tween 13

3 and 12 respectively; on the other hand, theapproximation ratios of SPT and BAIP are at leastn2 and 4n

ln n − o (1) respectively, where n is the num-ber of nodes. We then discuss in detail of their prooftechniques in next subsection.

The Iterative Maximum-Branch Minimization(IMBM) algorithm was another effort [32] to constructpower-efficient broadcast trees. It begins with a basicbroadcast tree in which the source directly transmits toall other nodes. Then it attempts to approximate theminimum-energy broadcast tree by iteratively replac-ing the maximum branch with less-power, more-hopalternatives.

Both BIP and IMBM operate under the assump-tion that the transmission power of each node is uncon-strained, i.e., every node can reach every other node.Both algorithms are centralized in the sense that theyrequire: (a) the source node needs to know the posi-tion/distance of every other node; and (b) each nodeneeds to know its downstream, on-tree neighbors so asto propagate broadcast messages. As a result, it maybe difficult to extend both algorithms into distributedversions, as a significant amount of information is re-quired to be exchanged among nodes.

2.3 Theoretical Analysis

Any broadcast routing is viewed as an arborescence(a directed tree) T , rooted at the source node of thebroadcasting, that spans all nodes. Let fT (p) denotethe transmission power of the node p required by T .For any leaf node p of T , fT (p) = 0. For any internal

node p of T ,

fT (p) = maxpq∈T

‖pq‖β ,

in other words, the β-th power of the longest dis-tance between p and its children in T . The to-tal energy required by T is

∑

p∈P fT (p). Thus theminimum-energy broadcast routing problem is differ-ent from the conventional link-based minimum span-ning tree (MST) problem. Indeed, while the MSTcan be solved in polynomial time by algorithms suchas Prim’s algorithm and Kruskal’s algorithm [37], theminimum-energy broadcast routing problem cannot besolved in polynomial time unless P=NP [26]. In itsgeneral graph version, the minimum-energy broadcastrouting can be shown to be NP-hard [38], and evenworse, it can not be approximated within a factorof (1− ε) log ∆, unless NP ⊆ DTIME

[

nO(log log n)]

,where ∆ is the maximal degree and ε is any arbitrarysmall positive constant. However, this hardness of itsgeneral graph version does not necessarily imply thesame hardness of its geometric version. In fact, asshown later in the chapter, its geometric version canbe approximated within a constant factor. Neverthe-less, this suggests that the minimum-energy broadcastrouting problem is considerably harder than the MSTproblem. Recently, Clementi et al. [26] proved thatthe minimum-energy broadcast routing problem is aNP-hard problem and obtained a parallel but weakerresult to those of [35].

Wan et al. [35] gave some lower bounds on the ap-proximation ratios of MST and BIP by studying somespecial instances in [35]. Their deriving of the upperbounds relies extensively on the geometric structuresof Euclidean MSTs. A key result in [35] is an upperbound on the parameter

∑

e∈mst(P ) ‖e‖2 for any finite

point set P of radius one. Note that the supreme ofthe total edge lengths of mst (P ),

∑

e∈mst(P ) ‖e‖, overall point sets P of radius one is infinity. However, theparameter

∑

e∈mst(P ) ‖e‖2 is bounded from above by a

constant for any point set P of radius one. They use cto denote the supreme of

∑

e∈mst(P ) ‖e‖2 over all point

sets P of radius one. The constant c is at most 12; see[35]. The proof of this theorem involves complicatedgeometric arguments; see [35] for more detail. Notethat for any point set P of radius one, the length ofeach edge in mst (P ) is at most one. Therefore, for anypoint set P of radius one and any real number β ≥ 2,

∑

e∈mst(P )

‖e‖β ≤∑

e∈mst(P )

‖e‖2 ≤ c ≤ 12.

The next theorem proved in [35] explores a rela-tion between the minimum energy required by a broad-

6

casting and the energy required by the Euclidean MSTof the corresponding point set.

Lemma 1 [35] For any point set P in the plane, thetotal energy required by any broadcasting among P isat least 1

c

∑

e∈mst(P ) ‖e‖β.

Proof. Let T be an arborescence for a broadcastingamong P with the minimum energy consumption. Forany none-leaf node p in T , let Tp be an Euclidean MSTof the point set consisting p and all children of p in T .Suppose that the longest Euclidean distance betweenp and its children is r. Then the transmission powerof node p is rβ , and all children of p lie in the diskcentered at p with radius r. From the definition of c,we have

∑

e∈Tp

(

‖e‖r

)β

≤ c,

which implies that

rβ ≥ 1c

∑

e∈Tp

‖e‖β .

Let T ∗ denote the spanning tree obtained by su-perposing of all Tp’s for non-leaf nodes of T . Then thetotal energy required by T is at least 1

c

∑

e∈T∗ ‖e‖β ,

which is further no less than 1c

∑

e∈mst(P ) ‖e‖β . This

completes the proof.

Consider any point set P in a two-dimensionalplane. Let T be an arborescence oriented from somemst (P ). Then the total energy required by T is atmost

∑

e∈Tp‖e‖β . From Lemma 1, this total energy is

at most c times the optimum cost. Thus the approxi-mation ratio of the link-based MST heuristic is at mostc. Together with c ≤ 12, this observation leads to thefollowing theorem.

Theorem 2 [35] The approximation ratio of the link-based MST heuristic is at most c, and therefore is atmost 12.

In addition, they derived an upper bound on theapproximation ratio of the BIP heuristic. Once again,the Euclidean MST plays an important role.

Lemma 3 [35] For any broadcasting among a pointset P in a two-dimensional plane, the total energy re-quired by the arborescence generated by the BIP algo-rithm is at most

∑

e∈mst(P ) ‖e‖β.

2.4 Centralized Clustering

We then study the non-adjustable power model case.The set of nodes that rebroadcast message in a reli-able broadcasting scheme define a connected dominat-ing set. A subset S of V is a dominating set if eachnode u in V is either in S or is adjacent to some nodev in S. Nodes from S are called dominators, whilenodes not is S are called dominatees. A subset C ofV is a connected dominating set (CDS) if C is a domi-nating set and C induces a connected subgraph. Con-sequently, the nodes in C can communicate with eachother without using nodes in V −C. A dominating setwith minimum cardinality is called minimum domi-nating set, denoted by MDS. A connected dominat-ing set with minimum cardinality is denoted by mini-mum connected dominating set (MCDS). A broadcast-ing based on connected dominating set only uses thenodes in CDS to relay the message. We first reviewseveral methods in the literature to build a connecteddominating set.

If every nodes cannot adjust its transmissionpower accordingly, then we need find the minimumconnected dominating set to save the total powerconsumption of the broadcasting protocol. Unfortu-nately, the problem of finding connected dominatingset of minimal size is NP-complete even for unit diskgraphs. Guha and Khuller [39] studied the approxima-tion of the connected dominating set problem for gen-eral graphs. They gave two different approaches, bothof them guarantee approximation ratio of Θ(H(∆)).As their approaches are for general graphs and thusdo not utilize the geometry structure if applied to thewireless ad hoc networks. One approach is to growa spanning tree that includes all nodes. The inter-nal nodes of the spanning tree is selected as the finalconnected dominating set. This approach has approx-imation ratio 2(H(∆)+1). The other approach is firstapproximating the dominating set and then connect-ing the dominating set to a connected dominating set.They [39] proved that this approach has approxima-tion ratio ln ∆ + 3.

One can also use the Steiner tree algorithm toconnect the dominators. This straightforward methodgives approximation ratio c(H(∆) + 1), where c is theapproximation ratio for the unweighted Steiner treeproblem. Currently, the best ratio is 1 + ln 3

2 ' 1.55,due to Robins and Zelikovsky [40].

By definition, any algorithm generating a maxi-mal independent set is a clustering method. We firstreview the methods that approximates the maximumindependent set, the minimum dominating set, and theminimum connected dominating set. Hunt et al. [41]

7

and Marathe et al. [42] studied the approximationof the maximum independent set and the minimumdominating set for unit disk graphs. They gave thefirst PTASs for MDS in UDG. The method is basedon the following observations: a maximal independentset is always a dominating set; given a square Ω witha fixed area, the size of any maximal independent setis bounded by a constant C. Assume that there aren nodes in Ω. Then, we can enumerate all sets withsize at most C in time Θ(nC). Among these enumer-ated sets, the smallest dominating set is the minimumdominating set. Then, using the shifting strategy pro-posed by Hochbaum [43], they derived a PTAS for theminimum dominating set problem.

Since we have PTAS for minimum dominating setand the graph V irtG connecting every pair of dom-inators within at most 3 hops is connected [44], wehave an approximation algorithm (constructing a min-imum spanning tree V irtG) for MCDS with approxi-mation ratio 3+ε. Notice that, Berman et al. [45] gavean 4

3 approximation method to connect a dominatingset and Robins et al. [40] gave an 4

3 approximationmethod to connect an independent set. Thus, we caneasily have an 8

3 approximation algorithm for MCDS,which was reported in [46]. Recently, Cheng et al. [47]designed a PTAS for MCDS in UDG. However, it isdifficult to distributize their method efficiently.

