Correct-basepaper-efficient Broadcasting in Mobile Ad Hoc Networks

7/31/2019 Correct-basepaper-efficient Broadcasting in Mobile Ad Hoc Networks

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Efficient Broadcasting in MobileAd Hoc Networks

Majid Khabbazian, Student Member, IEEE, and Vijay K. Bhargava, Fellow, IEEE

AbstractThis paper presents two efficient broadcasting algorithms based on 1-hop neighbor information. In the first part of the

paper, we consider sender-based broadcasting algorithms, specifically the algorithm proposed by Liu et al. In their paper,

Liu et al. proposed a sender-based broadcasting algorithm that can achieve local optimality by selecting the minimum number of

forwarding nodes in the lowest computational time complexity On log n, where n is the number of neighbors. We show that this

optimality only holds for a subclass of sender-based algorithms. We propose an efficient sender-based broadcasting algorithm based

on 1-hop neighbor information that reduces the time complexity of computing forwarding nodes to On. In Liu et al.s algorithm,n nodes are selected to forward the message in the worst case, whereas in our proposed algorithm, the number of forwarding nodes in

the worst case is 11. In the second part of the paper, we propose a simple and highly efficient receiver-based broadcasting algorithm.

When nodes are uniformly distributed, we prove that the probability of two neighbor nodes broadcasting the same message

exponentially decreases when the distance between them decreases or when the node density increases. Using simulation, we

confirm these results and show that the number of broadcasts in our proposed receiver-based broadcasting algorithm can be even less

than one of the best known approximations for the minimum number of required broadcasts.

Index TermsWireless ad hoc networks, flooding, broadcasting, localized algorithms.

1 INTRODUCTION

BROADCASTING is a fundamental communication operationin which one node sends a message to all other nodes inthe network. Broadcasting is widely used as a basicmechanism in many ad hoc network protocols. Forexample, ad hoc on-demand routing protocols such asAODV [1] and DSR [2] typically use broadcasting in theirroute discovery phase. Broadcasting is also used fortopology updates, for network maintenance, or simply for

sending a control or warning message. The simplestbroadcasting algorithm is flooding, in which every nodebroadcasts the message when it receives it for the first time.Using flooding, each node receives the message from all itsneighbors in a collision-free network. Therefore, the broad-cast redundancy significantly increases as the averagenumber of neighbors increases. High broadcast redundancycan result in high power and bandwidth consumption inthe network. Moreover, it increases packet collisions, whichcan lead to additional transmissions. This can cause severenetwork congestion or significant performance degradation,a phenomenon called the broadcast storm problem [3].Consequently, it is crucial to design efficient broadcastingalgorithms to reduce the number of required transmissions

in the network.A set of nodes is called a Dominating Set (DS) if any node

in the network either belongs to the set or is a 1-hop

neighbor of a node in the set. The set of broadcasting nodesforms a Connected DS (CDS). Therefore, the minimumnumber of required broadcasts is not less than the size ofthe minimum CDS. Unfortunately, finding the minimumCDS is NP-hard, even for the unit disk graphs [4], [5].However, there are some distributed algorithms that canfind a CDS whose size is smaller than a constant factor of

the size of the minimum CDS [6], [7]. These algorithms canbe employed to find a small-sized CDS that can be used as avirtual backbone for broadcasting in ad hoc networks.However, this approach is not efficient in networks withfrequent topology changes, as maintaining a CDS is oftencostly [8].

The main objective of efficient broadcasting algorithms isto reduce the number of broadcasts while keeping thebandwidth and computational overhead as low as possible.One approach to classify broadcasting algorithms is basedon the neighbor information they use. Some broadcastingalgorithms such as flooding and probabilistic broadcastingalgorithms [9], [10] do not rely on neighborhood knowl-

edge. These algorithms cannot typically guarantee fulldelivery and/or effectively reduce the number of broad-casts. Moreover, to decide whether or not to broadcast, theymay use a threshold (such as probability of broadcast),which may not be easy to find for different networksituations. In the second category, broadcasting algorithmsrequire having 2-hop or more neighbor information. Thebroadcasting algorithms in this category can reduce thenumber of broadcasts in the network and guarantee fulldelivery [11], [12], [13]. However, they may induce highoverhead in highly dynamic networks as they need tomaintain 2-hop network connectivity.

In this paper, we propose two broadcasting algorithms

based on 1-hop neighbor information. The first proposed

IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 2, FEBRUARY 2009 231

. M. Khabbazian is with the Department of Electrical and ComputerEngineering, University of British Columbia, 2523 Pearkes Lane,Vancouver, BC V6T 2C3, Canada. E-mail: [email protected].

. V.K. Bhargava is with the Department of Electrical and ComputerEngineering, University of British Columbia, 2356 Main Mall, Vancouver,BC V6T 1Z4, Canada. E-mail: [email protected].

Manuscript received 3 Jan. 2007; revised 25 Aug. 2007; accepted 9 Apr. 2008;published online 19 June 2008.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TMC-0002-0107.

Digital Object Identifier no. 10.1109/TMC.2008.94. 2009 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS


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algorithm is a sender-based algorithm. In sender-basedalgorithms, the broadcasting nodes select a subset of theirneighbors to forward the message. We compare ourproposed broadcasting algorithm to one of the bestsender-based broadcasting algorithms that use 1-hopinformation [8]. In [8], Liu et al. propose a broadcastingalgorithm that reduces the number of broadcasts and

achieves local optimality by selecting the minimum numberof forwarding nodes with minimum time complexityOn log n, where n is the number of neighbors. We showthat this optimality only holds for a subclass of sender-based broadcasting algorithms employing 1-hop informa-tion and prove that our proposed sender-based algorithmcan achieve full delivery with time complexity On.Moreover, Liu et al.s algorithm selects n forwarding nodesin the worst case, while our proposed algorithm selects11 nodes in the worst case. Based on our simulation results,our sender-based algorithm results in fewer broadcasts thandoes Liu et al.s algorithm. All these interesting propertiesare achieved at the cost of a slight increase in end-to-end

delay. Thus, our first proposed algorithm is preferred toLiu et al.s algorithm when the value of n is typically large,and it is important to bound the packet size.

