Local broadcast algorithms in wireless ad hoc networks reducing the number of transmissions.bak

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Local broadcast algorithms in wireless ad hoc networks reducing the number of transmissions.bak

  1. 1. 402 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 3, MARCH 2012 Local Broadcast Algorithms in Wireless Ad Hoc Networks: Reducing the Number of Transmissions Majid Khabbazian, Member, IEEE, Ian F. Blake, Fellow, IEEE, and Vijay K. Bhargava, Fellow, IEEE Abstract—There are two main approaches, static and dynamic, to broadcast algorithms in wireless ad hoc networks. In the static approach, local algorithms determine the status (forwarding/nonforwarding) of each node proactively based on local topology information and a globally known priority function. In this paper, we first show that local broadcast algorithms based on the static approach cannot achieve a good approximation factor to the optimum solution (an NP-hard problem). However, we show that a constant approximation factor is achievable if (relative) position information is available. In the dynamic approach, local algorithms determine the status of each node “on-the-fly” based on local topology information and broadcast state information. Using the dynamic approach, it was recently shown that local broadcast algorithms can achieve a constant approximation factor to the optimum solution when (approximate) position information is available. However, using position information can simplify the problem. Also, in some applications it may not be practical to have position information. Therefore, we wish to know whether local broadcast algorithms based on the dynamic approach can achieve a constant approximation factor without using position information. We answer this question in the positive—we design a local broadcast algorithm in which the status of each node is decided “on-the-fly” and prove that the algorithm can achieve both full delivery and a constant approximation to the optimum solution. Index Terms—Mobile ad hoc networks, distributed algorithms, broadcasting, connected dominating set, constant approximation. Ç1 INTRODUCTIONW IRELESS ad hoc networks have emerged to support either in the set or has a neighbor in the set. A DS is called a http://ieeexploreprojects.blogspot.com applications, in which it is required/desired to have Connected Dominating Set (CDS) if the subgraph inducedwireless communications among a variety of devices by its nodes is connected. Clearly, the forwarding nodes,without relying on any infrastructure or central manage- together with the source node, form a CDS. On the otherment. In ad hoc networks, wireless devices, simply called hand, any CDS can be used for broadcasting a messagenodes, have limited transmission range. Therefore, each (only nodes in the set are required to forward). Therefore,node can directly communicate with only those within its the problems of finding the minimum number of requiredtransmission range (i.e., its neighbors) and requires other transmissions (or forwarding nodes) and finding a Mini-nodes to act as routers in order to communicate with out-of- mum Connected Dominating Set (MCDS) can be reduced torange destinations. each other. Unfortunately, finding a MCDS (and hence One of the fundamental operations in wireless ad hoc minimum number of forwarding nodes) was proven to benetworks is broadcasting, where a node disseminates a NP hard even when the whole network topology is knownmessage to all other nodes in the network. This can be [1], [2]. A desired objective of many efficient broadcast algorithms is to reduce the total number of transmissions toachieved through flooding, in which every node transmits preferably within a constant factor of its optimum. For localthe message after receiving it for the first time. However, algorithms and in the absence of global network topologyflooding can impose a large number of redundant transmis- information, this is commonly believed to be very difficultsions, which can result in significant waste of constrained or impossible [3], [4].resources such as bandwidth and power. In general, not The existing local broadcast algorithms can be classifiedevery node is required to forward/transmit the message in based on whether the forwarding nodes are determinedorder to deliver it to all nodes in the network. A set of nodes statically (based on only local topology information) orform a Dominating Set (DS) if every node in the network is dynamically (based on both local topology and broadcast state information) [5]. In the static approach, the distin-. M. Khabbazian is with the Department of Applied Computer Science, guishing feature of local algorithms over other broadcast University of Winnipeg, 515 Portage Avenue, Winnipeg, MB, Canada. algorithms is that using local algorithms any local topology E-mail: m.khabbazian@uwinnipeg.ca. changes can affect only the status of those nodes in the. I.F. Blake and V.K. Bhargava are with the Department of Electrical and Computer Engineering, University of British Columbia, 2332 Main Mall, vicinity. Therefore, local algorithms can provide scalability Kaiser 4112, Vancouver, BC V6T 1Z4, Canada. as the constructed CDS can be updated, efficiently. The E-mail: {ifblake, vijayb}@ece.ubc.ca. existing local algorithms in this category typically use aManuscript received 21 July 2009; revised 17 Dec. 2010; accepted 10 Jan. priority function known by all nodes in order to determine2011; published online 17 Mar. 2011. the status of each node [5]. In this paper we show that,For information on obtaining reprints of this article, please send e-mail to:tmc@computer.org, and reference IEEECS Log Number TMC-2009-07-0297. using only local topology information and a globally knownDigital Object Identifier no. 10.1109/TMC.2011.67. priority function, the local broadcast algorithms based on 1536-1233/12/$31.00 ß 2012 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS
  2. 2. KHABBAZIAN ET AL.: LOCAL BROADCAST ALGORITHMS IN WIRELESS AD HOC NETWORKS: REDUCING THE NUMBER OF... 403the static approach are not able to guarantee a good constant approximation to the optimum solution. Theapproximation factor to the optimum solution (i.e., MCDS). proposed algorithm is both neighbor-designating and self-On the other hand, we show that local algorithms based on pruning, i.e., the status of each node is determined by itselfthe static approach can achieve interesting results such as a and/or other nodes. In particular, using our proposedconstant approximation factor and shortest path preserva- algorithm, each broadcasting node selects at most one of itstion if the nodes are provided with position information. neighbors to forward the message. If a node is not selected In the dynamic approach, the status of each node (hence to forward, it has to decide, on its own, whether or not tothe CDS) is determined “on-the-fly” during the broadcast forward the message.progress. Using this approach, the constructed CDS may The rest of the paper is organized as follows: In Section 2,vary from one broadcast instance to another even when the we describe our system model and assumptions. In Sectionswhole network topology and the source node remain 3 and 4, we analyze the power of local broadcast algorithmsunchanged. Consequently, the broadcast algorithms based based on the static and dynamic approach, respectively. Inon the dynamic approach typically have small maintenance Section 5, we use simulation to confirm the analyticalcost and are expected to be robust against node failures and results presented in Section 4. Finally, we conclude thenetwork topology changes. Many local broadcast algo- paper in Section 6. Most of the proofs are placed in therithms in this category use local neighbor information to Appendix, which can be found on the Computer Societyreduce the total number of transmissions and to guarantee Digital Library at http://doi.ieeecomputersociety.org/full delivery (assuming no loss at the MAC/PHY layer). 