3 Localized Methods

3.1 Based on Distributed CDS

A natural structure for broadcasting is connected dom-inating set. Many distributed clustering (or dominat-ing set) algorithms have been proposed in the litera-ture [48, 49, 50, 51, 52, 53]. All algorithms assumethat the nodes have distinctive identities (denoted byID hereafter).

In the rest of section, we will interchange theterms cluster-head and dominator. The node that isnot a cluster-head is also called dominatee. A node iscalled white node if its status is yet to be decided bythe clustering algorithm. Initially, all nodes are white.The status of a node, after the clustering method fin-ishes, could be dominator with color black or domi-natee with color gray. The rest of this section is de-voted for the distributed methods that approximatesthe minimum dominating set and the minimum con-nected dominating set for unit disk graph.

3.1.1 Clustering without Geometry Property

For general graphs, Jia et al. [54] described and an-alyzed some randomized distributed algorithms forthe minimum dominating set problem that run inpolylogarithmic time, independent of the diameter ofthe network, and that return a dominating set ofsize within a logarithmic factor from the optimumwith high probability. Their best algorithm runs inO(log n log ∆) rounds with high probability, and ev-ery pair of neighbors exchange a constant numberof messages in each round. The computed dominat-ing set is within O(log ∆) in expectation and withinO(log n) with high probability. Their algorithm worksfor weighted dominating set also.

The method proposed by Das et al. [55, 56] con-tains three stages: approximating the minimum domi-nating set, constructing a spanning forest of stars, ex-panding the spanning forest to a spanning tree. Herethe stars are formed by connecting each dominateenode to one of its dominators. The approximationmethod of MDS is essentially a distributed variationof the the centralized Chvatal’s greedy algorithm [36]for set cover. Notice that the dominating set prob-lem is essentially the set cover problem which is well-studied. It is then not surprise that the method byDas et al. [55, 56] guarantees a H(∆) for the MDSproblem, where H is the harmonic function and ∆ isthe maximum node degree.

While the algorithm proposed by Das et al.[55, 56] finds a dominating set and then grows it toa connecting dominating set, the algorithm proposedby Wu and Li [57, 58] takes an opposite approach.They first find a connecting dominating set and thenprune out certain redundant nodes from the CDS. Theinitial CDS C contains all nodes that have at least twonon-adjacent neighbors. A node u is said to be lo-cally redundant if it has either a neighbor in C withlarger ID which dominates all other neighbors of u,or two adjacent neighbors with larger ID which to-gether dominates all other neighbors of u. Their algo-rithm then keeps removing all locally redundant nodesfrom C. They showed that this algorithm works wellin practice when the nodes are distributed uniformlyand randomly, although no any theoretical analysis isgiven by them both for the worst case and for the av-erage approximation ratio. However, it was shown byAlzoubi et al. [48] that the approximation ratio of thisalgorithm could be as large as n

2 .Recently, Dai and Wu have proposed a distributed

dominant pruning algorithm [59]. Each node has apriority which can be simply its unique identifier ora combination of remaining battery, degree or iden-

8

tifier. A node u is “fully covered” by a subset S ofits neighboring nodes if and only if the following threeconditions hold:

• the subset S is connected,

• any neighbor of u is neighbor of at least one nodefrom S,

• all nodes in S have higher priority than u.

A node belongs to the dominating set if and onlyif there is no subset that fully covers it. The advan-tage of using CDS as defined in [57, 58, 59] is that eachnode can decide whether or not it is in dominating setwithout any additional communication steps involved,other than those needed to maintain neighborhood in-formation. The neighborhood information needed iseither 2-hop neighbors knowledge, or 1-hop neighborknowledge with their position.

Stojmenovic et al. [60] observed that distributedconstructions of connecting dominating set can be ob-tained following the clustering scheme of Lin and Gerla[49]. Connecting dominating set consists of two typesof nodes: clusterhead and border-nodes (also calledgateway or connectors elsewhere). The clusterheadnodes are decided as follows. At each step, all whitenodes which have the lowest rank among all whiteneighbors are colored black, and the white neighborsare colored gray. The ranks of the white nodes areupdated if necessary. The clustering method usestwo messages which can be called IamDominator andIamDominatee. A white node claims itself to be a dom-inator if it has the smallest ID among all of its whiteneighbors, if there is any, and broadcasts IamDomi-nator to its 1-hop neighbors. A white node receivingIamDominator message marks itself as dominatee andbroadcasts IamDominatee to its 1-hop neighbors. Theset of dominators generated by the above method isactually a maximal independent set. Here, we assumethat each node knows the IDs of all its 1-hop neigh-bors, which can be achieved if each node broadcastsits ID to its neighbors initially. This approach of con-structing MIS is well-known. The following rankingsof a node are used in various methods: the ID only[50, 49], the ordered pair of degree and ID [61], and anordered pair of degree and location [60]. In [62, 63],Basagni et al., used a general weight as a ranking cri-terion for selecting the node as the clusterhead, wherethe weight is a combination of mentioned criteria andsome new ones, such as mobility or remaining energy.After the clusterhead nodes are selected, border-nodesare selected to connect them. A node is a border-node

if it is not a clusterhead and there are at least two clus-terheads within its 2-hop neighborhood. It was shownby [48] that the worst case approximation ratio of thismethod is also n

2 , although it works well in practice.

3.1.2 Clustering with Geometry Property

Notice that none of the above algorithm utilizes thegeometry property of the underlying unit disk graph.Recently, several algorithms were proposed with a con-stant worst case approximation ratio by taking ad-vantage of the geometry properties of the underlyinggraph. It is used to connect the clusterheads con-structed as described above into a CDS with feweradditional nodes. During this second step of backboneformation, some connectors (also called gateways) arefound among all the dominatees to connect the domi-nators. Then the connectors and the dominators forma connected dominating set. Recently, Wan, et al. [64]and Wu and Lou [15] proposed a communication ef-ficient algorithm to find connectors based on the factthat there are only a constant number of dominatorswithin k-hops of any node. The following observationis a basis of several algorithms for CDS. After clus-tering, one dominator node can be connected to manydominatees. However, it is well-known that a domina-tee node can only be connected to at most five domi-nators in the unit disk graph model. Generally, it wasshown in [64, 44, 15] that for each node v (dominatoror dominatee), the number of dominators inside thedisk centered at v with radius k-units is bounded by aconstant `k < (2k + 1)2.

Given a dominating set S, let V irtG be the graphconnecting all pairs of dominators u and v if thereis a path in UDG connecting them with at most 3hops. Graph V irtG is connected. It is natural to forma connected dominating set by finding connectors toconnect any pair of dominators u and v if they areconnected in V irtG. This strategy is also adopted byWan, et al. [64] and Wu and Lou [15]. Notice that, inthe approach by Stojmenovic et al. [60], they set anydominatee node as the connector if there are two dom-inators within its 2-hop neighborhood. This approachis very pessimistic and results in very large number ofconnectors in the worst case [48]. Instead, Wan et al.[16] suggested to find only one unique shortest pathto connect any two dominators that are at most threehops away.

We briefly review their basic idea of forming aCDS in a distributed manner. Let ΠUDG(u, v) be thepath connecting two nodes u and v in UDG with thesmallest number of hops. Let’s first consider how toconnect two dominators within 3 hops. If the path

9

ΠUDG(u, v) has two hops, then u finds the dominateewith the smallest ID to connect u and v. If the pathΠUDG(u, v) has three hops, then u finds the node, sayw, with the smallest ID such that w and v are twohops apart. Then node w selects the node with thesmallest ID to connect w and v. Wang and Li [44] andAlzoubi et al. [64] discussed in detail some approachesto optimize the communication cost and the memorycost.

The graph constructed by this algorithm is calleda CDS graph (or backbone of the network). If we alsoadd all edges that connect all dominatees to their dom-inators, the graph is called extended CDS, denoted byCDS’. Let opt be the size of the minimum connecteddominating set. It was shown [42] that the size of thecomputed maximal independent set has size at most4∗opt+1. We already showed that the size of the con-nected dominating set found by the above algorithmis at most `3k + k, where k is the size of the maximalindependent set found by the clustering algorithm. Itimplies that the found connected dominating set hassize at most 4(`3 + 1) ∗ opt + `3 +1. Consequently, thecomputed connected dominating set is at most 4(`3+1)factor of the optimum (with an additional constant`3 + 1). It was shown in [16, 44] that the CDS’ graphis a sparse spanner in terms of both hops and length,meanwhile CDS has a bounded node degree.

3.2 Localized Low Weight Structures

The centralized algorithms do not consider computa-tional and message overheads incurred in collectingglobal information. Several of them also assume thatthe network topology does not change between tworuns of information exchange. These assumptions maynot hold in practice, since the network topology maychange from time to time, and the computational andenergy overheads incurred in collecting global infor-mation may not be negligible. This is especially truefor large-scale wireless networks where the topologyis changing dynamically due to the changes of posi-tion, energy availability, environmental interference,and failures, which implies that centralized algorithmsthat require global topological information may not bepractical.