We also propose a receiver-based broadcasting algorithmin this paper. In receiver-based algorithms, the receiverdecides whether or not to broadcast the message. Theproposed receiver-based algorithm is a novel broadcastingalgorithm that can significantly reduce the number ofbroadcasts in the network. We show that using ourproposed receiver-based algorithm, two close neighborsare not likely to broadcast the same message. In otherwords, we prove that the probability of broadcast for a nodeNA exponentially decreases when the distance between NAand its broadcasting neighbor decreases or when thedensity of nodes increases. Based on our experimentalresults, the number of broadcasts using our receiver-basedalgorithm is less than one of the best known approxima-tions for the minimum number of required broadcasts.

The rest of this paper is organized as follows: In Section 2,we describe the system model and network assumptions. InSection 3, we discuss our proposed sender-based broad-casting algorithm and its characteristics. We propose asimple and highly efficient receiver-based broadcastingalgorithm in Section 4 and prove an interesting property ofthe algorithm. We also relax some system model assump-tion in this section. In Section 5, we verify the theoreticalresults using simulation and compare the number of

forwarding nodes of our proposed broadcasting algorithmswith that of one of the best existing broadcasting algorithmsand an approximated lower bound of the optimal solution.Finally, we provide conclusions in Section 6.

2 SYSTEM MODEL

Our system model is very similarto that used by Liu et al. [8].We assume that all nodes are located in a 2D plane and havea transmission range of R. Therefore, the topology of thenetwork can be represented by a unit disk graph. We assumethat the network is connected. Two nodes are consideredneighbors if they are in the transmission range of each other.

We suppose that each node knows its location via a

localization technique such as Global Positioning System(GPS) or the lightweight techniques summarized in [14].Each node periodically broadcasts a very short Hellomessage, which includes its ID and position. Thus, eachnode gets the position of its neighbors as well.In the mediumaccess control (MAC) layer, we assume that scheduling isdone according to the p-persistent CSMA/CA protocol,

which is based on IEEE 802.11 in the broadcast mode. Inthe p-persistent CSMA/CA protocol, when a node has amessage to transmit, it initiates a defer timer by a randomnumber and starts listening to the channel. If the channel isbusy, it continues to listen until the channel becomes idle.When the channel is idle, it starts decrementing the defertimer at the end of each time unit. The message is broadcastwhen the timer expires.

3 AN EFFICIENT SENDER-BASED BROADCASTINGALGORITHM

3.1 Algorithm Structure

Our first proposed broadcasting algorithm is a sender-based algorithm, i.e., each sender selects a subset of nodesto forward the message. Each message can be identified byits source ID and a sequence number incremented for eachmessage at the source node. Algorithm 1 is a generalsender-based broadcasting algorithm and indicates thestructure of our proposed sender-based broadcastingalgorithm. Upon expiration of the timer, the algorithmrequests the MAC layer to schedule a broadcast. Themessage scheduled in the MAC layer is buffered and thenbroadcast with a probability p. This adds another delay (i.e.,the MAC-layer delay) in broadcasting the message. TheMAC-layer delay in IEEE 802.11 is a function of severalfactors including the network traffic. Note that there is achance that a node changes its decision (regarding theselected nodes or regarding whether to broadcast) duringthe MAC-layer delay due to receiving other copies of themessage. This chance is not negligible when the delay in theMAC layer is comparable to the average value of the timerset in the broadcasting algorithm. As stated in [15], onesolution to this problem is a cross-layer design in which thenetwork layer is given the ability to modify or removepackets that are present in the MAC-layer queue. Thissolution allows the broadcasting algorithms to performclose to their ideal performance even for very small averagetimer values [15]. In the entire paper, we assume that eitherthe MAC-layer delay is negligible compared to the averagedelay set by the algorithm or the network layer (hence, thealgorithm) is able to modify or remove packets buffered inthe MAC-layer queue (in this case, the algorithm does notrequire to set a defer timer).

The sender-based broadcasting algorithms can be di-vided into two subclasses. In the first subclass, each nodedecides whether or not to broadcast solely based on the firstreceived message and drops the rest of the same messagesthat it receives later. Liu et al.s algorithm falls in thissubclass and achieves local optimality by selecting theminimum number of forwarding nodes in the lowest

computational time complexity.

232 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 2, FEBRUARY 2009


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In the second subclass of sender-based broadcastingalgorithms, each node can decide whether or not to

broadcast after each message reception. However, if a nodebroadcasts a message, it will drop the rest of the samemessages that it receives in the future. Therefore, eachmessage is broadcast once at most by a node using thebroadcasting algorithms in both subclasses. Our firstproposed broadcasting algorithm falls in this subclass ofsender-based broadcasting algorithms. We show that theproposed algorithm can reduce both the computationalcomplexity of selecting the forwarding nodes and themaximum number of selected nodes in the worst case.

Algorithm 1 shows the basic structure of our proposedsender-based broadcasting algorithm. As shown inAlgorithm 1, each node schedules a broadcast for areceived message if the node is selected by the senderand if it has not scheduled the same message before.Clearly, each message is broadcast once at most by anode, which is similar to Liu et al.s algorithm. However,in Liu et al.s algorithm, each node may only schedule abroadcast when it receives a message for the first time.In contrast, in Algorithm 1, a broadcast schedule can beset at any time. For example, a message can be droppedafter the first reception but scheduled for broadcastthe second time. Clearly, the main design issue inAlgorithm 1 is how to select the forwarding nodes.