10.1109/TMC.2011.67.Others, such as probability-based and counter-based algo-rithms [6], [7], [8], do not rely on neighbor information. 2 SYSTEM MODEL AND ASSUMPTIONSThese algorithms typically cannot guarantee full deliverybut eliminate the overhead imposed by broadcasting We assume that the network consists of a set of nodes V ,“Hello” messages or exchanging neighbor information. jV j ¼ N. Each node is equipped with omnidirectional Many of the existing neighbor-information-based broad- antennas. Every node u 2 V has a unique id, denoted idðuÞ, and every packet is stamped by the id of its sourcecast algorithms in this category can be further classified as node and a nonce, a randomly generated number by theneighbor-designating and self-pruning algorithms. In neigh- source node. For simplicity, we assume that all nodes arebor-designating algorithms [9], [10], [11], each forwarding located in two-dimensional space. However, all the resultsnode selects some of its local neighbors to forward the presented in this paper can be readily extended to three-message. Only the selected nodes are then required to dimensional ad hoc networks.forward the message in the next http://ieeexploreprojects.blogspot.com step. For example, a To model the network, we assume two different nodesforwarding node u may select a subset of its 1-hop neighbors u 2 V and v 2 V are connected by an edge if and only ifsuch that any 2-hop neighbor of u is a neighbor of at least juvj R, where juvj denotes the euclidean distance betweenone of the selected nodes [9]. In self-pruning algorithms [3], nodes u and v and R is the transmission range of the nodes.[12], [13], on the other hand, each node decides by itself Thus, we can represent the communication graph bywhether or not to forward a message. The decision is made GðV ; RÞ, where V is the set of nodes and R is the trans-based on a self-pruning condition. For example, a simple mission range. This model is, up to scaling, identical to theself-pruning condition employed in [12] is whether all unit disk graph model, which is a typical model for two-neighbors have been covered by previous transmissions. In dimensional ad hoc networks. In reality, however, theother words, a node can avoid forwarding/rebroadcasting a transmission range can be of arbitrary shape as the wirelessmessage if all of its neighbors have received the message by signal propagation can be affected by many unpredictableprevious transmissions. factors. Finally, we assume that the network is connected In [14], it was shown that neither neighbor-designating and static during the broadcast and that there is no loss atnor self-pruning algorithms can guarantee both full the MAC/PHY layer. These assumptions are necessary indelivery and a constant approximation if they use only 1- order to prove whether or not a broadcast algorithm canhop neighbor information and do not piggyback informa- guarantee full delivery. Note that without these assump-tion into the broadcast packets. The authors then proposed tions even flooding cannot guarantee full delivery.a self-pruning algorithm based on partial 2-hop neighborinformation and proved that the algorithm achieves aconstant approximation to the optimum solution and 3 BROADCASTING USING THE STATIC APPROACHguarantees full delivery. However, in their proposed Let the k-neighborhood of a node u, denoted Gk ðuÞ, be thealgorithm, each node was assumed to have its (approx- subgraph induced by u and nodes at most k hops awayimate) position information, which is not practical in some from u. Suppose each node is given a globally fixed priorityapplications/scenarios. Also, having position information function P rðidðwÞ; Gh0 ðwÞÞ which gets a node’s id, idðwÞ, andcan provide nontrivial information in wireless ad hoc its local topology information, Gh0 ðwÞ, as inputs and returnsnetworks and can greatly simplify the problem. As such, a real number that determines the priority of w. Forwe wish to know whether similar results can be obtained example, priority of a node can be determined by its id,without using position information. In this paper, we by its degree (i.e., the number of its 1-hop neighbors) or byanswer this remaining question in the positive—we its neighbor connectivity ratio (i.e., ratio of pairs ofpropose a local broadcast algorithm based on 2-hop neighbors that are not directly connected to all possibleneighbor information and prove that it guarantees a pairs of neighbors).
  3. 3. 404 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 3, MARCH 2012Fig. 1. Distributing nodes on a line segment with length 2ðh þ h0 Þ þ 1. In local broadcast algorithms based on the staticapproach, the status (forwarding/nonforwarding) of eachnode u, StatðuÞ, is a function of idðuÞ, Gh ðuÞ, and P rðidðvÞ;Gh0 ðvÞÞ, where v 2 Gh ðuÞ and the parameters h and h0 arefixed constant numbers.1 Note that the status of each nodedoes not depend on that of other nodes. Therefore, any localtopology change can only affect the status of the nodes inthe vicinity. In designing local broadcast algorithms, we arelooking for status functions that not only guarantee Fig. 2. Network partitioning and possible neighbors of a cell Ci for the pffifficonstructing a CDS (hence full delivery) but also ensure case ¼ 22 .that the constructed CDS has small size, preferably within a constant factor of the optimum. In the following, we show N À1 þ 2ðh þ h0 À 1Þ:that no such status function exists. The idea is to find a 2ðh þ h0 Þ þ 1graph in which any status function fails either in construct- Since h and h0 are constant, the approximation factor woulding a CDS or finding a CDS whose size is smaller than be at leastðNÞ, where N is the total number of nodes in the network.Our approach is to construct a graph with a large number of b2ðhþh0 Þþ1c þ 2ðh þ h0 À 1Þ NÀ1nodes for which both the local topology, Gh ð:Þ, and the 2 ðNÞ: 2  ð2ðh þ h0 Þ þ 1Þrelative priority of the nodes in Gh ð:Þ are the same. Without loss of generality, we can assume R ¼ 1. As Consequently, local broadcast algorithms based on theshown in Fig. 1, let us distribute N nodes on the x-axis static approach are not able to guarantee a good approx- http://ieeexploreprojects.blogspot.combetween the coordinates 0 and 2ðh þ h0 Þ þ 1 such that the imation factor in the worst case. Note that this result does ðiÀ1Þcoordinate of the ith node is ðNÀ1Þ Â ð2ðh þ h0 Þ þ 1Þ, where not imply that local broadcast algorithms cannot achieve a1 i N and N ) 2ðh þ h Þ þ 1. It is easy to see that good bound on average. 