Santivanez et al. [65] show that flooding is agood solution for the sake of scalability and simplicity.Several flooding techniques for wireless networks havebeen proposed [66, 67, 60], each with respect to certainoptimization criterion. However, none of them takesadvantage of the feature that the transmission powerof a node can be adjusted.

Some distributed heuristics are proposed, such as

[68, 69, 70]. Most of them are based on distributedMST method. A possible drawback of these dis-tributed method is that it may not perform well underfrequent topological changes as it relies on informa-tion that is multiple hops away to construct the MST.Refer to [71] for more detail. The relative neighbor-hood graph, the Gabriel graph and the Yao graph allhave O(n) edges and contain the Euclidean minimumspanning tree. This implies that we can construct theminimum spanning tree using O(n log n) messages.

Localized minimum energy broadcast algorithmsare based on the use of a locally defined geometricstructure, such as RNG (relative neighborhood graph),proposed by Toussaint [90]. RNG consists of all edgesuv such that uv is not the longest edge in any triangleuvw. That is, uv belongs to RNG if there is no nodew such that uw < uv and vw < uv.

Cartigny et al. [72] proposed a localized algo-rithm, called RBOP [72] that is built upon the notionof relative neighborhood graph (RNG). In RBOP, thebroadcast is initiated at the source and propagated,following the rules of neighbor elimination [60], onthe topology represented by RNG. Simulation resultsshow that the energy consumption could be as high as100% as compared to BIP. However, the communica-tion overhead due to mobility and changes in activitystatus in BIP are not considered, therefore RBOP issuperior to BIP in dynamic ad hoc networks. Li andHou [71], and Cartigny et al. [96] proposed anotherlocalized algorithms, which applies LMST (localizedminimal spanning tree) instead of RNG as the broad-cast topology. In LMST, proposed in [75], each nodecalculates local minimum spanning tree of itself andits 1-hop neighbors. A node uv is in LMST if andonly if u and v select each other in their respectivetrees. The simulations [71, 96] show that the perfor-mance of LMST based schemes is significantly betterthan the performance of RBOP, and with about 50%more energy consumption than BIP in static scenarios.Cartigny et al. [92] demonstrated that, when c > 0 inpower-attenuation model where energy consumptionfor transmitting over an edge uv is ‖uv‖β + c, thereexists an optimal ’target’ transmission radius, so thatfurther energy savings can be obtained if transmissionradii are selected near target radius.

However, as shown in [21] (also by Figure 1), thetotal weights of RNG and LMST could still be as largeas O(n) times of the total weight of MST. Here, inFigure 1, ‖uivi‖ = 1 and ‖uiui+1‖ = ‖vivi+1‖ = ε fora very small positive real number ε. Given a graphG, let ωb(G) =

∑

e∈G ‖e‖b. Then ω1(RNG) = Θ(n) ·ω1(MST ) and ω1(LMST ) = Θ(n) · ω1(MST ).

10

n

v1

ui v i

u vn

u1

Figure 1: An instance of wireless nodes that everynetwork structures described previously (except MST)have an arbitrarily large total weight.

In [21, 73], we described three low weight planarstructures that can be constructed by localized meth-ods with total communication costs O(n). The energyconsumption of broadcast based on those structuresare within O(nβ−1) of the optimum, i.e., ωβ(H) =O(nβ−1) · ωβ(MST ), ωβ(LMST2) = O(nβ−1) ·ωβ(MST ), ωβ(IMRG) = O(nβ−1) ·ωβ(MST ) for anyβ ≥ 1. This improves the previously known “light-est” structure RNG by O(n) factor since in the worstcase ω(RNG) = Θ(n) · ω(MST ) and ωβ(RNG) =Θ(nβ) · ωβ(MST ).

We will now review in detail these three struc-tures.

3.2.1 Structure Based on RNG’

Although RNG is very sparse structure (the averagenumber of neighbors per node is about 2.5), in somedegenerate cases a particular node may have arbitrar-ily large degree. This motivated Stojmenovic [22] todefine a modified structure where each node will havedegree bounded by 6. The same structure was inde-pendently proposed by Li in [21], with an additionalmotivation. Li proved that the modified RNG is thefirst localized method to construct a structure H withweight O(ω(MST )) using total O(n) local-broadcastmessages. Note that, if each node already knows thepositions and IDs of all its neighbors, then no mes-sages are needed to decide which of its edges belongto (modified) RNG. Notice that, traditionally, the rel-ative neighborhood graph will always select an edgeuv even if there is some node on the boundary oflune(u, v). Here lune(u, v) is the intersection of twodisks centered at nodes u and v with radius ‖uv‖ re-spectively. Thus, RNG may have unbounded node de-gree, e.g., considering n− 1 points equally distributedon the circle centered at the nth point v, the degree of vis n−1. Notice that for the sake of lowing the weight ofa structure, the structure should contain as less edgesas possible without breaking the connectivity. Li [21]and Stojmenovic [22] then naturally extended the tra-ditional definition of RNG as follows.

We need to make distinct edge lengths. This canbe achieved by adding the secondary, and if necessary,the ternary keys for comparing two edges. Each nodeis assumed to have an unique ID. Then consider therecord (‖uv‖), ID(u), ID(v), where ID(u) < ID(v)(otherwise u and v are exchanged for given edge). Twoedges compare their lengths first to decide which oneis longer. If same, they then compare their secondarykey, which is their respective lower endpoint node’s ID.If this is also same, then the ternary key resolves thecomparison (otherwise we are comparing edge againstitself). This simple method for making distinct edgelength was proposed in [23, 75]. The edge lengths, sodefined, are then used in the regular definition of RNG.It is easy to show that two RNG edges uv and uw goingout of the same node must have angle between themat least π/6, otherwise vw < uv or vw < uw, and oneof the two edges becomes the longest in the triangleand consequently could not be in RNG. Li denotedmodified RNG structure by RNG’. Obviously, RNG’is a subgraph of traditional RNG. It was proved in[21, 22] that RNG’ still contains a MST as a subgraph.However, RNG’ is still not a low weight structure.

Notice that it is well-known that the communi-cation complexity of constructing a minimum span-ning tree of a n-vertex graph G with m edges isO(m + n log n); while the communication complexityof constructing MST for UDG is O(n log n) even underthe local broadcasting communication model in wire-less networks. It was shown in [21] that it is impossibleto construct a low-weighted structure using only onehop neighbor information.

The localized algorithm given in [21] that con-structs a low-weighted structure using only some twohops information is as follows.

Algorithm 1 Construct Low Weight Structure H

1. All nodes together construct the graph RNG’ ina localized manner.

2. Each node u locally broadcasts its incident edgesin RNG’ to its one-hop neighbors. Node u listensto the messages from its one-hop neighbors.

3. Assume node u received a message informing exis-tence of edge xy ∈ RNG′ from its neighbor x. Foreach edge uv ∈ RNG′, if uv is the longest amonguv, xy, ux, and vy, node u removes edge uv. Tiesare broken by the label of the edges. Here assumethat uvyx is the convex hull of u, v, x, and y.

4. Let H be the final structure formed by all remain-ing edges in RNG’.

11

Obviously, if an edge uv is kept by node u, thenit is also kept by node v. The following theorem wasproved in [21].

Theorem 4 [21] The total edge weight of H is withina constant factor of that of the minimum spanning tree.

This was proved by showing that the edges in Hsatisfy the isolation property (defined in [74]). They[21] also showed that the final structure contains MSTof UDG as a subgraph.

Clearly, the communication cost of Algorithm 1is at most 7n: initially each node spends one messageto tell its one-hop neighbors its position information,then each node uv tells its one-hop neighbors all itsincident edges uv ∈ RNG′ (there are at most total6n such messages since RNG′ has at most 3n edges).The computational cost of Algorithm 1 could be highsince for each link uv ∈ RNG′, node u has to testwhether there is an edge xy ∈ RNG′ and x ∈ N1(u)such that uv is the longest among uv, xy, ux, and vy.Then [73] presents some new algorithms that improvethe computational complexity of each node while stillmaintains low communication costs.

3.2.2 Structure Based on LMSTk

The first new method in [73] uses a structure calledlocal minimum spanning tree, let us first review itsdefinition. It is first proposed by Li, Hou and Sha[75]. Each node u first collects its one-hop neighborsN1(u). Node u then computes the minimum spanningtree MST (N1(u)) of the induced unit disk graph onits one-hop neighbors N1(u). Node u keeps a directededge uv if and only if uv is an edge in MST (N1(u)).They call the union of all directed edges of all nodesthe local minimum spanning tree, denoted by LMST1.If only symmetric edges are kept, then the graph iscalled LMST−1 , i.e., it has an edge uv iff both directededge uv and directed edge vu exist. If ignoring the di-rections of the edges in LMST1, they call the graphLMST+

1 , i.e., it has an edge uv iff either directed edgeuv or directed edge vu exists. They prove that thegraph is connected, and has bounded degree 6. In[73], Li et al. also showed that graph LMST−1 andLMST+

1 are actually planar. Then they extend thedefinition to k-hop neighbors, the union of all edges ofall minimum spanning tree MST (Nk(u)) is the k lo-cal minimum spanning tree, denoted by LMSTk. Forexample, the 2 local minimum spanning tree can beconstructed by the following algorithm.

