1072 ieee transactions on computers, vol. 57, no. 8...

Localized Broadcasting withGuaranteed Delivery and Bounded

Transmission RedundancyMajid Khabbazian, Student Member, IEEE, and Vijay K. Bhargava, Fellow, IEEE

Abstract—The common belief is that localized broadcast algorithms are not able to guarantee both full delivery and a good bound on

the number of transmissions. In this paper, we propose the first localized broadcast algorithm that guarantees full delivery and a

constant approximation ratio to the minimum number of required transmissions in the worst case. The proposed broadcast algorithm is

a self-pruning algorithm based on one round of information exchange. Using the proposed algorithm, each node determines its

forwarding status in Oð�G log �GÞ, where �G is the maximum node degree of the network. By extending the proposed algorithm, we

show that localized broadcast algorithms can achieve both full delivery and a constant approximation ratio to the optimum solution with

message complexity OðNÞ, where N is the total number of nodes in the network and each message contains a constant number of

bits. We also show how to save bandwidth by reducing the size of piggybacked information. Finally, we relax several system-model

assumptions, or replace them with practical ones, in order to improve the practicality of the proposed broadcast algorithm.

Index Terms—Wireless ad hoc networks, broadcasting, localized algorithms, minimum connected dominating set, approximation

algorithms.

Ç

1 INTRODUCTION

BROADCASTING is a fundamental communication primitivein which a message is sent from a source node to all

other nodes in the network. Broadcasting has manyapplications, including route discovery in wireless ad hocrouting protocols, and is frequently used to adapt networkchanges caused by the dynamic nature of ad hoc networks.The simplest broadcast mechanism is flooding, in whichevery node retransmits the first copy of the receivedmessage to all of its 1-hop neighbors. Despite its simplicity,flooding can cause a large number of redundant transmis-sions, wasting valuable network resources such as band-width and power. It can also lead to significantperformance degradation and network congestion. There-fore, it is essential to design efficient broadcast algorithmsin order to reduce the number of redundant transmissions.Unfortunately, it is not practical to design a delivery-guaranteed broadcast algorithm that eliminates all redun-dant transmissions. This is because the problem of findingthe minimum Connected Dominating Set (CDS) in a unitdisk graph can be reduced to the problem of eliminating allof the redundant transmissions. A set of nodes is called aDominating Set (DS) if every node in the network eitherbelongs to the set or is a 1-hop neighbor of a node in the set.It is well known that finding the Minimum CDS (MCDS) of

a unit disk graph is NP-hard in general [1], [2]. Note thatevery CDS can be used as a backbone of the network tobroadcast the message. On the other hand, the forwardingnodes in the delivery-guaranteed broadcast algorithmsform a CDS. Therefore, broadcast algorithms can be usedto find a CDS if a source node is selected to initiate thebroadcast. Consequently, the problems of finding a mini-mum CDS and of designing an optimum broadcastalgorithm can be reduced to each other.

To reduce the number of redundant transmissions, broad-cast algorithms typically impose some bandwidth overheadto, for example, collect neighbor information via messageexchanges. It is desirable to reduce this bandwidth overheadto improve the practicality of the broadcast algorithm for adhoc networks with frequent topology changes. In [3], theauthors prove that every distributed algorithm for construct-ing a nontrivial CDS has the lower message complexitybound of �ðN logNÞ, whereN is the number of nodes and themessage size is a constant multiple of the number of bitsrepresenting the node IDs (a CDS is said to be trivial if itconsists of all nodes). Let us define a distributed broadcastalgorithm as a nontrivial broadcast algorithm if the forward-ing nodes always form a nontrivial CDS. In this paper, weshow that the same lower bound does not hold for nontrivialbroadcast algorithms, although they are closely related to theproblem of finding a nontrivial CDS. In fact, we show that anextension of our proposed nontrivial broadcast algorithmrequires OðNÞ messages, where the message size is aconstant. Moreover, we show that the number of forwardingnodes using the extended algorithm is within a constantfactor of the optimum solution (i.e., minimum CDS).