Algorithm 1. A general sender-based algorithm1: Extract information from the received message M

2: if M has been scheduled for broadcast or does notcontain nodes ID then

3: drop the message4: else

5: set a defer timer6: end if

7: When defer timer expires8: Select a subset of neighbors to forward the message9: Attach the list of forwarding node to the message

10: Schedule a broadcast

3.2 Forwarding-Node Selection Algorithm

Let us consider point A as the node NA and a circle CA;R

centered at A with a radius R as the transmission range

of NA. We use AB to denote the distance between twopoints A and B. Before delving into the algorithmdescription and proofs, we need to define the followingterms:

Definition 1 (bulged slice). As illustrated in Fig. 1, we define abulged slice around A as the intersection area of three circleswith radius R and centers A, M, and N, where AM R,AN R, and MN R. Note that in any bulged slice AMN,we have ffMAN

3.

Definition 2 (right/left bulged slice). As shown in Fig. 2, letA and Bbe two points such that 0 < AB R and AMN be abulged slice around A. Suppose that the point B is on one ofthe arcs AM

_or AN

_of the bulged slice AMN. In this case,

AMN is called the right bulged slice of B around A if itcontains the

3clockwise rotation of point B around A and is

called its left bulged slice around A otherwise.

Definition 3 (bulged angle). Let B1 and B2 be two bulgedslices around A. The bulged angle ffAB1;B2 is defined to beequal to 0 < 2 ifB2 is an counterclockwise rotation ofB1 around A.

Definition 4 (B-coverage set). A subset of neighbors of NA iscalled a B-coverage set of NA if any nonempty bulged slicearound A contains at least one node from the set. A bulgedslice is empty if there is no node inside it.

Definition 5 (slice-based selection algorithm). Aforwarding-node selection algorithm is called a slice-based selectionalgorithm (or slice-based algorithm) if for any node NA, itselects a B-coverage set of it.

A node can have several different B-coverage sets. There-fore, there is more than one slice-based selection algorithm.For example, a trivial slice-based selection algorithm wouldbe one that selects all of the neighbors as the B-coverage set.Clearly, this algorithm will result in flooding if it is used asthe forwarding-node selection scheme in Algorithm 1. Inthis section, we first show that Algorithm 1 can achieve fulldelivery if it uses any slice-based algorithm to select the

forwarding nodes. We then present an efficient slice-based

KHABBAZIAN AND BHARGAVA: EFFICIENT BROADCASTING IN MOBILE AD HOC NETWORKS 233

Fig. 1. A bulged slice around A.

Fig. 2. Left bulged slice of B and right bulged slice of C around A.


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algorithm that selects 11 nodes in the worst case and has

computational complexity On, where n is the number of

neighbors.

Lemma 1. For any two points P1 and P2 inside a bulged slice,we have

P1P2 R:

Proof. As shown in Fig. 3, the line passing through P1 and

P2 intersects the bulged slice AMN at P01 and P

02. Clearly,

P1P2 P01P02. Therefore, to prove the lemma, it is

sufficient to show that P01P02 R. This is easy to show

if both P01 and P02 are on the same arc of the bulged slice.

Thus, without loss of generality, we can assume that P01and P02 are on the arcs AM

_and AN

_, respectively. Let us

consider the perpendicular bisector of the line segment

P01M (line L). Line L passes through N becauseNM NP01 R. Since the point P

02 is on the arc AN

_,

the line segment MP02 will cross the line L at a point Q.

Using triangle inequality, we have

P01P02 QP

02 QP

01

QP02 QM

P02M R:

Note that Q is on the line L; hence, QP01 QM. tu

Consider two points A and B such that R < AB 2R. As

shown in Fig. 4, the line segment AB intersects the circle

CA;R at point Q. Let AQM and AQN be the left and right

bulged slices of Q around A, respectively. The following

lemmas hold:

Lemma 2. A point P is inside the bulged slice AQM or AQN if

AP R and BP R.

Proof. It is easy to show that for any triangle 4ABC,

AM AB or AM AC, where M is a point on the line

segment BC. Consequently, in the triangle 4P AB

(shown in Fig. 4), we have

P Q AP R or P Q BP R:

Therefore, P Q R. Thus, based on the bulged slice

definition, the point P is inside the bulged slice AQM

orAQN

. tu

Lemma 3. For any point P 6 A inside the bulged slice AQM orAQN, we have

BP < BA:

Proof. Using triangle inequality, we get

BP BQ QP BQ R BA:

Note that this equality holds only when BP BQ QPand QP R or simply when P A. tu

Theorem 1. In a collision-free network, Algorithm 1 can achievefull delivery if it uses a slice-based selection algorithm to selectthe forwarding nodes.

Proof. Using Algorithm 1, each node broadcasts the

message at most once. Therefore, broadcasting willeventually terminate. By contradiction, suppose there isat least one node that has not received the message afterthe broadcasting termination. Let us consider thefollowing set:

fNX; NY; NZjNX has broadcast the message;

NZ has not received the message; and

NY is the neighbor of both NX and NZg:

Suppose NS is the node that initiated broadcasting, andNT is a node that has not received the message. Thenetwork is connected; thus, there is a path between NS andNT. Clearly, we can find two neighbor nodes NC and NBalong the path from NT to NS such that NC has notreceived the message, while NB has received it. Supposethat NB has received the message from NA. Consequently,NA; NB; NC 2 ; thus, 6 ;. As a result, we have

9NA0 ; NB0 ; NC02 s:t:8NX; NY; NZ2 : A0C0XZ: 1

Obviously, NA0 and NC0 are not neighbors, because NC0

has not received the message. Thus, A0C0 > R. UsingLemma 2, B0 is inside the bulged slice A0P1P2 or A

0P1P3,where P1 is the intersection of line segment A

0C0 and thecircle CA0;R, and A0P1P2 and A0P1P3 are the left and theright bulged slices ofP1 around A

0, respectively. Without

loss of generality, assume that B

0

is inside the bulged


Fig. 3. Upper bound on the distance between nodes inside a bulged

slice.