Algorithm 2 Construct Low Weight StructureLMST2 by 2-hop Neighbors

1. Each node u collects its two hop neighbors in-formation N2(u) using a communication efficientprotocol described in [76].

2. Each node u computes the Euclidean minimumspanning tree MST (N2(u)) of all nodes N2(u),including u itself.

3. For each edge uv ∈ MST (N2(u)), node u tellsnode v about this directed edge.

4. Node u keeps an edge uv if uv ∈ MST (N2(u))or vu ∈ MST (N2(v)). Let LMST+

2 be the finalstructure formed by all edges kept. It keeps anedge if either node u or node v wants to keep it.Another option is to keep an edge only if bothnodes want to keep it. Let LMST−2 be the struc-ture formed by such edges.

In [73], they prove that structures LMST2

(LMST+2 and LMST−2 ) are connected, planar, low-

weighted, and have bounded node degree at most6. In addition, MST is a subgraph of LMSTk andLMSTk ⊆ RNG′. Although the constructed struc-ture LMST2 has several nice properties such as beingbounded degree, planar, and low-weighted, the com-munication cost of Algorithm 2 could be very largeto save the computational cost of each node. Thelarge communication costs are from collecting the twohop neighbors information N2(u) for each node u. Al-though the total communication of the protocol de-scribed in [76] is O(n), the hidden constant is large.

3.2.3 Combining RNG’ and LMSTk

We could improve the communication cost of collectingN2(u) by using a subset of two hop information with-out sacrificing any properties. Define NRNG′

2 (u) =w | vw ∈ RNG′ and v ∈ N1(u) ∪ N1(u). We de-scribe our modified algorithm as follows.

Algorithm 3 Construct Low Weight StructureIMRG by 2-hop Neighbors in RNG’

1. Each node u tells its position information to itsone-hop neighbors N1(u) using a local broadcastmodel. All nodes together construct the graphRNG’ in a localized manner.

2. Each node u locally broadcasts its incident edgesin RNG’ to its one-hop neighbors. Node u listensto the messages from its one-hop neighbors.

3. Each node u computes the Euclidean minimumspanning tree MST (NRNG′

2 (u)) of all nodesNRNG′

2 (u), including u itself.

12

4. For each edge uv ∈ MST (NRNG′2 (u)), node u tells

node v about this directed edge.

5. Node u keeps an edge uv if uv ∈MST (NRNG′

2 (u)) or vu ∈ MST (NRNG′2 (v)).

Let IMRG+ be the final structure formed byall edges kept. Similarly, the final structureis called IMRG− when edge uv ∈ RNG′

is kept iff uv ∈ MST (NRNG′2 (u)) and

uv ∈ MST (NRNG′2 (v)). Here IMRG is the

abbreviation of Incident MST and RNG Graph.

Notice that in the algorithm, node u constructsthe local minimum spanning tree MST (NRNG′

2 (u))based on the induced UDG of the point sets NRNG′

2 (u).It is obvious that the communication cost of Algorithm3 is at most 7n.

We can show that structures IMRG+ andIMRG− are still connected, planar, bounded degree,and low-weighted. They are obviously planar, andwith bounded degree since both structures are stillsubgraphs of the modified relative neighborhood graphRNG’. Clearly, the constructed structures are super-graphs of the previous structures, i.e., LSMT2+ ⊆IMRG+ and LSMT−2 ⊆ IMRG−, since Algorithm 3uses less information than Algorithm 2 in construct-ing the local minimum spanning tree. It is proved in[73] that Algorithm 3 constructs structures IMRG−

or IMRG+ using at most 7n messages. The struc-tures IMRG− or IMRG+ are connected, planar,bounded degree, and low-weighted. Both IMRG− andIMRG+ have node degree at most 6.

Recall that until now there is no efficient localizedalgorithm that can achieve all following desirable fea-tures: bounded degree, planar, low weight and span-ner. It is still an open problem.

3.2.4 A Negative Result

In [21, 73], Li et al. proposed several meth-ods to construct structures in a localized mannersuch that the total edge lengths of these structuresare within a constant factor of MST. They alsoshowed that the energy consumption of broadcast-ing based on those structures are within O(nβ−1) ofthe optimum, i.e., ωβ(H) = O(nβ−1) · ωβ(MST ),ωβ(LMST2) = O(nβ−1) · ωβ(MST ), ωβ(IMRG) =O(nβ−1) · ωβ(MST ) for any β ≥ 1.

They also show that it is impossible to design adeterministic localized method that constructs a struc-ture such that the broadcasting based on this structureconsumes energy within a constant factor of the opti-mum. Assume that there is a deterministic localized

algorithm to do so: it uses k-hop information of everynode u to select the edges incident on u, and the energyconsumption is no more than C times of the optimum.They construct two set of nodes configurations suchthat the k-hop information collected in a special nodeu is same for both configurations. In addition, there isan edge uv in both UDGs such that if node u decidesto keep edge uv (then edge uv is kept in both configu-rations), the energy consumption of one configurationis already more than C times of the optimum; if nodeu decides to remove edge uv (then edge uv is removedin both configurations), then the structure constructedfor another configuration is disconnected.

3.3 Combining Clustering and LowWeight

Seddigh, Gonzalez and Stojmenovic [77] specify twomore location based broadcasting algorithms thatcombine RNG and internal node concept (connecteddominating set) as follows. PI-broadcast algorithm ap-plies the planar subgraph construction first, and thenapplies the internal nodes concept on the subgraph.The result is different from the internal nodes appliedon the whole graph. IP-broadcast algorithm changesthe order of concept application compared to the pre-vious algorithm. Internal nodes are first identified inthe whole graph, and then the obtained subgraph (con-taining only internal nodes) is further reduced to pla-nar one by the RNG construction.

The solution in [77] is for one-to-one communi-cation model, where message sent from one node isreceived by only the targeted neighbor. In [78], Li etal. combine the low-weighted structures and the con-nected dominating set for energy efficient broadcastingin traditional one-to-many (omnidirectional antenna)networks. Similarly, they proposed two approachesfor combining them as in [77]. Notice that the con-structed low-weighted structures are a subgraph ofRNG, thus, they are still planar graphs. For simplic-ity, they also call these two combinations, PI-broadcastand IP-broadcast respectively. They found that theenergy consumption of the IP-broadcast schemes indense networks is significantly less than that of thePI-broadcast schemes. The reason is that, in the IP-broadcast schemes, one retransmission by the internalnodes will be received by many non-internal nodes indense networks, thus, energy consumption is reduced.Several localized improvement heuristics are also ap-plied after the IP-broadcast or PI-broadcast schemesto further improve the energy consumption. Ingelrest(cf. [92]) in his master thesis also combined LMST

13

with dominating sets, and target radius idea to derivenew minimum energy broadcast protocols.

Following Figure 2 illustrates the different struc-tures used for broadcasting. All figures are computedon a set of 500 nodes randomly distributed in a squareregion with side length 7500m. Each node has trans-mission range 500m. Here the CDS is generated us-ing ID as criterion for selecting dominator/conector;IMRG-CDS is the subgraph constructed by applyingIMRG on the induced unit disk graph of all CDSnodes, i.e., dominators and connectors. Our simu-lation results also confirms that CDS structure andIMRG-CDS consumes the least energy when the nodepower is non-adjustable, while the IMRG consumesthe least energy when each node can adjust its poweraccording to the longest incident link.

3.4 Flooding Based Methods

3.4.1 Selecting Forwarding Neighbors

The simplest broadcasting mechanism is to let everynode retransmit the message to all its one-hop neigh-bors when receiving the first copy of the message,which is called flooding in the literature. Despite itssimplicity, flooding is very inefficient and can resultin high redundancy, contention, and collision. Oneapproach to reducing the redundancy is to let a nodeonly forward the message to a subset of one-hop neigh-bors who together can cover the two-hop neighbors. Inother words, when a node retransmits a message to itsneighbors, it explicitly asks a subset of its neighborsto relay the message.

In [79], Lim and Kim proposed a broadcastingscheme that chooses some or all of its one-hop neigh-bors as rebroadcasting node. When a node receives abroadcast packet, it uses a Greedy Set Cover algorithmto determine which subset of neighbors should re-broadcast the packet, given knowledge of which neigh-bors have already been covered by the sender’s broad-cast. The Greedy Set Cover algorithm recursivelychooses 1-hop neighbors which cover the most 2-hopneighbors and recalculates the cover set until all 2-hopneighbors are covered.