0Gh0 ðuÞ and Gh0 ðvÞ are isomorphic if u; v 2 ½h0 ; 2h þ h0 þ 1Š. 3.1 Using Position InformationBased on the definition of priority function, the relative In the context of broadcast algorithms based on the staticpriority of two nodes u and v only depends on their ids if approach, we may wish to know whether using positionGh0 ðuÞ and Gh0 ðvÞ are isomorphic. Therefore, we can information can help us to yield a small approximationdistribute all nodes in the interval ½h0 ; 2h þ h0 þ 1Š such that factor. In this section, we show that a constant approxima-their priorities increase as their coordinates (their distance tion factor is achievable if position information is available.to the origin) increase. Using this distribution, all nodes in To show this, we partition the network area into square cellsthe unit interval ½h0 þ h; h0 þ h þ 1Š will have similar views as illustrated in Fig. 2 and present simple local algorithmsof the local topology Gh ð:Þ and priority relationship that select a constant number of nodes in each cell.between the nodes in Gh ð:Þ. Therefore, the output of the The approach of network partitioning to use geographi-status function is the same for all nodes in this unit interval, cal information has been used for different purposesi.e., either all or none of the nodes in the unit interval will be including saving energy in sensor networks, and handlingselected. Clearly, the latter case is not possible since at least mobility in broadcast algorithms for ad hoc networks. In theone node in every unit interval has to be selected (otherwise primary work by Xu et al. [15], the authors propose anthe graph induced by the selected nodes will be discon- algorithm that partitions the network area into square cells.nected). On the other hand, selecting all nodes in the To save energy, at each time, the algorithm selects one nodeinterval ½h0 þ h; h0 þ h þ 1Š will result in a large set of in each cell as active and puts other nodes in the cell toselected nodes. Note that every MCDS consists of at most sleep. The set of active nodes must form a CDS in order totwo nodes in each unit interval. Therefore, for the simple maintain network connectivity and coverage. To guaranteenetwork shown in Fig. 1, we have jMCDSj 2  this, the proposed algorithm in [15] assumes that the sideð2ðh þ h0 Þ þ 1), where jMCDSj denotes the size of a MCDS.2 length of each square cell is pffiffi , where R is the radio R 5If all nodes in a unit interval are selected, the size of the transmission range. However, this is not sufficient toobtained CDS would be at least guarantee connectivity, because some square cells may not contain any node.3 In this section, we show under what 1. For the sake of generality, we consider two separate parameters hand h0 . 3. In [15], the authors implicitly assume that there exists at least one node 2. To be exact, jMCDSj ¼ dbNÀ1c l e , where l ¼ 2ðh þ h0 Þ þ 1. lÀ1 in each square cell. This assumption guarantees connectivity. l NÀ1
  4. 4. KHABBAZIAN ET AL.: LOCAL BROADCAST ALGORITHMS IN WIRELESS AD HOC NETWORKS: REDUCING THE NUMBER OF... 405conditions the set of nodes constructed by selecting one any of the two conditions that we call high-connectivitynode in each nonempty set form a CDS. We also show that condition and high-transmission condition.these conditions can be relaxed at the price of selecting We say a network satisfies the high-connectivity conditionmore than one yet a constant number of nodes in some if there exists a constant c, 0 c 1, such that the networknonempty cells. Therefore, the result of this section can be remains connected if the transmission ranges of all nodes isused to extend the work in [15] to the case where some cells reduced from R to cR. Dense networks typically satisfy the high-connectivity condition. For example, when the densitymay contain no (working) sensor due to, for example, a of nodes is high, it is expected that the network remainsnonuniform distribution of sensors, sensor mobility, or connected if all nodes reduce their transmission range to,sensor failure. say, 90 percent of the original. Notice that the high- In [16], authors use a network partitioning approach in connectivity condition does not require nodes to reducedesigning broadcast protocols for dense mobile ad hoc their transmission ranges. It just states that the networknetworks. To handle mobility, the proposed broadcast remains connected if the transmission range of nodes isprotocols in [16] rely on the distribution of nodes as reduced by a constant.opposed to the majority of existing broadcast protocols that The high-transmission condition is an alternative condi-rely on the frequently changing communication topology. tion that can guarantee the constructed DSsq ðÞ is a CDS.In this section, we focus on reducing the number of We say a network satisfies the high-transmission condition iftransmissions in a general static ad hoc network. Readers there exists a constant c, c 0, such that every selected nodemay refer to [16] for a discussion on how to handle mobility by the algorithm can increase its transmission range from Rwhen network partitioning is used. to at least ð1 þ cÞR by increasing its transmission power. In the following, we show that, if is selected carefully, thenTheorem 1. Let 0 and m ! 1 be constant numbers. Let S be high-connectivity and high-transmission conditions can a CDS. Suppose there are at most m nodes of S in any square guarantee that the constructed DSsq ðÞ (by, for example, of size R Â R. Then, we have the simple local algorithm explained above) is a CDS. The 2 ! following lemma is useful in proving these results. 2 jSj m jMCDSj; Lemma 1. Let be a positive real number. Let u 2 Ci and v 2 Cj be two nodes located in two different square cells Ci and Cj , where the size of each cell is R Â R. For any pair of nodes that is, the size of S is within a constant factor of jMCDSj. u0 2 Ci and v0 2 Cj we have For the proof, see the Appendix, which can be found in pffiffiffi http://ieeexploreprojects.blogspot.com0 j juvj þ 2 2R: ju0 vthe online supplemental material. As shown in Fig. 2, assume that all nodes in the network For the set of nodes V and any positive real number r, letare located in a square area of size L Â L. Considering GðV ; rÞ denote the graph constructed by connecting twoTheorem 1, it is natural to partition the network area into nodes in V if and only if their euclidean distance is at most r.small square cells of size R Â R, where is a positive real Using this notation, the high-connectivity condition impliesnumber, and search for algorithms that guarantee a that GðV ; cRÞ is connected for some constant c, 0 c 1.constant number of selected nodes in each cell. One simple The following theorem shows that, for some range of valuesalgorithm is to select exactly one node in each nonempty of , DSsq ðÞ is a CDS if the network satisfies the high-cell. A node can be selected using local neighbor informa- connectivity condition.tion as follows: Suppose each node knows the id and Theorem 2. Let , 0 1Àc , be a real number. Note that pffiffi pffiffi 2 2position of itself and its 1-hop neighbors. Also, assume that pffiffi 2 . Let DSsq ðÞ be a constructed DS by selecting one 2 2 2 . Therefore, the maximum distance between any two node in each nonempty set. If GðV ; cRÞ is connected, thenpoints in a cell is at most R, hence all the nodes in a cell are DSsq ðÞ is a CDS whose size is at most de2 jMCDSj. 