Fig. 4. Other properties of bulge slice from Lemmas 2 and 3.


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slice A0P1P2. Since NA0 has at least one neighbor (i.e., NB0)in this slice, there must be a selected node ND in the slicethat has forwarded the message. Using Lemma 1, we getB0D R; hence, nodes ND and NB0 are neighbors.Therefore, ND; NB0 ; NC0 2 . However, this contradicts(1) because using Lemma 3, we have

DC0 < A 0C0:

tu

Algorithm 2 shows our proposed slice-based selectionalgorithm. Suppose that node NA uses the proposedalgorithm to select the forwarding nodes from its neighbors.Let us assume that NA stores all of its neighbors IDs andlocations in an array of length n, where n is the number ofneighbors. The algorithm selects the first node NS1randomly from the array. The first node can also be selecteddeterministically by, for example, selecting the node that isthe farthest away from NA. Let LBAP and RBAP denotethe left bulged slice and right bulged slice of P around A,

respectively. Suppose that NSi is the last node selected bythe algorithm. To select the next node, the algorithm iteratesthrough the array and selects the node NSi1 such that it isinside the slice LBASi, ffALBASi;LBASi1 6 0, and

8NB inside LBASi :

ffA LBASi;LBAB ffA LBASi;LBASi1 :2

If there is no such node, the algorithm selects NSi1 such that

8NB inside CA;R :

ffA LBASi;RBASi1 ffA LBASi;RBAB :3

The algorithm terminates by selecting the last node NSm if

NSm is inside LBAS1 or NS1 is inside LBASm orSm1 S1.

Algorithm 2 A slice-based selection algorithmInput: ListA1 . . . n: List of all neighbors of NAOutput: A B-coverage set of NA : fNSig

1: ind 1; i 02: repeat

3: ang max 0; ang min 24: i i 15: NSi ListAind6: chk false7: for j 1; j lengthListA; j++ do8: if ListAj is in LBASi then9: if ffALBASi;LBAListAj > ang max

then

10: chk true11: ind max j12: ang max ffALBASi;LBAListAj13: end if

14: else

15: if ffALBASi;RBAListAj < ang minthen

16: ind min j17: ang min ffALBASi;RBAListAj18: end if

19: end if

20: end for

21: if chk then

22: ind ind max23: else

24: ind ind min25: end if

26: until S1 is in LBAListAind OR ListAind is in

LBAS127: if ind6 1 then28: NSi1 ListAind29: end if

Lemma 4. Suppose the proposed algorithm selects m nodesfNS1 ; NS2 ; . . . ; NSmg. For any 1 i < m 2, we have

ffA LBASi;LBASi2 >

3:

Proof. Based on (2), (3), and the algorithm terminationcondition, we can show that

ffA LBASi;LBASi2 > ffA LBASi;LBASi1 for any 1 i < m 2. By contradiction, assume that

ffA LBASi;LBASi2

3:

Therefore, Si2 is inside LBASi. Thus, using (2), wehave

ffA LBASi;LBASi2 ffA LBASi;LBASi1 ;

which is a contradiction. tu

Theorem 2. The proposed slice-based selection algorithm willselect at most 11 nodes.

Proof. By contradiction, assume that the algorithm selectsmore than 11 nodes. Therefore, S11 is not in LBAS1.Using Lemma 4, we get

ffA LBAS1;LBAS11

X5i1

ffA LBAS2i1;LBAS2i1 > 5

3:

T h e r e f o r e , ffALBAS11;LBAS1 < 2 5 3

3.

Consequently, S1 is inside LBAS11; thus, the proposedslice-based algorithm will terminate after selecting S11.tu

The above theorem gives an upper bound on the number of

nodes selected by the proposed selection algorithm. InSection 5, using simulation, we show that the averagenumber of selected nodes (when the nodes are distributeduniformly) is less than six.

Theorem 3. Time complexity of the proposed slice-based selectionalgorithm is On, where n is the number of neighbors.

Proof. The algorithm selects the first node in O1. To selecteach of the other nodes, the algorithm performs Onoperations by checking all the neighbors in the array.Therefore, the complexity of the algorithm is Om n,where m is the number of selected nodes. UsingTheorem 2, we have m 11; thus, the time complexity

of algorithm isO

n. tu



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3.3 Reducing the Number of Forwarding Nodes

In the sender-based broadcasting algorithms, each broad-casting node attaches a list of its selected forwarding nodesto the message before broadcasting it. This procedure willincrease the bandwidth and power required to broadcastthe message. As shown earlier, our proposed slice-basedselection algorithm reduces the number of selected for-

warding nodes to 11 in the worst case. In this section, weshow how to further reduce the number of selected nodes.

Recall that the proposed slice-based algorithm selects asubset of NAs neighbors such that there is at least oneselected node in any nonempty bulged slice around A.Suppose NA extracts the list of the forwarding nodes fromeach message it receives. Let LA be a subset of NAsneighbors that has broadcast the message or been selectedby other nodes to forward it. Since all of the selectedforwarding nodes are required to broadcast the message, itis sufficient for NA to find a subset of its neighbors SA suchthat any nonempty bulged slice around A contains at leastone node from SA [ LA. Algorithm 2 can be simply

extended to achieve this in On. Note that the extendedalgorithm can start with a node from LA and select anynode in LA as soon as it appears in the left bulged slice ofthe previously selected node. Finally, the extended algo-rithm removes all of the nodes in LA from the set ofselected nodes.

3.4 Maximizing the Minimum Node Weight ofB-Coverage Set

Suppose node NA assigns a weight to each of its neighbors.The weight can represent the neighbors battery lifetime, itsdistance to NA, the average delay of the node, the level oftrust, or a combination of them. In some scenarios, we may

desire to find a B-coverage set such that its minimum nodeweight is the maximum or its maximum node weight is theminimum among that of all B-coverage sets. For example,assume that the weight of each node represents its batterylifetime in a wireless network. It may be desirable to selectthe nodes with a higher battery lifetime to forward themessage in order to keep the nodes with a lower batterylifetime alive. Algorithm 3 shows how to find a B-coverageset such that its minimum node weight is the maximumamong that of all B-coverage sets. A similar approach can beused to find a B-coverage set such that its maximum nodeweight is the minimum.