Calinescu et al. [80] gave two practical heuris-tics for this problem (they called selecting forwardingneighbors). The first algorithm runs in time O(n log n)and returns a subset with size at most 6 times of theminimum. The second algorithm has an improved ap-proximation ratio 3, but with running time O(n2).Here n is the number of total two-hop neighbors ofa node. When all two-hop neighbors are in the samequadrant with respect to the source node, they gave

an exact solution in time O(n2) and a solution withapproximation factor 2 in time O(n log n). Their al-gorithms partition the region surrounding the sourcenode into four quadrants, solve each quadrants usingan algorithm with approximation factor α, and thencombine these solutions. They proved that the com-bined solution is at most 3α times of the optimumsolution.

Their approach assumes that every node u cancollect its 2-hop neighbors N2(u) efficiently. Noticethat, the 1-hop neighbors of every node u can be col-lected efficiently by asking each node to broadcast itsinformation to its 1-hop neighbors. Thus all nodes gettheir 1-hop neighbors information by using total O(n)messages. However, until recently, it was not knownhow to collect the 2-hop neighbors information withO(n) communications. The simplest broadcasting of1-hop neighbors N1(u) to all neighbors u does let allnodes in N1(u) to collect their corresponding 2-hopneighbors. However, the total communication cost ofthis approach is O(m), where m is the total numberof links in UDG. Recently, Calinescu [76] proposed anefficient approach to collect N2(u) using the connecteddominating set [64] as forwarding nodes. Assume thatthe node position is known. He proved that the ap-proach takes total communications O(n), which is op-timum within a constant factor.

3.4.2 Gossip and Probabilistic Schemes

Probabilistic Scheme: The Probabilistic schemefrom [12] is similar to Flooding, except that nodesonly rebroadcast with a predetermined probability. Indense networks multiple nodes share similar transmis-sion coverages. Thus, randomly having some nodes notrebroadcast saves node and network resources with-out harming delivery effectiveness. In sparse networks,there is much less shared coverage; thus, nodes wontreceive all the broadcast packets with the Probabilis-tic scheme unless the probability parameter is high.When the probability is 100%, this scheme is identi-cal to Flooding. Cartigny and Simplot [94] appliedprobability which is a function of the distance to thetransmitting neighbor.

Counter-Based Scheme: Tseng et al. [12] showan inverse relationship between the number of times apacket is received at a node and the probability ofthat node being able to reach additional area on arebroadcast. This result is the basis of their Counter-Based scheme. Upon reception of a previously unseenpacket, the node initiates a counter with a value of oneand sets a RAD (which is randomly chosen between 0and Tmax seconds). During the RAD, the counter

14

MST BIP RNG CDS

LMST1 LMST2 IMRG IMRG-CDS

Figure 2: Different structures generated from a UDG used for broadcasting.

is incremented by one for each redundant packet re-ceived. If the counter is less than a threshold valuewhen the RAD expires, the packet is rebroadcast. Oth-erwise, it is simply dropped. From [12], threshold val-ues above six relate to little additional coverage areabeing reached.

The overriding compelling features of theCounter-Based scheme are its simplicity and its inher-ent adaptability to local topologies. That is, in a densearea of the network, some nodes won’t rebroadcast; insparse areas of the network, all nodes rebroadcast. Thedisadvantage of all counter and probabilistic schemesis that delivery is not guaranteed to all nodes even ifideal MAC is provided. In other words, they are notreliable.

3.4.3 Area Based Decision

In either probabilistic schemes or the counter-basedschemes a node decides whether to rebroadcast a re-ceived packets purely based on its own information.Tseng et al. [12] proposed several other criteria basedon the additional coverage area to decide whether thenode will rebroadcast the packet. These coverage-areabased methods are similar to the methods of selectingforwarding neighbors, which tries to select a set of one-hop neighbors sufficient to cover all its two-hop neigh-

bors. While area based methods only consider thecoverage area of a transmission; they don’t considerwhether nodes exist within that area. Two coverage-area based methods are proposed in [12]: Distance-Based Scheme and Location Based Scheme.

In Distance-Based Scheme, a node compares thedistance between itself and each neighbor node thathas previously rebroadcast a given packet. Upon re-ception of a previously unseen packet, a RAD is ini-tiated and redundant packets are cached. When theRAD expires, all source node locations are examinedto see if any node is closer than a threshold distancevalue. If true, the node doesn’t rebroadcast.

The Location-Based scheme uses a more preciseestimation of expected additional coverage area in thedecision to rebroadcast. In this method, each nodemust have the means to determine its own location,e.g., a GPS. Whenever a node originates or rebroad-casts a packet it adds its own location to the headerof the packet. When a node initially receives a packet,it notes the location of the sender and calculates theadditional coverage area obtainable were it to rebroad-cast. If the additional area is less than a thresholdvalue, the node will not rebroadcast, and all futurereceptions of the same packet will be ignored. Other-wise, the node assigns a RAD before delivery. If thenode receives a redundant packet during the RAD, it

15

recalculates the additional coverage area and comparesthat value to the threshold. The area calculation andthreshold comparison occur with all redundant broad-casts received until the packet reaches either its sched-uled send time or is dropped.

We will review also some upcoming work relatedto dominating sets and broadcasting problem. In [96],a beaconless broadcasting method is proposed. Allnodes have the same transmission radius, and nodesare not aware of their neighborhood. That is, no bea-cons or hello messages are sent in order to discoverneighbors prior to broadcasting process. The sourcetransmits the message to all neighbors. Upon receiv-ing the packet (together with geographic coordinatesof the sender), each node calculates the portion of itsperimeter, along circle of transmission radius, that isnot covered by this and previous transmissions of thesame packet. Node then sets or updates its timeoutinterval, which inversely depends on the size of the un-covered perimeter portion. If the perimeter becomesfully covered, the node cancels retransmissions. Oth-erwise, it retransmits at the end of timeout interval.The method is reliable, as opposed to other area basedmethods.

3.4.4 Neighbor Coverage Based Decision

The method presented in the previous subsection werebased on covering an area where nodes could be lo-cated. Instead of covering area, one could simply coverneighboring nodes, assuming their location, or exis-tence of their link to a previous transmitting node, areknown. The basic method was independently and al-most simultaneously (August 2000) proposed in twoarticles [81, 82]. The methods were called NeighborElimination by Stojmenovic and Seddigh [82], while asimilar method, called Scalable Broadcast Algorithm,was proposed by Peng and Lu [81]. Two-hop neigh-bors information is used to determine whether a nodewill rebroadcast the packet. Suppose that a node u re-ceives a broadcast data packet from its neighbor nodev. Node u knows all the neighbors of node v, and thusall nodes that are common neighbors of them (alreadyreceived the data from v). If node u has additionalneighbors not reached by node v’s broadcast, node uschedules the packet for delivery with a RAD. How-ever, if node u receives a redundant broadcast packetfrom some other neighbors within RAD, node u willrecalculate whether it needs rebroadcast the packet.This process is continued until either the RAD ex-pires and the packet is then sent, or the packet isdropped (when all its neighbors are already coveredby the broadcasts of some of its neighbors).

Lipman, Boustead and Judge [95] described thefollowing broadcasting protocol. Upon receiving abroadcast message(s) from a node h, each node i (thatwas determined by h as a forwarding node) determineswhich of its one-hop neighbors also received the samemessage. For each of its remaining neighbors j (whichdid not receive a message yet, based on i’s knowledge),node i determines whether j is closer to i than any one-hop neighbors of i (that are also forwarding nodes ofh) who received the message already. If so, i is respon-sible for message transmission to j, otherwise it is not.Node i then determines a transmission range equal tothat of the farthest neighbor it is responsible for.

4 Scheduling Active and SleepPeriods

In ad hoc wireless networks, the limitation of powerof each host poses a unique challenge for power-awaredesign. There has been an increasing focus on lowcost and reduced node power consumption in ad hocwireless networks. Even in standard networks suchas IEEE 802.11, requirements are included to sacri-fice performance in favor of reduced power consump-tion. In order to prolong the life span of each nodeand, hence, the network, power consumption should beminimized and balanced among nodes. Unfortunately,nodes in the dominating set in general consume moreenergy in handling various bypass traffic than nodesoutside the set. Therefore, a static selection of domi-nating nodes will result in a shorter life span for certainnodes, which in turn result in a shorter life span of thewhole network.