21-hop neighbors of each other. Thus, a node can verifywhether or not it has the minimum id in the cell in which it For the proof, see the Appendix, which can be foundis located. Consequently, in every cell, using local neighbor in the online supplemental material.information, the node with minimum id can select itself. The next theorem proves that, for some range of , theNote that the set of selected nodes form a dominating set as constructed DSsq ðÞ is a CDS whose size is within aevery node in the network is either in the set or within the constant factor of jMCDSj if the selected nodes increasetransmission range of the selected node in its cell. their transmission range by a constant. Let be a positive real number. Let Ssq ðÞ denote a set Theorem 3. Let , 0 minð2;cÞ , and c 0 be real numbers. pffiffi pffiffi 2 2constructed by dividing the network area into small square 2 Note that 2 . Let DSsq ðÞ be a constructed DS by selectingcells of size R Â R (as shown in Fig. 2) and selecting one node in each nonempty set. If every node in DSsq ðÞexactly one node in each nonempty cell. As explained pffiffi 2 increases its transmission range to at least ð1 þ cÞR, thenearlier, the set Ssq ðÞ is a dominating set if 2 . DSsq ðÞ will be CDS whose size is at most de2 jMCDSj. 2Therefore, we pffiffi the notation DSsq ðÞ instead of Ssq ðÞ, use 2wherever 2 . In general, the graph induced by DSsq ðÞ For the proof, see the Appendix, which can be foundis not connected even for small values of . Nevertheless, it in the online supplemental material.can be proven that the graph induced by DSsq ðÞ is There are optimization techniques to further reduce theconnected (hence, DSsq ðÞ is a CDS) if the network satisfies size of a constructed DSsq ðÞ or to relax the requirement of
  5. 5. 406 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 3, MARCH 2012transmitting at higher power for many selected nodes. For For example, consider a network with N 3 nodes locatedexample, suppose that node us is the selected node in R on a circle with radius 2 sinð Þ such that the distance between Ncell Ci . The node us is not required to increase its any two neighbors is R. In the corresponding cycle graph,transmission power if every node v within transmission every MCDS is a simple path of length N À 3. Let s and d berange of any node u 2 Ci is within the transmission range of the nodes at the two ends of the path. Nodes s and d are onlyeither us or a selected neighbor of us . This condition can be 3 hops away from each other. However, lMCDS ðs; dÞ, theformally expressed as length of the shortest path between s and d in the graph induced by the nodes in MCDS, is N À 3. Thus 8u; v s:t: u 2 Ci ^ juvj R: ð1Þ 9ws 2 DSsq ðÞ s:t: jus ws j R ^ jws vj R: lMCDS ðs; dÞ N ¼ À 1: lðs; dÞ 3Both the high-connectivity and high-transmission condi-tions can be relaxed if we allow selecting more than one yet Consequently, even a MCDS cannot provide a gooda constant number of nodes in each nonempty cell. In the approximation of a shortest path in the original graph.following, we describe a simple algorithm that achieves a Theorem 4, on the other hand, shows that the length of aconstant approximation factor in any network GðV ; RÞ shortest path in CDSsq ðÞ constructed based on the high-without relying on any condition. transmission condition is at most one more than that of the Suppose pffiffi divide the network into square cells with pffiffi we original shortest path.size 22 R Â 22 R. All nodes in a cell are 1-hop neighbors of Theorem 4. For the CDS constructed based on the high-each other as the maximum distance between any two point transmission condition we havein the cell is R. Two different cells Ci and Cj are calledneighbors if and only if lCDS ðs; dÞ lðs; dÞ þ 1: 9u 2 Ci ; v 2 Cj s:t: juvj R: Employing a symmetric criteria to construct the CDS or using (1) will change the shortest path approximation toIn this case, nodes u and v are called connectors of theneighbor cells Ci and Cj , and u is referred to as a connector lCDS ðs; dÞ 2 Â lðs; dÞ þ 1:to the cell Cj through the node v. As shown in Fig. 2, eachcell is a neighbor ofpat most 20 other cells if the side of each ffiffi For the proof, see the Appendix, which can be foundsquare cell is set to 22 R. Assume that each node has a list of in the online supplemental material.its 2-hop neighbors together with their positions. Suppose u http://ieeexploreprojects.blogspot.comis a node located in the cell Ci . The node u selects itself as a 4 BROADCASTING USING THE DYNAMIC APPROACHmember of CDS if, based on a criteria, it is the selectedconnector to a neighboring cell Cj through a node v 2 Cj . Using the dynamic approach, the status (forwarding/The designed criteria must be symmetric in the sense that u nonforwarding) of each node is determined “on-the-fly”selects itself as a connector to the cell Cj through the node as the broadcasting message propagates in the network. Inv 2 Cj if and only if v selects itself as a connector to the cell particular, in neighbor-designating broadcast algorithms,Ci through the node u 2 Cj . As an example of a symmetric each forwarding node selects a subset of its neighbors tocriteria, a node u 2 Ci can select itself if and only if it has a forward the packet and in self-pruning algorithms eachneighbor v in a neighboring cell Cj such that juvj is node determines its own status based on a self-pruningminimum among all the possible connectors of the cells Ci condition after receiving the first or several copies of theand Cj . Any tie can be broken using, for example, nodes’ message. It was recently proved that self-pruning broadcastids. Note that node u has a list of its 2-hop neighbors (as algorithms (hence broadcast algorithms based on thewell as their positions) and therefore it can compute the set dynamic approach) are able to guarantee both full deliveryof all possible connectors between its own cell and any and a constant approximation factor to the optimumneighboring cell. By Theorem 1, the constructed set is a CDS solution (MCDS) [14]. However, the proposed algorithmwhose size is at most 20 Â de2 ¼ 180 times of its optimum. in [14] uses position information in order to design a strong 2 self-pruning condition. In the previous section, we observedNote that, to construct a smaller CDS, many of the selected that position information can simplify the problem ofnodes can be pruned using similar conditions as (1). An important application of