Algorithm 3. Maximizing the minimum node weightInput: ListA1 . . . n: List of all neighbors of NAOutput: A B-coverage set of NA with highest minimumnode weight

1: SListA sortListA {Sort the neighbor nodes bytheir weights} fSListi ! SListj , i jg

2: H n; T 1; m bn2e

3: St Algorithm 2SList14: if St is a B-coverage set for NA then

5: return SList[1]6: end if

7: while H > T 1 do8: St Algorithm 2SList1 . . . m {Pass m nodes

with the highest weights to Algorithm 2 as the input}

9: if St is a B-coverage set for NA then

10: H m11: m dTm

2e

12: else

13: T m14: m dHm

2e

15: end if

16: end while17: return Algorithm 2SList1 . . . H

Algorithm 3 first sorts the nodes by their weights in

decreasing order. Then, in each step, it passes m nodes with

the highest weights to Algorithm 2 as input and gets a set of

(at most 11) nodes as output, where 1 m n is an integer

initially set to dn2e. If the output set is a B-coverage set,

Algorithm 3 sets H to m and decreases m to dTm2e, where T

and H are variables initially set to 1 and n, respectively.

Otherwise, it sets T to m and increases m to dHm2e. After a

finite number of steps, we get H T 1. Algorithm 3 then

returns the output of Algorithm 2SList1 . . . H.

Corollary 1. Algorithm 3 will select at most 11 nodes.

Proof. The proof is clear, as Algorithm 3 returns an outputof Algorithm 2 (Line 17). tu

Theorem 4. The time complexity of Algorithm 3 is On log n.

Proof. Algorithm 3 requires On log n operations to sort thelist of neighbors ListA1 . . . n. The computational com-plexity of Algorithm 2 is On. Therefore, Algorithm 3performs On operations in each iteration of the whileloop. The while loop terminates after Olog n iterationsbecause it uses a binary search approach to find theminimum value of H. Consequently, the order ofAlgorithm 3 is On log n log n n. tu

Theorem 5. The minimum weight of nodes of the B-coverageset selected by Algorithm 3 is the maximum among that ofall B-coverage sets.

Proof. Suppose that Stmin is a B-coverage set such that theminimum weight of nodes in Stmin is greater than orequal to that of other B-coverage sets. Let NX 2 Stmin bethe node with the minimum weight in Stmin. Assumethat NA has K neighbors with weights greater than orequal to the weight of NX. Therefore, the output ofAlgorithm 2SList1 . . . K is a B-coverage set. Note thatAlgorithm 3 finds the minimum H such that the outputof Algorithm 2SList1 . . . H is a B-coverage set for NA.Therefore, H K, and thus, the minimum weight of

nodes of the B-coverage set selected by Algorithm 3 isgreater than or equal to the weight of NX. tu

3.5 Similarity with a Topology Control Algorithm

In [16] and [17], the authors proposed a cone-basedtopology control algorithm, where each node makes localdecisions about its transmission power. The objective of thealgorithm is to minimize the transmission power of eachnode without violating the network connectivity. In order todo that, each node NA transmits with the minimum powerP such that in every nonempty cone of degree aroundNA, there is some node that NA can reach with power P. Acone is nonempty if there is at least a node in the cone that

NA can reach using its maximum power. For

2

3 , they



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proved that the network remains connected if the cone-

based algorithm is employed.Suppose that we use cones instead of bulged slices in the

proposed forwarding-node selection algorithm. Therefore,the algorithm will select the forwarding node set such thatany nonempty cone of degree around NA contains at leastone node from the forwarding node set. Surprisingly, thisalgorithm will not guarantee full delivery. Fig. 5 shows acounterexample for the case where 2

3. Fig. 6 shows that

even for 3

, full delivery cannot be guaranteed. In bothFigs. 5 and 6, the node NA initiates broadcasting and selectsonly NB and NC to forward the message. Suppose that ND isclose enough to the point M such that it is the only nodethat can reach NE. In this case, NE will not receive the

message because ND is not selected by neither NB nor NC toforward the message. Note that the cone-based and theforwarding-node selection algorithms use different ap-proaches. In the cone-based algorithm, a node NA increasesits power from zero until there is a node in each nonemptycone around NA. However, in the forwarding-node selec-tion algorithm, a node NA selects some nodes (the forward-ing nodes) until there is a selected node in each nonemptybulged slice around NA.

4 A HIGHLY EFFICIENT RECEIVER-BASEDBROADCASTING ALGORITHM

In this section, we propose a novel receiver-based broad-casting algorithm that can significantly reduce redundantbroadcasts in the network. As mentioned earlier, inreceiver-based broadcasting algorithms, the receiver of themessage decides whether or not to broadcast the message.Therefore, a potential advantage of receiver-based broad-casting algorithms over sender-based ones is that they donot increase the size of the message by adding a list offorwarding nodes.

4.1 Algorithm Structure

Algorithm 4 shows a general approach used in severalreceiver-based broadcasting algorithms [13], [18]. Our

proposed receiver-based broadcasting algorithm employs

this approach. Clearly, the main design challenge ofAlgorithm 4 is to determine whether or not to broadcast areceived message. A trivial algorithm is to refrain broad-

casting if and only if all the neighbors have received themessage during the defer period. Although this algorithm issimple to implement, it has limited effect in reducing thenumber of redundant broadcasts. Suppose NAs defer timeexpires at t0. Using the above strategy, node NA willbroadcast if some of its neighbors (at least one) have notreceived the message by t0. However, this broadcast isredundant if all such neighbors receive the message fromother nodes after time t0. This scenario typically occurswhen t0 is small compared to the maximum defer time. Inthe next section, we introduce a responsibility-basedscheme (RBS) that further reduces the redundant broadcastswithout any changes in the MAC-layer defer-time design.