Wu, Wu, and Stojmenovic [83] study dynamicselection of dominating nodes, also called activityscheduling. Activity scheduling deals with the wayto rotate the role of each node among a set of givenoperation modes. For example, one set of operationmodes is sending, receiving, idles, and sleeping. Dif-ferent modes have different energy consumptions. Ac-tivity scheduling judiciously assigns a mode to eachnode to save overall energy consumptions in the net-works and/or to prolong life span of each individualnode. Note that saving overall energy consumptionsdoes not necessarily prolong life span of a particular in-dividual node. Specifically, they propose to save over-all energy consumptions by allowing only dominatingnodes (i.e., gateway nodes) to retransmit the broadcastpacket. In addition, in order to maximize the lifetimeof all nodes, an activity scheduling method is used thatdynamically selects nodes to form a connected domi-

16

nating set. Specifically, in the selection process of agateway node, we give preference to a node with ahigher energy level. The effectiveness of the proposedmethod in prolonging the life span of the network isconfirmed through simulation. Source dependent for-warding sets appear to be more energy balanced. How-ever, it was experimentally confirmed in [84] that thedifference in energy consumption between an idle nodeand a transmitting node is not major, while the majordifference exists between idle and sleep states of nodes.Therefore the most energy efficient methods will selectstatic dominating set for a given round, turning all re-maining nodes to a sleep state. Depending on energyleft, changes in activity status for the next round willbe made. The change can therefore be triggered bychanges of power status, in addition to node mobility.From this point of view, internal nodes based domi-nating sets provide static selection for a given roundand more energy efficiency than forwarding set basedmethod that requires all nodes to remain active in allthe rounds.

In [85], the key for deciding dominating set statusis a combination of remained energy and node degree.Xu, Heidemann, and Estrin [86] discuss the followingsensor sleep node schedule. The tradeoff between net-work lifetime and density for this cell-based schedulewas investigated in [87]. The given 2-D space is parti-tioned into a set of squares (called cells), such as anynode within a square can directly communicate withany nodes in an adjacent square. Therefore, one repre-sentative node from each cell is sufficient. To prolongthe life span of each node, nodes in the cell are selectedin a alternative fashion as a representative. The adja-cent squares form a 2-D grid and the broadcast processbecomes trivial. Note that the selected nodes in [86]make a dominating set, but the size of it is far fromoptimal, and also it depends on the selected size ofsquares. On the other hand, the dominating set con-cept used here has smaller size and is chosen withoutusing any parameter (size of square, which has to becarefully selected and propagated with node relativepositioning in solution [86]).

The Span algorithm [88] selects some nodes ascoordinators. These nodes form dominating set. Anode becomes coordinator if it discovers that two ofits neighbors cannot communicate with each other di-rectly or through one or two existing coordinators.Also, a node should withdraw if every pair of its neigh-bors can reach each other directly or via some othercoordinators (they can also withdraw if each pair ofneighbors is connected via possibly non-coordinatingnodes, to give chance to other nodes to become coordi-

nators). Since coordinators are not necessarily neigh-bors, three-hop neighboring topology knowledge is re-quired. However, the energy and bandwidth requiredfor maintenance of three-hop neighborhood informa-tion is not taken into account in experiments [88]. Onthe other hand, if the coordinators are restricted to beneighboring nodes, then the dominating set definition[88] becomes equivalent to one given by Wu and Li [58].Next, protocol [88] heavily relies on proactive periodicbeacons for synchronization, even if there is no pend-ing traffic or node movement. The recent research onenergy consumption [84] indicates that the use of suchperiodic beacons or hello messages is an energy expen-sive mechanism, because of significant start up cost forsending short messages. Finally, [87] observed that theoverhead required for coordination with SPAN tendsto explode with node density, and thus counterbalancesthe potential savings achieved by the increased density.

5 Conclusion

In this chapter, we reviewed several methods for effi-cient broadcasting for wireless ad hoc networks. Al-though the structures IMRG and LMSTk (k ≥ 2) hastotal edge length within a constant factor of the MST,the broadcasting based on these locally constructedstructures could still consumes energy arbitrarily largethan the optimum, when we assume that the powerneeded to support a link uv is ‖uv‖β . It has beenproved that the broadcasting based on MST consumesenergy within a constant factor of the optimum, butMST cannot be constructed locally. We also show byexample that there is no structure that can be con-structed locally and the broadcasting based on it con-sumes energy within a constant factor of the optimum.

In the minimum energy broadcasting problem,each node can adjust its transmission power in orderto minimize total energy consumption but still enablea message originated from a source node to reach allthe other nodes in an ad-hoc wireless network. Theproblem is known to be NP-complete [89]. There exista number of approximate solutions in literature whereeach node requires global network information (includ-ing distances between any two neighboring nodes inthe network) in order to decide its own transmissionradius. Three greedy heuristics were proposed in [29]for the minimum-energy broadcast routing problem:MST (minimum spanning tree), SPT (shortest-pathtree), and BIP (broadcasting incremental power). Itwas shown that the total energy consumed by MST orBIP methods are no more than 12 times larger thanthe optimum [35]. Cartigny, Simplot and Stojmenovic

17

[72] described a localized protocol where each node re-quires only the knowledge of its distance to all neigh-boring nodes and distances between its neighboringnodes (or, alternatively, geographic position of itselfand its neighboring nodes). In addition to using onlylocal information, the protocol is shown experimen-tally to even competitive with the best-known global-ized BIP solution [29], which a variation of Dijkstra’sshortest path algorithm. The solution [72] is based onthe use of RNG that preserves connectivity and is de-fined in localized manner. The transmission range foreach node is equal to the distance to its furthest RNGneighbor, excluding the neighbor from which the mes-sage came from. Localized energy efficient broadcastfor wireless networks with directional antennas are de-scribed in [91], and are also based on RNG. Messagesare sent only along RNG edges, requiring about 50%more energy than BIP based [29] globalized solution.However, when the communication overhead for main-tenance is added, localized solution becomes superior.Localized minimum spanning tree can replace RNGto improve energy efficiency, as proposed in [71, 92].Their simulations show the performance is compara-ble to that of BIP. Li et al. [21, 73] recently proposedseveral methods with further improvements. They de-scribed three low weight planar structures that can beconstructed by localized methods with total communi-cation costs O(n) and the simulations showed a signif-icant improvement of energy consumption comparedwith [91, 71].

References

[1] Deborah Estrin, Ramesh Govindan, John S. Hei-demann, and Satish Kumar, “Next century chal-lenges: Scalable coordination in sensor networks,”in Mobile Computing and Networking (Mobicom),1999, pp. 263–270.

[2] J. M. Kahn, R. H. Katz, and K. S. J. Pister, “Nextcentury challenges: Mobile networking for ”smartdust”,” in International Conference on MobileComputing and Networking (MOBICOM), 1999,pp. 271–278.

[3] Seapahn Megerian and Miodrag Potkonjak,“Wireless sensor networks,” Book Chapter in Wi-ley Encyclopedia of Telecommunications, Editor:John G. Proakis, 2002.

[4] G.J. Pottie and W.J. Kaiser, “Wireless integratednetwork sensors,” Communications of the ACM,vol. 43, no. 5, pp. 551–558, 2000.

[5] Chalermek Intanagonwiwat, Ramesh Govindan,and Deborah Estrin, “Directed diffusion: a scal-able and robust communication paradigm for sen-sor networks,” in Mobile Computing and Net-working (Mobicom), 2000, pp. 56–67.

[6] Seapahn Meguerdichian, Farinaz Koushanfar,Miodrag Potkonjak, and Mani Srivastava, “Cov-erage problems in wireless ad-hoc sensor net-work,” in IEEE INFOCOM’01, 2001, pp. 1380–1387.

[7] S. Capkun, M. Hamdi, and J.P. Hubaux, “Gps-free positioning in mobile ad-hoc networks,” inProc. Hawaii Int. Conf. on System Sciences, 2001.

[8] C. Chiang and M. Gerla, “demand multicast inmobile wireless networks,” 1998.

[9] Jorjeta G. Jetcheva, Yih-Chun Hu, David A.Maltz, and David B. Johnson, “A simple proto-col for multicast and broadcast in mobile ad hocnetworks,” 2001.

[10] M. Liu, R. Talpade, and A. McAuley, “Amroute:Adhoc multicast routing protocol,” 1999.

[11] Charles E. Perkins, Elizabeth M. Belding-Royer,and Samir Das, “Ad hoc on demand distancevector (aodv) routing,” 2002.

[12] Y.-C. Tseng, S.-Y. Ni, Y.-S. Chen, and J.-P. Sheu,“The broadcast storm problem in a mobile ad hocnetwork,” Wireless Networks, vol. 8, pp. 153–167,2002, Short version in MOBICOM 99.

[13] B. Williams and T. Camp, “Comparison of broad-casting techniques for mobile ad hoc networks,”in Proceedings of the ACM International Sympo-sium on Mobile Ad Hoc Networking and Comput-ing (MOBIHOC), 2002, pp. 194–205.

[14] C. W. Wu, Y. C. Tay, and C.-K Toh, “Ad hocmulticast routing protocol utilizing increasing id-numbers (amris) functional specification,” 1998.

[15] J. Wu and W. Lou, “Forward-node-set-basedbroadcast in clustered mobile ad hoc networks,”Wireless Communications and Mobile Comput-ing, 2003, 3, 2, 2003, 155-173.

[16] Khaled Alzoubi, Peng-Jun Wan, and OphirFrieder, “Message-optimal connected-dominating-set construction for routing inmobile ad hoc networks,” in 3rd ACM Interna-tional Symposium on Mobile Ad Hoc Networkingand Computing (MobiHoc’02), 2002.