Computational overhead also plays an important role indesigning efficient broadcast algorithms. Flooding is anideal broadcast algorithm in terms of computational over-head since each node requires almost no computation (it

1072 IEEE TRANSACTIONS ON COMPUTERS, VOL. 57, NO. 8, AUGUST 2008

. The authors are with the Department of Electrical and ComputerEngineering, University of British Columbia, 2356 Main Mall, Vancouver,BC, Canada V6T 1Z4. E-mail: {majidk, vijayb}@ece.ubc.ca.

Manuscript received 12 Mar. 2007; revised 23 Oct. 2007; accepted 14 Jan.2008; published online 20 Mar. 2008.Recommended for acceptance by S.H. Son.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TC-2007-03-0082.Digital Object Identifier no. 10.1109/TC.2008.31.

0018-9340/08/$25.00 � 2008 IEEE Published by the IEEE Computer Society

simply broadcasts the first copy of the received message).The existing broadcast algorithms typically require somecomputation for selecting forwarding nodes or for self-pruning. In this paper, we show how to efficientlyimplement the proposed broadcast algorithm using somecomputational geometry techniques to reduce the compu-tational overhead.

1.1 Related Work

Different approaches can be used to classify the existingbroadcast algorithms. One approach is based on whetheror not they use a previously constructed backbone. Onesolution for reducing the number of redundant transmis-sions is to form a CDS of nodes and use them as abackbone to broadcast the message. As mentioned earlier,finding the MCDS is an NP-hard problem. However,there are distributed algorithms to find a small-sized CDSwith constant approximation ratio to the MCDS [3], [4].The main drawback of this solution is that maintaining aCDS is often costly in networks with frequent topologychanges [5].

The second approach to classifying the existing broad-cast algorithms is based on whether global or localinformation is employed by the algorithm. An algorithmis global if it uses whole or partial global state information.On the other hand, a distributed algorithm is localized if itis based solely on local information. It is clear that globalbroadcast algorithms are not appropriate for mobile ad hocnetworks due to the dynamic nature of the network. Inaddition, they may not scale well when the number ofnodes in the network increases.

The localized broadcast algorithms typically usek � 0 rounds of information exchange to collect k-hopneighbor information. Therefore, they can be classifiedbased on the number of rounds of information exchanges.These algorithms can be further categorized based onwhether or not they use any “side information” piggy-backed in the packets. Our proposed algorithm is based onone round of information exchange, but uses partial 2-hopneighbor information obtained by extracting the informa-tion piggybacked in the packets.

Some broadcast algorithms do not require any informa-tion exchange (i.e., k ¼ 0). Flooding and probabilisticbroadcast algorithms such as [6] and [7] fit in this category.Probabilistic algorithms typically cannot guarantee fulldelivery. However, they can reduce the number ofredundant transmissions at low communication overheadand have great potential in unreliable communicationenvironments. To decide whether or not to broadcast,probabilistic broadcast algorithms often use a threshold(such as probability of broadcast). Choosing a correct valueof threshold is difficult. Moreover, the optimal value of thethreshold may change due to network topology changes.An adaptive approach to this problem was presented in [8].

Most existing localized broadcast algorithms usek rounds of neighbor information, where k � 1 is a smallnumber. These algorithms can be further divided intoneighbor-designating (sender-based) and self-pruning (re-ceiver-based). In neighbor-designating algorithms [5], [9],the forwarding status of each node is determined by othernodes. In other words, each broadcasting node selects asubset of its k-hop neighbors to forward the message. Themain design challenge for neighbor-designating algorithms