Algorithm 4. A general receiver-based algorithm1: Extract information from the received message M2: if M has been received before then

3: drop the message4: else5: set a defer timer

6: end if

7: When defer timer expires8: decide whether or not to schedule a broadcast

4.2 Responsibility-Based Scheme

Algorithm 5 shows the proposed RBS. The main idea ofAlgorithm 5 is that a node avoids broadcasting if it is notresponsible for any of its neighbors. A node NA is notresponsible for a neighbor NB if NB has received themessage or if there is another neighbor NC such that NC hasreceived the message and NB is closer to NC than it is to NA.Suppose NA stores IDs of all its neighbors that havebroadcast the message during the defer period. Whenexecuted by a node NA, Algorithm 5 first uses thisinformation to determine which neighbors have notreceived the message (Lines 1-9 of Algorithm 5). It thenreturns false if and only if it finds a neighbor NB that has notreceived the message and

AB BC


Fig. 5. A counterexample for 23

.

Fig. 6. A counterexample for 3 .


8/15

for any NAs neighbor NC that has received the message.

The output of RBS determines whether or not the broadcast

is redundant.

Algorithm 5. RBSInput: ListA: List of all neighbors of NA, and ListB: List ofbroadcasting neighborsOutput: true or false

1: ListC ListA2: for i 1; i lengthListC; i++ do3: for j 1; j lengthListB; j++ do4: if distListCi; ListBj R then5: removeElementListCi; ListC6: break7: end if8: end for

9: end for10: ListD ListA ListC11: for i 1; i lengthListC; i++ do12: check true13: for j 1; j lengthListD; j++ do14: if distListCi; ListDj < distListCi; NA

then

15: check false16: break17: end if18: end for19: if check then

20: return (false)

21: end if22: end for

23: return (true)

Example 1. As shown in Fig. 7, NA has five neighbors.

Suppose that NA has received a message from NF. Note

that NA has the position of all its neighbors. Therefore, it

can find that NE and ND have received the message but

NB and NC have not. As shown in Fig. 7, NA is not

required to broadcast because

BE < BA and CD < CA:

Theorem 6. In a collision-free network, Algorithm 4 can achievefull delivery if it uses the proposed RBS to determine whetheror not to broadcast.

Proof. Using Algorithm 5, each node broadcasts a messageat most once. Therefore, broadcasting will eventuallyterminate. By contradiction, suppose there is at least onenode that has not received the message after the

broadcasting termination. Let us consider the set

fNX; NYjNX and NY are neighbors;

NX has received the message; and

NY has not received the messageg:

Suppose NS is the node that initiated broadcasting,and NT is a node that has not received the message. Thenetwork is connected; thus, there is a path between NSand NT. Clearly, we can find two neighbor nodes NB andNA along the path from NT to NS such that NB has notreceived the message, while NA has. Consequently,NA; NB 2 ; thus, 6 ;. As a result, we have

9NA0 ; NB0 2 s:t: 8NX; NY 2 : A0B0 XY : 4

Clearly, NA0 has not broadcast since NB0 has notreceived the message. Therefore, there must be a nodeNC0 such that NC0 has received the message and C0B0 R and BX Rg:

Note that the nodes positions have a Poisson distribu-tion. Therefore, using Lemma 6, we get

P rb 9NP inside IDA;R;DB;R;DQ;QB 1 e

IDA;R;DB;R ;DQ;QB

! 1 e

Rd2

3 :9

Thus,

P rbBrd 1 P rb

1 X1k0

Prob jj k Prb j jj k ;

where jj is the cardinality of the set . Therefore, using(9) and Lemma 5, we get

P rbBrd 1

X1

k0

ke

k!1 e

Rd2

3

k

1 ee

Rd2

3;

where is the area of the hatched crescent in Fig. 8(collection of all points Q, QA > R, and QB R):

DB;R IDA;R;DB;R

R2 2 arccosd

2R

d

ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiR2

d

2

2s:

tu

Corollary 2. Using Theorem 8, we get

P rbBrd eRAB2

3 :

Proof. Consider the function

fx x ex 1:

It is easy to show that fx has a global minimum atx 0. Therefore, we have

1 ex x

for any real number x. As a result, we get

P rbBrd 1 ee

Rd2

3 eRd2

3 :

tu

It is also possible that node NB receives the message from

more than one neighbor in its defer period. In this case, the

number of NBs neighbors that have received the message

increases, and the number of that have not received the

message decreases. Consequently, the probability that NB is

required to broadcast the message further decreases

compared to the case where NB receives the message from

only one neighbor. It is worth mentioning that RBS can

guarantee that the number of forwarding nodes is within a

constant factor of the optimal solution (minimum CDS) if it

is provided with 2-hop neighbor information [21].

4.4 Relaxing Some System Model Assumptions

We assumed in Section 2 that the nodes are placed in a 2D

plane. However, this assumption is not used in the proof of

Theorem 6. Therefore, the proposed receiver-based algo-

rithm can also achieve full delivery when the nodes are

distributed in a 3D space. Note that in this case, RBS uses

3D node positions.

We can also relax the assumption that all the nodes havethe same transmission range R. When the nodes transmis-

sion ranges are different, the topology graph should be

defined as a directed graph for which there is a link from

NA to NB ifNB is in the transmission range of NA. Suppose

G is an undirected graph obtained by removing unidirec-

tional links of the topology graph. We assume that G is

connected and define two nodes as neighbors if there is a

link between them in G (i.e., they are in the transmission

range of each other). Note that many wireless MAC

protocols such as IEEE 802.11 require bidirectional links.