constructing a CDS is to reducing the total number of broadcasting nodes. Moreover, having position information may not be practical in someemploy it as a backbone for routing. When a CDS is applications. Therefore, it is interesting to know if both fullconstructed, only the nodes in the set are required to delivery and a constant approximation factor can beforward packets toward the destination. Therefore, each achieved when position information is not available. In thispath between the source node s and destination node d can section, we design a hybrid (i.e., both neighbor-designatingbe represented as and self-pruning) broadcast algorithm and show that the s; w1 ; w2 ; . . . ; wk ; d; algorithm can achieve both full delivery and constant approximation only using connectivity information.where wi are in the CDS. Let lðs; dÞ and lCDS ðs; dÞ denote thelength of shortest paths between s and d in the original graph 4.1 The Proposed Local Broadcast Algorithmand the graph induced by CDS [ fs; dg, respectively. In Suppose each node has a list of its 2-hop neighbors (i.e.,general, the ratio lCDS ðs;dÞ can be very large, in the worst case. nodes that are at most 2 hops away). This can be achieved in lðs;dÞ
  6. 6. KHABBAZIAN ET AL.: LOCAL BROADCAST ALGORITHMS IN WIRELESS AD HOC NETWORKS: REDUCING THE NUMBER OF... 407two rounds of information exchange. In the first round, not place the packet in the queue if a copy of it is already ineach node broadcasts its id to its 1-hop neighbors (simply the queue. The second function is called to remove a packetcalled neighbors). Thus, at the end of the first round, each form the queue (it does not do anything if the packet is notnode has a list of its neighbors. In the second round, each in the queue). Note that to remove a packet from the queue,node transmits its id together with the list of its neighbors. the algorithm needs to have access to the MAC layer queue. The proposed broadcast algorithm is a hybrid algorithm, This requires a cross-layer design. In the absence of anyhence every node that broadcasts the message may select cross-layer design, the broadcast algorithm can use a timersome of its neighbors to forward the message. In our at the network layer. We refer the reader to [17] (Section 4)proposed broadcast algorithm, every broadcasting node for a discussion on how to use a timer to avoid this cross-selects at most one of its neighbors. A node has to broadcast layer problem.the message if it is selected to forward. Other nodes that arenot selected have to decide whether or not to broadcast on Algorithm 1. The proposed hybrid algorithm executed by utheir own. This decision is made based on a self-pruning 1: Extract ids of the broadcasting node and the selectedcondition called the coverage condition. node from the received message m To evaluate the coverage condition, every node u 2: if u has broadcast the message m before thenmaintains a list Listcov ðmÞ for every unique message m. u 3: Discard the messageUpon receiving a message m for the first time, Listcov ðmÞ is u 4: Returncreated and filled with the ids of all neighbors of u and then 5: end ifupdated as follows: Suppose u receives m from its neighbor 6: if u receives m for the first time thenv and assume that v selects w 6¼ u to forward the message. 7: Create and fill the list Listcov ðmÞ uNote that w may not be a neighbor of u. However, since w is 8: end ifa neighbor of v, it is at most 2 hops away from u. Having id’s 9: Update the list Listcov ðmÞ uof v and w (included in the message), node u updates 10: Remove the information added to the message by theListcov ðmÞ by removing all nodes in Listcov ðmÞ that are a u u previous broadcasting nodeneighbor of either v or w. This update can be done because u 11: if Listcov ðmÞ 6¼ ; then uhas a list of its 2-hop neighbors. Since w will eventually 12: Select an id from Listcov ðmÞ and add it to the ubroadcast the message, by updating the list, u removes messagethose neighbors that have received the message or will 13: Schedule the message {(*only update the selected idreceive it, eventually. Every time u receives a copy of if m is already in the queue*)}message m it updates Listcov ðmÞ as explained earlier. If w ¼ 14: else {(ÃListcov ðmÞ ¼ ; in this case*)} u http://ieeexploreprojects.blogspot.com uu (i.e., u is selected by v to forward the message), node u 15: if u was selected thenupdates Listcov ðmÞ by removing only neighbors of v from 16: u Schedule the message {(*only remove the id of thethe list. Note that in this case, u must broadcast the message. selected neighbor if m is already in the queue*)}However, u has to update Listcov ðmÞ as it needs to select 17: u elseone of its neighbors from the updated list (if it is not empty) 18: Remove the message form the queue if u has notto forward the message. been selected by any node beforeDefinition 1 (coverage condition). We say the coverage 19: end if condition for node u is satisfied at time t if Listcov ðmÞ ¼ ; 20: end if u at time t. The proposed algorithm obeys the following: Algorithm 1 shows our proposed hybrid broadcast 1. u discards a received message m if it has broadcastalgorithm. When a node u receives a message m, it creates m before.a list Listcov ðmÞ if it is not created yet and updates the list as u 2. If u is selected to forward the message, it schedules aexplained earlier. Then, based on whether u was selected to broadcast (regardless of the coverage condition) andforward or whether the coverage condition is satisfied, u never removes the messages from the queue inmay schedule a broadcast by placing a copy of m in its future. However, u may change or remove theMAC layer queue. There are at least two sources of delay in selected node’s id from the scheduled message everythe MAC layer. First, a message may not be at the head of time it receives a new copy of the message andthe queue so it has to wait for other packets to be updates Listcov ðmÞ. utransmitted. Second, in contention based channel access 3. Suppose u has not been selected to forward themechanisms such as CSMA/CA, to avoid collision, a packet message by time t and the Listcov ðmÞ becomes empty uat the head of the queue has to wait for a random amount of at time t after an update. Then at time t, it removes thetime before getting transmitted. In this paper, we assume message from the MAC layer queue (if the messagethat a packet can be removed from the MAC layer queue if has been scheduled before and is still in the queue).it is no longer required to be transmitted. Therefore, the 4. If Listcov ðmÞ 6¼ ; then u selects a node from ubroadcast algorithm has access to two functions to manip- Listcov ðmÞ 6¼ ; to forward the message and adds the uulate the MAC layer queue. The first function is the id of the selected node in the message. The selectionscheduling/placing function, which is used to place a can be done randomly or based on a criteria. Formessage in the MAC layer queue. We assume that the example, u can select the node with the minimum idscheduling function handles duplicate packets, i.e., it does or the one with maximum battery life-time.