18

[17] Khaled Alzoubi, Xiang-Yang Li, Yu Wang, Peng-Jun Wan, and Ophir Frieder, “Geometric span-ners for wireless ad hoc networks,” IEEE Trans-actions on Parallel and Distributed Processing,2002, To appear. Short version in IEEE ICDCS2002.

[18] Xiang-Yang Li, Peng-Jun Wan, and OphirFrieder, “Coverage in wireless ad-hoc sensor net-works,” IEEE Transactions on Computers, 2002,Short version in IEEE ICC 2002.

[19] Xiang-Yang Li, Gruia Calinescu, Peng-Jun Wan,and Yu Wang, “Localized delaunay triangulationwith application in wireless ad hoc networks,”IEEE Transaction on Parallel and DistributedProcessing, 2003, Short version appeared at IEEEINFOCOM, 2002.

[20] Yu Wang and Xiang-Yang Li, “Localized con-struction of bounded degree planar spanner forwireless networks,” in DialM-POMC 2003, Ac-cepted for publication, 2003.

[21] Xiang-Yang Li, “Approximate MST for UDG lo-cally,” in COCOON 2003, Big Sky, MT, 2003.

[22] Ivan Stojmenovic, “Degree limited RNG,” inWorkshop on Wireless Networks, Cocoyoc, Mex-ico, Feb. 25, 2003.

[23] Xiang-Yang Li, Ivan Stojmenovic, and YuWang, “Partial Delaunay Triangulation and De-gree Limited Localized Bluetooth Multihop Scat-ternet Formation,” in IEEE Transactions on Par-allel and Distributed Systems, to appear, 2003.

[24] Xiang-Yang Li, Wen Zhen Song, and Yu Wang,“Efficient topology control for wireless networkswith non-uniform transmission ranges”, ACMWireless Networks, Accepted for publication,2003.

[25] Wen-Zhan Song, Xiang-Yang Li, Yu Wang, andWeizhao Wang, “dbblue: Low diameter and self-routing bluetooth scatternet,” in DialM-POMC2003, Accepted for publication, 2003.

[26] A. Clementi, P. Crescenzi, P. Penna, G. Rossi,and P. Vocca, “On the complexity of comput-ing minimum energy consumption broadcast sub-graphs,” in 18th Annual Symposium on Theo-retical Aspects of Computer Science, LNCS 2010,2001, pp. 121–131.

[27] A. Clementi, P. Penna, and R. Silvestri, “Onthe power assignment problem in radio networks,”Electronic Colloquium on Computational Com-plexity, 2001, To approach. Preliminary resultsin APPROX’99 and STACS’2000.

[28] L. M. Kirousis, E. Kranakis, D. Krizanc, andA. Pelc, “Power consumption in packet radio net-works,” Theoretical Computer Science, vol. 243,pp. 289–305, 2000.

[29] J. Wieselthier, G. Nguyen, and A. Ephremides,“On the construction of energy-efficient broadcastand multicast trees in wireless networks,” in Proc.IEEE INFOCOM 2000, 2000, pp. 586–594.

[30] Weifa Liang, “Constructing minimum-energybroadcast trees in wireless ad hoc networks,” inProceedings of the ACM International Symposiumon Mobile Ad Hoc Networking and Computing(MOBIHOC), 2002, pp. 112–122.

[31] Arindam K. Das, Robert J. Marks, Mohamed El-Sharkawi, Payman Arabshahi, and Andrew Gray,“Minimum power broadcast trees for wireless net-works: An ant colony system approach,” in Proc.IEEE Int. Symp. on Circuits and Systems, 2002.

[32] Fulu Li and Ioanis Nikolaidis, “On minimum-energy broadcasting in all-wireless networks,” inProc. IEEE 26th Annual IEEE Conference on Lo-cal Computer Networks (LCN’01), 2001.

[33] R. J. Marks, A.K. Das, M. El-Sharkawi, P. Arab-shahi, and A. Gray, “Minimum power broadcasttrees for wireless networks: optimizing using theviability lemma,” in Proc. IEEE Int. Symp. onCircuits and Systems, 2002, pp. 245–248.

[34] Arindam K. Das, Robert J. Marks, Mohamed El-Sharkawi, Payman Arabshahi, and Andrew Gray,“Minimum power broadcast trees for wireless net-works: Integer programming formulations,” inProc. of IEEE INFOCOM 2003, 2003.

[35] Peng-Jun Wan, G. Calinescu, Xiang-Yang Li, andOphir Frieder, “Minimum-energy broadcast rout-ing in static ad hoc wireless networks,” ACMWireless Networks, 2002, Preliminary version ap-peared in IEEE INFOCOM 2000.

[36] V. Chvatal, “A greedy heuristic for the set-covering problem,” Mathematics of OperationsResearch, vol. 4, no. 3, pp. 233–235, 1979.

19

[37] T. J. Cormen, C. E. Leiserson, and R. L. Rivest,Introduction to Algorithms, MIT Press andMcGraw-Hill, 1990.

[38] M. R. Garey and D. S. Johnson, Computers andIntractability, W.H. Freeman and Co., NY, 1979.

[39] Sudipto Guha and Samir Khuller, “Approxima-tion algorithms for connected dominating sets,”in European Symposium on Algorithms, 1996, pp.179–193.

[40] G. Robins and A. Zelikovsky, “Improved steinertree approximation in graphs,” in Proceedings ofACM/SIAM Symposium on Discrete Algorithms,2000, pp. 770–779.

[41] Harry B. Hunt III, Madhav V. Marathe,Venkatesh Radhakrishnan, S. S. Ravi, Daniel J.Rosenkrantz, and RichardE. Stearns, “NC-approximation schemes for NP- and PSPACE -hard problems for geometric graphs,” J. Algo-rithms, vol. 26, no. 2, pp. 238–274, 1999.

[42] Madhav V. Marathe, Heinz Breu, Harry B. HuntIII, S. S. Ravi, and Daniel J. Rosenkrantz, “Sim-ple heuristics for unit disk graphs,” Networks, vol.25, pp. 59–68, 1995.

[43] Dorit S. Hochbaum and Wolfgang Maass, “Ap-proximation schemes for covering and packingproblems in image processing and vlsi,” Journalof ACM, vol. 32, pp. 130–136, 1985.

[44] Yu Wang and Xiang-Yang Li, “Geometric span-ners for wireless ad hoc networks,” in Proc.of 22nd IEEE International Conference on Dis-tributed Computing Systems (ICDCS), 2002.

[45] P. Berman, M. Furer, and A. Zelikovsky, “Appli-cations of matroid parity problem to approximat-ing steiner trees,” Tech. Rep. 980021, ComputerScience, UCLA, 1998.

[46] Khaled M. Alzoubi, Virtual Backbone in WirelessAd Hoc Networks, Ph.D. thesis, Illinois Instituteof Technology, 2002.

[47] Xiuzhen Cheng, “A polynomial time approxima-tion algorithm for connected dominating set prob-lem in wireless ad hoc networks,” 2002.

[48] Khaled M. Alzoubi, Peng-Jun Wan, and OphirFrieder, “New distributed algorithm for con-nected dominating set in wireless ad hoc net-works,” in HICSS, Hawaii, 2002.

[49] Chunhung Richard Lin and Mario Gerla, “Adap-tive clustering for mobile wireless networks,”IEEE Journal of Selected Areas in Communica-tions, vol. 15, no. 7, pp. 1265–1275, 1997.

[50] I. Chlamtac and A. Farago, “A new approach todesign and analysis of peer to peer mobile net-works,” Wireless Networks, vol. 5, pp. 149–156,1999.

[51] Alan D. Amis, Ravi Prakash, Dung Huynh, andThai Vuong, “Max-min d-cluster formation inwireless ad hoc networks,” in Proc. of theNineteenth Annual Joint Conference of the IEEEComputer and Communications Societies INFO-COM, 2000, vol. 1, pp. 32–41.

[52] Ji-Cherng Lin, Shi-Nine Yang, and Maw-ShengChern, “An efficient distributed algorithm forminimal connected dominating set problem,” inProc. of the Tenth Annual International PhoenixConference on Computers and Communications1991, 1991, pp. 204–210.

[53] Alan D. Amis and Ravi Prakash, “Load-balancingclusters in wireless ad hoc networks,” in Proc. ofthe 3rd IEEE Symposium on Application-SpecificSystems and Software Engineering Technology,2000.

[54] Lujun Jia, Rajmohan Rajaraman, and TorstenSuel, “An efficient distributed algorithm for con-structing small dominating sets,” in ACM PODC,2000.

[55] B. Das and V. Bharghavan, “Routing in ad-hocnetworks using minimum connected dominatingsets,” in 1997 IEEE International Conference onon Communications (ICC’97), 1997, vol. 1, pp.376–380.

[56] R. Sivakumar, B. Das, and V. Bharghavan, “Animproved spine-based infrastructure for routing inad hoc networks,” in IEEE Symposium on Com-puters and Communications, Athens, Greece,June 1998.