Let us assume that nodes put not only their ID and position

but also their transmission range into the hello messages

that they periodically broadcast. Therefore, the neighbors ofa node know both its position and transmission range. In

this case, nodes can use Algorithm 6 to decide whether or

not to broadcast.Algorithm 6 is a modified version of RBS. When

executed by a node NA, Algorithm 6 uses the position and

transmission range of the broadcasting nodes to determine

which neighbors have not received the message. It then

returns false if and only if it finds a neighbor NB that has not

received the message and

AB BC or BC > transmission range of NC

for any NAs neighbor NC that has received the message.

Algorithm 6. Modified RBSInput: ListA: List of all neighbors of NA, and ListB: List of

broadcasting neighborsOutput: true or false

1: ListC ListA2: for i 1; i lengthListC; i++ do3: for j 1; j lengthListB; j++ do4: if distListCi; ListBj RListBj then5: {RListBj: Transmission range of node ListBj}6: removeElementListCi; ListC7: break

8: end if



11/15

9: end for

10: end for

11: ListD ListA ListC12: for i 1; i lengthListC; i++ do13: check true14: for j 1; j lengthListD; j++ do15: if distListCi; ListDj < distListCi; NA

then16: if distListCi; ListDj RListDj then17: check false18: break

19: end if

20: end if

21: end for

22: if check then

23: return (false)24: end if25: end for

26: return (true)

Corollary 3. In a collision-free network, Algorithm 4 can achievefull delivery if it uses the modified RBS to determine whetheror not to broadcast.

Proof. The proof is similar to the proof of Theorem 6. Bycontradiction, suppose there is at least one node that hasnot received the message after the broadcasting termina-tion. Let us consider the set

NX; NYjNX and NY are neighbors;

NX has received the message; and

NY has not received the message

:

Suppose NS is the node that initiated broadcasting, and

NT is a node that has not received the message. There is apath between NS and NT in G; thus, we can find twoneighbors NA and NB along the path from NS to NT suchthat NA has received the message and NB has notreceived it. Consequently, NA; NB 2 ; thus, 6 ;. Asa result, we have

9NA0 ; NB0 2 s:t: 8NX; NY 2 : A0B0 XY : 10

Clearly, NA0 has not broadcast, since NB0 has not received

the message. Therefore, there must be a node NC0 such

that NC0 has received the message and

C0B0 < A0B0 and C0B0 Transmission range of NC0 :

This result contradicts (10) because NC0 ; NB0 2 . tu

We can relax the assumption of having precise position

information as well. The reader is referred to [21] for

more details about broadcasting under uncertain position

information.

5 SIMULATION

5.1 Average Number of Nodes Selected by theProposed Sliced-Based Algorithm

In Section 3, we proved that the proposed forwarding-node

selection algorithm selects 11 nodes in the worst case. In

practice, the number of selected nodes is typically less than

11. To avoid the complexity of mathematical analysis, we

used a simulation to find the average number of selected

nodes. For a given number of neighbors 1 n 160, werandomly put n points inside a circle with radius R. We then

ran the proposed selection algorithm and obtained thenumber of selected nodes. To get the average number of

selected nodes, we ran simulation 106 times for each given n.As shown in Fig. 9, the average number of selected nodes is

less than six and approaches five when n increases. Note that

the proposed sliced-based selection algorithm does not

necessarily select a B-coverage with a minimum number ofnodes. However, there is a sliced-based selection algorithm

that can find a B-coverage with a minimum number of nodes

in On log n and can consequently reduce the averagenumber of selected nodes. It is worth mentioning that Fig. 9

shows the average number of selected nodes by the sourcenode (the node that initiates the broadcasting). For the rest ofbroadcasting nodes, the average number of selected nodes is

at least one less than that for the source node because of the

optimization technique introduced in Section 3.

5.2 Probability of Broadcast Using the ProposedRBS

Suppose that the proposed receiver-based algorithm is used

for broadcasting in the network. Assume that node NBreceives a message from NA for the first time. It has been

proven that the probability ofNB broadcasting the message

P rbBrdB exponentially decreases when the distance AB

decreases or when the node density increases. We usedsimulation to confirm this theoretical result. For the

simulation, we considered two nodes NA and NB with

distance 0 < d R from each other. We uniformly placed

nodes with density inside the network and checked

whether or not NB was required to broadcast the message.

We ran simulation 106 times for a given and R. We then

estimated P rbBrdB by the ratio of the number of times NBwas required to broadcast over the total number of runs.

Figs. 10, 11, 12, and 13 show the simulation results for

several values of, d, and R. As shown in these figures, theprobability of broadcast exponentially decreases when d

decreases or when increases. For example, when R 300 m


Fig. 9. Average number of nodes selected by the proposed slice-based

algorithm.


12/15

and 4 104, the probability of broadcast is 0.1 ford 250 m and reduces to 104 for d 200 m. This propertycan justify why the proposed receiver-based algorithm cansignificantly reduce the number of broadcasts in the net-work. Fig. 14 illustrates an instance of using RBS for the casewhere R 300 m, 4 104, and nodes are placed in asquare area of 1;000 1;000 m2. As shown in Fig. 14, onlynine nodes (represented by stars) among 400 nodes broad-

cast the message.

5.3 Performance of Proposed Sender-Based andReceiver-Based Algorithms

The main objective of efficient broadcasting algorithms is toreduce the number of broadcasts. Therefore, we consideredthe ratio of broadcasting nodes over the total number ofnodes as the metric to evaluate the performance of theproposed broadcasting algorithms. Using the ns-2 simula-tor, we evaluated this metric against two parameters:transmission range and node density. In each simulationrun, we uniformly distributed N nodes in a 1;000 1;000 m2 square area, where N network area. Arandomly generated topology was discarded if it led to adisconnected network. Only one broadcasting occurred in

each simulation run by a randomly selected node. Table 1


Fig. 11. Probability of broadcast for R 400 m.