  7. 7. 408 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 3, MARCH 2012 5. If u has been selected to forward and Listcov ðmÞ ¼ ; it u resides in DO;2R , that is the disk centered at O with radius does not select any node to forward the message. This 2R. It is because, the center of every Disk Di , 1 i k À 1, is the only case where a broadcasting node does not is inside DO;3R . Thus, by an area argument, we get 2 select any of its neighbors to forward the message. 2 ! R4.2 Analysis of the Proposed Broadcast Algorithm ðk À 1Þ ð2RÞ2 : 2In this section, we prove that the proposed broadcastalgorithm guarantees full delivery as well as a constant Hence, k 17.approximation to the optimum solution irrespective of the We first prove that the number of broadcasting nodesforwarding node selection criteria and the random delay in inside DO;R for which the coverage condition is not 2the MAC layer (hence, the random sequence of the satisfied is at most k À 1. We then prove the same upperbroadcasting nodes). In order to prove these properties, bound for the number of broadcasting nodes inside DO;R 2we assume that nodes are static during the broadcast, the for which the coverage condition is satisfied. Conse-network is connected and there is no loss at the MAC/PHY quently, the total number of broadcasting nodes insidelayer. Note that even flooding cannot guarantee full DO;R is bounded by 2k À 2 32. By contradiction, 2delivery without these assumptions. suppose that there are more than k À 1 broadcastingTheorem 5. Algorithm 1 guarantees full delivery. nodes inside DO;R for which the coverage condition is not 2 satisfied. Let u1 ; . . . ; uk be the first k broadcasting nodesProof. Every node broadcasts a message at most once. ordered chronologically based on their broadcast time, Therefore, the broadcast process eventually terminates. and a1 ; . . . ; ak the corresponding selected neighbor. Thus, By contradiction, assume that node d has not received for every i, 1 i k, we have ai 2 Listcov ðmÞ, where ui the message by the broadcast termination. Since the Listcov ðmÞ is the list of node ui at the time it broadcasts ui network is connected, there is a path from the source the message. Since u1 ; . . . ; uk are all in DO;R and for every node s (the node that initiates the broadcast) to node d. 2 i, 1 i k, jui ai j R, we get Clearly, we can find two nodes u and v on this path such that u and v are neighbors, u has received the message 8i; 1 i k: ai 2 DO;3R : 2 and v has not received it. The node u has not broadcast the message since v has not received it. Therefore, u has Thus, by the definition of k, there are two nodes ai , aj , i j not been selected to broadcast; thus, the coverage such that jai aj j R. The node ui is broadcast before uj and condition must have been satisfied for u. As the result, is a neighbor of it. Therefore, uj is aware of ui ’s selected http://ieeexploreprojects.blogspot.com v must have a neighbor w, which has broadcast the neighbor ai and removes aj from Listcov ðmÞ as soon as it uj message or was selected to broadcast. Note that all the receives the message from ui . This is a contradiction selected nodes will eventually broadcast the message. because aj 2 Listcov ðmÞ at the time uj broadcasts. uj This is a contradiction as, based on the assumption, v It remains to prove that the number of broadcasting cannot have a broadcasting neighbor. u t nodes inside DO;R for which the coverage condition is 2Lemma 2. Using Algorithm 1, the number of broadcasting nodes satisfied is at most k À 1. By contradiction, suppose that inside any disk DO;R centered at an arbitrary point O and 2 there are at least k broadcasting nodes inside DO;R for 2 with a radius R is at most 32. 2 which the coverage condition is satisfied. Let v1 ; . . . ; vk 2Proof. All nodes inside DO;R are neighbors of each other, DO;R be the first k broadcasting nodes, ordered 2 2 thus they receive each others messages. The broadcasting chronologically based on their broadcast time. Note nodes can be divided into two types based on whether or that a broadcasting node must have been selected (by not the coverage condition was satisfied for them just another node) to forward the message if its coverage before they broadcast the message. Recall that the condition is satisfied. Let b1 ; b2 ; . . . ; bk be the nodes that coverage condition may be satisfied for a broadcasting selected v1 ; . . . ; vk to forward the message. Clearly, for node if the node has been selected to forward the every i, 1 i k, we have bi 2 DO;3R . Also, for every i, 2 message. It is because a selected node has to broadcast 1 i k and every j, 1 j k and j 6¼ i, we get bi 6¼ bj , the message irrespective of the coverage condition. because each node can select at most one other node to Consider two disks centered at O with radii R and 3R , 2 2 forward. By the definition of k, there must exist two respectively. Suppose k is the minimum number such nodes bi and bj , i j such that jbi bj j R. This is a that for every set of k nodes wi 2 DO;3R , 1 i k, we have 2 contradiction because bi and bj are neighbors and bj 9i; j 6¼ i: jwi wj j R: receives the bi broadcast message, thus vj 62 Listcov ðmÞ bj as vi and vj are neighbors. u t Following, we find an upper bound on k. By the Corollary 1. Let u be any node in the network. Using minimality of k, there must exist k À 1 nodes wi 2 DO;3R , 2 Algorithm 1, the number of broadcasting nodes within the 1 i k À 1, such that transmission range of u is at most 224. 8i; j 6¼ i: jwi wj j R: ð2Þ Proof. All the nodes within the transmission range of u Consider k À 1 disks D1 ; . . . ; DkÀ1 with radius R centered (including u) are inside a disk with radius R. A disk with 2 at wi , 1 i k À 1, respectively. By (2), D1 ; . . . ; DkÀ1 are radius R can be covered with at most seven disks with nonoverlapping disks. Also, every disk Di , 1 i k À 1, radius R [18]. Thus, by Lemma 2, the number of 2
  8. 8. KHABBAZIAN ET AL.: LOCAL BROADCAST ALGORITHMS IN WIRELESS AD HOC NETWORKS: REDUCING THE NUMBER OF... 409 broadcasting nodes within the transmission range of u is Liststr ðmÞ and selected nodes can avoid broadcasting under u at most 7 Â 32 ¼ 224. u t the strong coverage condition. Alg-str guarantees full delivery.Theorem 6. Algorithm 1 has a constant approximation factor to the optimal solution (MCDS). Moreover, the approximation For the proof, see the Appendix, which can be found in factor is at most 224. the online supplemental material.Proof. Let SMCDS be a MCDS and SAlg be the set of 4.4 Extending the Network Model broadcasting nodes using Algorithm 1. Let u be any node The results presented in the paper can be extended to the in SMCDS . By Corollary 1, the number of broadcasting case where the nodes are distributed in three-dimensional nodes within the transmission range of u is at most 224. space. In other words, when the nodes are distributed in Note that every broadcasting node is within the three-dimensions it can be shown that local broadcast transmission range of at least one node in SMCDS , algorithms based on the static approach can provide a because SMCDS is a dominating set. Therefore constant approximation if nodes have their position information. By replacing circles with balls, it can be jSAlg j 224 Â jSMCDS j: similarly shown that Algorithm 1 can provide both full t u delivery and a constant approximation