[57] Jie Wu and Hailan Li, “Domination and its ap-plications in ad hoc wireless networks with uni-directional links,” in Proc. of the InternationalConference on Parallel Processing 2000, 2000, pp.189–197.

[58] Jie Wu and Hailan Li, “A dominating-set-basedrouting scheme in ad hoc wireless networks,”the special issue on Wireless Networks in the

20

Telecommunication Systems Journal, vol. 3, pp.63–84, 2001.

[59] F. Dai and J. Wu, “Distributed Dominant Prun-ing in Ad Hoc Networks,” Proc. IEEE ICC, 2003.

[60] Ivan Stojmenovic, Mahtab Seddigh, and JovisaZunic, “Dominating sets and neighbor elimina-tion based broadcasting algorithms in wirelessnetworks,” IEEE Transactions on Parallel andDistributed Systems, vol. 13, no. 1, pp. 14–25,2002.

[61] F. Garcia, J. Solano, and I. Stojmenovic, “Con-nectivity based k-hop clustering in wireless net-works,” in Telecommunication Systems, 22, 1-4,205-220, 2003.

[62] S. Basagni, “Distributed clustering for ad hoc net-works,” in Proceedings of the IEEE InternationalSymposium on Parallel Architectures, Algorithms,and Networks (I-SPAN), 1999, pp. 310–315.

[63] S. Basagni, I. Chlamtac, and A. Farago, “A gen-eralized clustering algorithm for peer-to-peer net-works,” in Workshop on Algorithmic Aspects ofCommunication, 1997.

[64] Peng-Jun Wan, Khaled M. Alzoubi, and OphirFrieder, “Distributed construction of connecteddominating set in wireless ad hoc networks,” inINFOCOM, 2002.

[65] C. Santivanez, B. McDonald, I. Stavrakakis, andR. Ramanathan, “the scalability of ad hoc routingprotocols,” in Proc. of IEEE INFOCOM 2002,2002.

[66] Christopher Ho, Katia Obraczka, Gene Tsudik,and Kumar Viswanath, “Flooding for reliablemulticast in multi-hop ad hoc networks,” inProceedings of the 3rd International Workshopon Discrete Algorithms and Methods for MobileComputing and Communications, Seattle, WA,1999, pp. 64–71.

[67] Amir Qayyum, Laurent Viennot, and AnisLaouiti, “Multipoint relaying: An efficient tech-nique for flooding in mobile wireless networks,”Tech. Rep. Research Report RR-3898, INRIA,Feb. 2000, Conference version in Proc. of IEEEHICSS’35, 2001.

[68] Ashwinder Ahluwalia, Eytan Modiano, andLi Shu, “On the complexity and distributed con-struction of energy efficient broadcast trees in

static ad hoc wireless networks,” in Proceedings of36th Annual Conference on Information Sciencesand Systems (CISS), 2002.

[69] Mario Cagalj, Jean-Pierre Hubaux, and ChristianEnz, “Minimum-energy broadcast in all-wirelessnetworks: Np-completeness and distribution is-sues,” in Proceedings of the ACM MOBICOM02, 2002.

[70] J.E. Wieselthier, G.D. Nguyen, andA. Ephremides, “The energy efficiency ofdistributed algorithms for broadcasting in adhoc networks,” in Proceedings of IEEE 5thInternational Symposium on Wireless PersonalMultimedia Communication (WPMC), 2002, pp.499–503.

[71] Ning Li and Jennifer C. Hou, “Blmst: a scalable,power efficient broadcast algorithm for wirelesssensor networks,” 2003, Submitted for publica-tion, a version as UIUC Computer Science De-partment Technical Report.

[72] J. Cartigny, D. Simplot, and I. Stojmenvic, “Lo-calized minimum-energy broadcasting in ad-hocnetworks,” in Proc. of IEEE INFOCOM 2003,2003.

[73] Xiang-Yang Li, Yu Wang, Peng-Jun Wan, andOphir Frieder, “Localized low weight graph andits applications in wireless ad hoc networks,”2003, Submitted for publication.

[74] Gautam Das, Giri Narasimhan, and Jeffrey Sa-lowe, “A new way to weigh malnourished eu-clidean graphs,” in ACM Symposium of DiscreteAlgorithms, 1995, pp. 215–222.

[75] Ning Li, Jennifer C. Hou, and Lui Sha, “Designand analysis of a mst-based topology control algo-rithm,” in Proc. of IEEE INFOCOM 2003, 2003.

[76] Gruia Calinescu, “Computing 2-hop neigh-borhoods in ad hoc wireless networks,” 2002,Manuscript.

[77] Mahtab Seddigh, J. Solano Gonzalez, and I. Stoj-menovic, “Rng and internal node based broad-casting algorithms for wireless one-to-one net-works,” ACM Mobile Computing and Commu-nications Review, vol. 5, no. 7, pp. 37–44, 2002.

[78] Xiang-Yang Li, Kousha Moaveninejad, andYu Wang, “Low weighted structures and inter-nal node based broadcasting schemes for wirelessad hoc networks,” Manuscript, 2003.

21

[79] H. Lim and C. Kim, “Multicast tree construc-tion and flooding in wireless ad hoc networks,” inProceedings of the ACM International Workshopon Modeling, Analysis and Simulation of Wirelessand Mobile Systems (MSWIM), 2000.

[80] Gruia Calinescu, Ion Mandoiu, Peng-Jun Wan,and Alexander Zelikovsky, “Selecting forwardingneighbors in wireless ad hoc networks,” in ACMDialM, 2001.

[81] W. Peng and X. Lu, “On the reduction of broad-cast redundancy in mobile ad hoc networks,” inProceedings of MOBIHOC, Aug. 2000.

[82] I. Stojmenovic, M. Seddigh, “Broadcasting algo-rithms in wireless networks,” in Proc. Int. Conf.on Advances in Infrastructure for Electronic Busi-ness, Science, and Education on the Internet SS-GRR, L’Aquila, Italy, July 31-Aug. 6, 2000.

[83] J. Wu, B. Wu, and I. Stojmenovic, “Power-aware broadcasting and activity scheduling in adhoc wireless networks using connected dominat-ing sets,” Wireless Communications and MobileComputing, 3, 2003, 425-438.

[84] L.M. Feeney and M. Nilson, “Investigating theenergy consumption of a wireless network inter-face in an ad hoc networking environment,” inIEEE INFOCOM, 2001, pp. 1548–1557.

[85] J. Shaikh, J. Solano,I. Stojmenovic, and J. Wu,“New metrics for dominating set based energy ef-ficient activity scheduling in ad hoc networks,”IEEE Conf. on Local Computer Networks/WLN,Bonn, Germany, Oct. 20-24, 2003.

[86] Y. Xu, J. Heidemann, and D. Estrin,“Geography-informed energy conservationfor ad hoc networks,” in Proc. MobiCom, 2001.

[87] D.M. Blough and P. Santi, “Investigating up-per bounds on network lifetime extension for cell-based energy conservation techniques in station-ary ad hoc networks,” in MOBICOM, 2002.

[88] Benjie Chen, Kyle Jamieson, Hari Balakrishnan,and Robert Morris, “Span: An energy-efficientcoordination algorithm for topology maintenancein ad hoc wireless networks,” in Mobile Comput-ing and Networking, 2001, pp. 85–96.

[89] N. F. Huang and T.H. Huang, “On the complex-ity of some arborescences finding problems on amulti-hop radio network,” BIT, vol. 29, pp. 212–216, 1989.

[90] G. Toussaint, “The relative neighborhood graphof finite planar set,” Pattern Recognition, vol. 12,no. 4, pp. 261–268, 1980.

[91] J. Cartigny, D. Simplot, and I. Stojmenovic,“Localized energy efficient broadcast for wirelessnetworks with directional antennas,” in Proc.IFIP Mediterranean Ad Hoc Networking Work-shop (MED-HOC-NET 2002), Sardegna, Italy,2002.

[92] J. Cartigny, F. Ingelrest, D. Simplot-Ryl, I.Stojmenovic, “Localized LMST and RNG basedminimum energy broadcast protocols in ad hocnetworks,” submitted for publication.

[93] F. Ingelrest, D. Simplot-Ryl, I. Stojmen-ovic, “Target transmission radius over LMST forenergy-efficient broadcast protocol in ad hoc net-works,” submitted for publication.

[94] J. Cartigny, and D. Simplot, “Border node re-transmission based probabilistic broadcast proto-cols in ad hoc networks,” Telecommunication Sys-tems, 22, 1-4, 2003, 189-204.

[95] J. Lipman, P. Boustead, and J. Judge, “Efficientand scalable information dissemination in mobilead hoc networks,” in Proc. of the First Int. Conf.on Ad-hoc networks and wireless ADHOC-NOW,2002, pp. 119–134.

[96] J. Carle, D. Simplot, I. Stojmenovic, “Areacoverage based beaconless broadcasting in adhocnetworks,” in preparation.

22