Fig. 14. Broadcasting nodes in a 1;000 1;000 m2 square area with

400 nodes.

TABLE 1Simulation Parameters



13/15


14/15

range was set to 250 m. Figs. 18 and 19 compare theaverage delivery ratio of the broadcasting algorithms for

different probabilities of message-reception failure. Asshown in these figures, the Edge Forwarding algorithm isthe most robust broadcasting algorithm against message-reception failure. This is because the impact of message lossis less when the broadcast redundancy is high. Interest-ingly, the robustness of the RBS algorithm significantlyimproves as the node density increases. The simulationresults indicate that the slice-based algorithm is the leastrobust broadcasting algorithm against message-receptionfailure. This is mainly due to the fact that the slice-basedalgorithm selects a small number of nodes (less than six onthe average) to forward the message. Therefore, when theprobability of message-reception failure is high, it is very

likely that most of the selected nodes fail to receive andthus forward the message.

Finally, we simulated the broadcasting algorithms in a

mobile wireless setting. The nodes were initially distributed

using a uniform distribution. In the simulation, we used a

random walk mobility model and set the maximum velocity

to 10 m/s. We fixed the transmission range to 250 m and

varied the total number of nodes within 25-1,000. The

simulation results indicate that all the broadcasting algo-

rithms considered in this paper can achieve a high delivery

ratio (above 95 percent on the average) for N! 50, where N

is the total number of nodes in the network. This is mainly

because the implemented broadcasting algorithms make

broadcasting decisions on the fly. For N 25, the

implemented algorithms failed to achieve a high delivery

ratio because the network could easily get disconnected due

to the nodes mobility. Clearly, broadcasting algorithms

cannot achieve a high delivery ratio in such scenarios. It is

worth mentioning that for N! 50, the ratio of broadcasting

nodes is almost the same as the case where there is no

mobility (see Fig. 15).

6 CONCLUSION AND FUTURE WORK

In the first part of this paper, we proposed a forwarding-

node selection algorithm that selects at most 11 nodes in

On, where n is the number of neighbors. This limited

number of nodes is an improvement over Liu et al.salgorithm, which selects n nodes in the worst case and has

time complexity On log n. Moreover, we showed that our

proposed forwarding-node selection algorithm results in

fewer broadcasts in the network. In the second part of the

paper, we proposed an efficient receiver-based algorithm

and showed why it significantly reduces the number of

forwarding nodes in the network. We also relaxed some

system model assumptions that are typically used in the

broadcasting algorithms. Interestingly, the 2-hop-based

version of our proposed receiver-based algorithm can

guarantee constant approximation to the optimal solution

(minimum CDS). As far as the authors know, this is the firstbroadcasting algorithm that constructs a CDS on the fly

and can guarantee both full delivery and a constant

approximation ratio to the optimal solution. As part of

our future work, we will investigate the necessary condi-

tions to guarantee both full delivery and constant approx-

imation ratio to the minimum CDS.

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Fig. 19. Average delivery ratio versus probability of message-reception

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Fig. 18. Average delivery ratio versus probability of message-reception

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[9] Y. Tseng, S. Ni, and E. Shih, Adaptive Approaches to RelievingBroadcast Storms in a Wireless Multihop Mobile Ad HocNetworks, Proc. 21st Intl Conf. Distributed Computing Systems(ICDCS 01), pp. 481-488, 2001.

[10] Y. Sasson, D. Cavin, and A. Schiper, Probabilistic Broadcast forFlooding in Wireless Mobile Ad Hoc Networks, Proc. IEEEWireless Comm. and Networking Conf. (WCNC 03), pp. 1124-1130,2003.

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Simple Reliable Broadcast Algorithm in Manets, Proc. IEEEINFOCOM 04, pp. 2084-2095, 2004.[12] J. Wu and F. Dai, Broadcasting in Ad Hoc Networks Based on

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Majid Khabbazian received the BSc degree incomputer engineering from the Sharif Universityof Technology, Tehran, Iran, in 2002 and theMASc degree in electrical and computer en-gineering from the University of Victoria, BritishColumbia, Canada, in 2004. He is currently aPhD candidate in the Department of Electricaland Computer Engineering, University of BritishColumbia, Vancouver, British Columbia, Cana-

da, where he is a university graduate fellowshipholder. He was a university graduate fellowship recipient at theUniversity of Victoria. His current research interests include cryptogra-phy and wireless network security. He is a student member of the IEEE.

Vijay K. Bhargava received the BSc, MSc, andPhD degrees from Queens University, Kingston,Ontario, Canada in 1970, 1972, and 1974,respectively. Currently, he is a professor andthe head of the Department of Electrical andComputer Engineering, University of BritishColumbia, Vancouver, British Columbia, Cana-da. He was with the University of Victoria from1984 to 2003 and with Concordia University,Montreal, from 1976 to 1984. He is a coauthor of

the book Digital Communications by Satellite (John Wiley & Sons,1981), a coeditor of Reed-Solomon Codes and Their Applications(IEEE,

1994), and a coeditor of Communications, Information and NetworkSecurity(Kluwer Academic Publishers, 2003). His research interests arein wireless communications. He is very active in the IEEE and wasnominated by the IEEE Board of Directors for the office of IEEEPresident-Elect. He has served on the Board of the IEEE InformationTheory Society and the IEEE Communications Society. He is a pastpresident of the IEEE Information Theory Society. He is the editor-in-chief for the IEEE Transactions on Wireless Communications. Hereceived the IEEE Centennial Medal in 1984, the IEEE CanadasMcNaughton Gold Medal in 1995, the IEEE Haraden Pratt Award in1999, the IEEE Third Millennium Medal in 2000, the IEEE GraduateTeaching Award in 2002, and the Eadie Medal of the Royal Society ofCanada in 2004. He is a fellow of the IEEE, the Engineering Institute ofCanada (EIC), the Canadian Academy of Engineering, and the RoyalSociety of Canada.

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