to the optimum solution.4.3 The Strong Coverage Condition Algorithm 1 (the proposed algorithm based on theAs proven, the proposed broadcast algorithm guarantees dynamic approach) can be extended to the case where thethat the total number of transmissions is always within a nodes have different transmission ranges. In this case, itconstant factor of the minimum number of required ones. can be proved that the algorithm guarantees a constant RMaxHowever, the number of transmissions may be further approximation factor if the ratio RMin is constant, wherereduced by slightly modifying the broadcast algorithm. As RMax is the maximum transmission range and RMin is theexplained earlier, in the proposed algorithm, a selected minimum transmission range of the nodes in the networknode has to broadcast the message even if its coverage and two nodes have a link iff both are in transmissioncondition is satisfied. Nevertheless, in some cases, a selected range of each other. Similar to the proof of Theorem 6, this can be proved by showing that the number of broadcastingnode can avoid broadcasting. For example, a selected node nodes inside any disk DO;RMin is constant. Also, we can useu can abort transmission (by removing the message from 2 the quasi unit disk graph to model the network [19]. In thisthe queue) at time t if by time t and based on its collected model, there is a link between two nodes if their euclideaninformation, all its neighbors have http://ieeexploreprojects.blogspot.com received the message. distance is less than R, 0 1, and there is no link ifThis idea can be generalized as follows: the euclidean distance is more than R. This model is closer Suppose, for each unique message m, every node u to reality than the unit disk graph model. Using the quasimaintains and updates an extra list Liststr ðmÞ. Similar to unit disk graphs model we can show that Algorithm 1 uListcov ðmÞ, Liststr ðmÞ is created and filled with the ids of u’s guarantees a constant approximation factor if is constant. u uneighbors upon the first reception of message m. Also, Similarly, the proof is by showing that the number ofevery time u receives m, it updates Liststr ðmÞ as follows: Let broadcasting nodes in any disk D R is constant. u O; 2v be the broadcasting node and w 6¼ u the selected node by strv. Node u first removes the nodes in Listu ðmÞ that areneighbors of v. If the priority of w (e.g., its id) is higher than 5 EXPERIMENTAL RESULTSu, it also removes the nodes in Liststr ðmÞ that are neighbors One of the major contributions of this work is the design of uof w. To further reduce the number of redundant transmis- a local broadcast algorithm based on the dynamic approachsions, a selected node can abort broadcasting m under the (Algorithm 1) that can achieve both full delivery and afollowing strong coverage condition. constant approximation factor to the optimum solution without using position information. To confirm the analy-Definition 2 (strong coverage condition). We say the strong tical results, we implemented Algorithm 1, Liu et al.’s coverage condition is satisfied for node u at time t if algorithm (a neighbor-designating algorithm) [9], edge Liststr ðmÞ ¼ ; at time t. u forwarding algorithm (a self-pruning algorithm) [20] and flooding in the network simulator ns-2 and evaluated the Note that the strong coverage condition is only used by ratio of broadcasting nodes (i.e., number of broadcastingselected nodes to check whether they need to broadcast. nodes/total number of nodes) and end-to-end delay forOther nodes make a decision based on the previously- each algorithm. Table 1 summarizes the parameters used indefined coverage condition (a weaker condition). The the ns-2 simulator. We also implemented the Wan-Alzoubi-following theorem states that the full delivery is guaranteed Frieder algorithm [21] in C++ and used it as an approxima-if the selected nodes abort transmissions when the strong tion of the minimum number of broadcasting nodescoverage condition is satisfied. Using a similar approach to required. Note that the Wan-Alzoubi-Frieder algorithmthat used in the proof of Lemma 2, it can be proven that this (referred to as ratio-6 approximation algorithm) is not aextension of the algorithm also achieves a constant local algorithm and is only used as a benchmark as it has anapproximation factor. approximation factor of at most 6 [22].Theorem 7. Suppose Alg-str is a modified version of Algorithm 1 Both Liu et al.’s broadcast algorithm and the edge- in which each node maintains two lists Listcov ðmÞ and forwarding algorithm use position information of nodes to u
  9. 9. 410 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 3, MARCH 2012 TABLE 1 Simulation Parametersreduce the total number of transmissions. Liu et al. showedthat the number of broadcasting nodes using their algo-rithm is significantly lower that that of previous notablebroadcast algorithms [20], [23]. In Liu’s algorithm, eachbroadcasting node selects a smallest group of its 1-hopneighbors such that any potential 2-hop neighbor is withintransmission range of at least one of the selected nodes. It Fig. 4. Ratio of broadcasting nodes versus transmission range.then piggybacks the list of the selected nodes in themessage before broadcasting it. Upon receiving a message (i.e., the total number of nodes or the transmission range).for the first time, a node schedules a broadcast if and only if To get the results shown in Fig. 3, we set the transmissionit is selected to forward; otherwise, it never broadcasts the range to 250 m and varied the total number of nodes frommessage. Using the edge-forwarding algorithm, each node 25 to 1,000. In Fig. 4, the number of nodes was fixed to 1,000divides its transmission coverage into six equal size sectors. and the transmission range was varied from 50 to 300 m.It then decides whether or not to broadcast based on The transmission range and the total number of nodes werethe sector in which the broadcasting node is located, and selected from a large interval so that the simulation coversthe position of its 1-hop neighbors that have received the very sparse and very dense networks as well as themessage. http://ieeexploreprojects.blogspot.comdiameters. Both Figs. 3 and 4 show networks with large To compute the number of broadcasting nodes, we that the ratio of broadcasting nodes using Algorithm 1 isuniformly distributed the nodes in a square of size significantly lower than that of Liu et al.’s algorithm and the1;000 Â 1;000 m2 . We allowed only one network-wide edge-forwarding algorithm and is very close to thebroadcast at each simulation run, selected the next for- estimated minimum number of required transmissionswarding node randomly, and used the strong coverage (computed using the Wan-Alzoubi-Frieder algorithm).condition in Algorithm 1 to further reduce the total number We computed another metric called the algorithm delay,of transmissions. which we define as the time between the first and the last Figs. 3 and 4 show the average ratio of broadcasting transmission in a single network-wide broadcast. Fig. 5,nodes for over 500 runs for each given value of the input shows the average delay of our proposed algorithm, Liu et al.’s algorithm, and the flooding algorithm. To get theseFig. 3. Ratio of broadcasting nodes versus total number of nodes. Fig. 5. Average delay versus total number of nodes.
  10. 10. KHABBAZIAN ET AL.: LOCAL BROADCAST ALGORITHMS IN WIRELESS AD HOC NETWORKS: REDUCING THE NUMBER OF... 411Fig. 6. Ratio of broadcasting nodes versus maximum speed. Fig. 7. Ratio of broadcasting nodes versus maximum speed.results, we fixed the transmission range to 250 m and varied Therefore, they do not take into consideration the overheadthe number of nodes from 25 to 1,000. Based on the of the 2-hop neighbor discovery messages. However, in thissimulation results, the average delay of our broadcast setting, nodes exchange Hello messages during the wholealgorithm is 86 to 54 percent of that of Liu et al.’s algorithm simulation run in order to keep the list of neighbors up-to-and 87 to 71 percent of that of the flooding algorithm. date. Fig. 6 shows the effect of mobility on the number of Since Algorithm 1 requires nodes to collect their 2-hop transmissions. The x-axis is the maximum speed set in theneighbor information, it is mainly suitable for networks in random waypoint mobility model and the y-axis is thewhich nodes have low mobility. The network-wide broad- average ratio of broadcasting nodes for at least 500 runs. Ascast may take from a fraction of second to a few seconds shown in Fig. 6, the number of transmissions slightly http://ieeexploreprojects.blogspot.comdepending on the MAC layer settings, the size of the decreases as mobility increases. Fig. 7, shows the averagenetwork, and the amount of contention. Therefore, if the delay of Algorithm 1 decreases as mobility increases.average distance that nodes move during the network- We also evaluated the performance of Algorithm 1 inwide broadcast is a small fraction of R, then we expect denser networks. Fig. 8 shows the average number ofAlgorithm 1 to achieve similar results as in the static transmissions when the total number of nodes varies fromnetworks. Otherwise, in networks with high-mobility rates, 50 to 1,000. To obtain the results shown in Fig. 8, we set theAlgorithm 1 cannot use the benefit of selecting nodes to maximum speed to 10 m=s and fixed the hello messageforward the message. It is because, a node does not know at transmission rate to one per second. The simulation resultswhat time a selected node will transmit the message if the show that for low-mobility rates, the performance (in termsselected node is not within its transmission range. There- of the number of transmissions) of both Algorithm 1 andfore, in high-mobility settings, a node may not knowwhether any of its neighbors is covered by the transmissionof a selected node as it does not know the list of 1-hopneighbors of the selected node at the time the selected nodetransmits the message. If nodes do not select other nodes toforward, then Algorithm 1 becomes similar to the proposedalgorithm in [12]. The proposed algorithm in [12] canprovide full delivery in static networks, but it cannotguarantee a constant approximation factor. To evaluate theimpact of low mobility on the number of transmissionsrequired by Algorithm 1, we performed a simulationexperiment. As before, we distributed the nodes in a squareof size 1;000 Â 1;000 m2 and allowed only one network-widebroadcast at each simulation run. We set the transmissionrange to 250 m and the number of nodes to 50. At thebeginning of each run, nodes start exchanging Hellomessages. After a period of time, a random node initiatesa broadcast. During the whole simulation run, nodes moveaccording to the random waypoint mobility model [24]. Inour previous simulations experiments, nodes do not Fig. 8. Ratio of broadcasting nodes versus total number of nodes; max.exchange Hello messages after the broadcast initiation. speed ¼ 10 m/s.
  11. 11. 412 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 3, MARCH 2012 Fig. 11. An instance of using Liu et al.’s broadcast algorithm;Fig. 9. Algorithm’s delay versus maximum back-off delay. Transmission range ¼ 250 m, #nodes ¼ 400.Liu et al.’s algorithm are very close to the case where the Finally, Figs. 10 and 11, respectively, show an instancenetwork is static. Also, for both algorithms, the average of using our proposed algorithm and Liu et al.’s algorithmnumber of nodes that do not receive a message in the for the same network where the total number of nodes isaforementioned settings is less than 2. 400 and the transmission range is set to 250 m (broad- As mentioned in Section 4, Algorithm 1 can use a back- casting nodes are shown by stars). As proven in [14], inoff timer in the network layer in the absence of a cross layerdesign. Each time a node receives a message, it sets the neighbor-designating algorithms based on 1-hop neighbortimer with a random number and decides upon timer information (and in Liu et al.’s algorithm, in particular)expiration whether to retransmit. Larger back-off delays most of the nodes around the boundary of the network http://ieeexploreprojects.blogspot.comdecrease the probability of collisions and reduce the broadcast the message. This undesirable property of Liunumber of transmissions as nodes can hear more transmis- et al.’s algorithm can be observed in Fig. 11. Note thatsions, hence make better decisions on whether or not to these snapshots of Algorithm 1 and Liu et al.’s algorithmretransmit. However, as shown in Fig. 9, algorithm’s end- are not necessarily representative of the overall perfor-to-end delay linearly increases with the maximum duration mance of the two algorithms.of back-off timer. To obtain the results shown in Fig. 9, weset the transmission range to 250 m and considered the 6 CONCLUSIONSnetwork static. The x-axis represents the maximum back-offdelay in seconds and the y-axis represents the delay of In this paper, we investigated capabilities of local broadcastAlgorithm 1 in seconds. algorithms in reducing the total number of transmissions that are required to achieve full delivery. As proven, local broadcast algorithms based on the static approach cannot guarantee a small sized CDS if the position information is not available. It was shown that having relative position information can greatly simplify the problem of reducing the total number of selected nodes using the static approach. In fact, we showed that a constant approximation factor is achievable using position information. Using the dynamic approach, it was recently shown that a constant approximation is possible using (approximate) position information [14]. In this paper, we showed that local broadcast algorithms based on the dynamic approach do not require position information to guarantee a constant approximation factor. The results presented in the paper can be extended to the case where nodes are distributed in three-dimensional space. Also, the proposed algorithm based on the dynamic approach can be extended to the case where nodes have different transmission ranges orFig. 10. An instance of using our broadcast algorithm; Transmission when the network is modeled using the quasi unit diskrange ¼ 250 m, #nodes ¼ 400. graph model.
  12. 12. KHABBAZIAN ET AL.: LOCAL BROADCAST ALGORITHMS IN WIRELESS AD HOC NETWORKS: REDUCING THE NUMBER OF... 413REFERENCES Majid Khabbazian received the undergraduate degree in computer engineering from the Sharif[1] M. Garey and D. Johnson, Computers and Intractability: A Guide to University of Technology, Iran, and the master’s the Theory of NP-Completeness. W.H. Freeman Co., 1990. and PhD degrees both in electrical and compu-[2] B. Clark, C. Colbourn, and D. Johnson, “Unit Disk Graphs,” ter engineering from the University of Victoria Discrete Math., vol. 86, pp. 165-177, 1990. and the University of British Columbia, Canada,[3] J. Wu and F. Dai, “Broadcasting in Ad Hoc Networks Based on respectively. From 2009 to 2010, he was a Self-Pruning,” Proc. IEEE INFOCOM, pp. 2240-2250, 2003. research fellow at the Computer Science and[4] J. Wu and W. Lou, “Forward-Node-Set-Based Broadcast in Artificial Intelligence Lab, MIT. 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Wu and H. Li, “On Calculating Connected Dominating Set for Efficient Routing in Ad Hoc Wireless Networks,” Proc. Int’l Workshop Discrete Algorithms and Methods for Mobile Computing and Comm. (DiaLM ’99), pp. 7-14, 1999.[24] T. Camp, J. Boleng, and V. Davies, “A Survey of Mobility Models for Ad Hoc Network Research,” Wireless Comm. and Mobile Computing (WCMC ’02), vol. 2, pp. 483-502, 2002.

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