is to choose a small subset of nodes to forward the message.For example, in [5], the authors propose a broadcastalgorithm based on 1-hop neighbor information that selectsthe smallest subset of its 1-hop neighbors with the maximumcoverage area, where the coverage area of a set of nodes is theunion of their transmission coverage. When 2-hop neighborinformation is available, it can be extended to select theminimum number of 1-hop neighbors that cover all 2-hopneighbors. This procedure is known as the minimumforwarding set problem. There are heuristic algorithms inthe literature that give a constant approximation ratio to theminimum forwarding set problem [10]. However, as shownin [9], even an optimal solution to this problem may not resultin a small-sized CDS in the network. In [9], the authors extendthe minimum forwarding set problem to the case where thecomplete 2-hop topology information is available. Note thattwo rounds of information exchange provide partial topol-ogy information for the 2-hop neighbor set. To get complete2-hop topology information, each node must either get theposition information of the 2-hop neighbors or perform threerounds of information exchange [9]. The authors show thattheir proposed algorithm can provide a probabilistic constantapproximation ratio to MCDS, but point out that it mayperform poorly when the network becomes extremely dense.

In the self-pruning algorithms, each node determines itsstatus (forwarding/nonforwarding) using local informa-tion [11], [12], [13]. Clearly, the main design challenge withregard to self-pruning algorithms is to find an effective self-pruning condition. Different self-pruning conditions havebeen proposed in the literature. For example, one simpleself-pruning condition is whether all the neighbors havereceived the message before the defer timer expiration.Similar to the neighbor-designating algorithms, the existingself-pruning algorithms do not provide a constant approx-imation ratio to the optimal solution (MCDS).

1.2 Our Contribution

In the first part of this paper, we consider two generalstructures commonly used in neighbor-designating and self-pruning algorithms based on 1-hop neighbor information.We show that, using these structures, we are not able toguarantee both full delivery and a good bound on the numberof forwarding nodes in the network. An open question iswhether localized broadcast algorithms can provide both fulldelivery and constant approximation to the optimal solution(MCDS). As mentioned earlier, finding MCDS is an NP-hardproblem. In the absence of global network information, thisproblem becomes even more challenging. It is a commonbelief that localized broadcast algorithms (e.g., self-pruningalgorithms) cannot guarantee a constant approximation ratioto the optimal solution [11], [14]. One contribution of thispaper is to show that localized broadcast algorithms are ableto guarantee a reasonable bound on the number of forward-ing nodes. In particular, we propose a self-pruning algorithmand prove that it provides full delivery as well as guarantee-ing a constant approximation ratio to the optimal solution(MCDS). We design efficient algorithms with nearly opti-mum computational complexity to compute the proposedself-pruning condition.

It is proven in [3] that every distributed algorithm forconstructing a nontrivial CDS has a lower messagecomplexity bound of �ðN logNÞ, where N is the numberof nodes and the message size is a constant multiple of the

KHABBAZIAN AND BHARGAVA: LOCALIZED BROADCASTING WITH GUARANTEED DELIVERY AND BOUNDED TRANSMISSION... 1073

number of bits representing the node IDs. We show that thesame message complexity lower bound does not apply tothe nontrivial broadcast algorithms. In fact, we show thatthe message complexity of an extension of our proposedalgorithm is OðNÞ, where N is the number of nodes and themessage size is a constant. We prove that the extendedbroadcast algorithm can also guarantee full delivery and aconstant approximation ratio to MCDS.

We show how to reduce the bandwidth requirements by

reducing the amount of piggybacked information in each

broadcast packet. We relax several system-model assump-

tions, or replace them with practical ones, to improve the

practicality of the proposed broadcast algorithm. Finally,

we verify the analytical results using simulation and

provide a comparison with one of the best broadcast

algorithms based on 1-hop information. The simulation

results show that the proposed broadcast algorithm per-

forms better than what was analytically guaranteed for the

worst case.

2 SYSTEM MODEL

In this section, we describe the system-model assumptions.

We discuss how to relax some of these assumptions in

Section 5.We consider a wireless network as a collection of

N nodes represented by points located at their positionsin a 2D plane. Each node is equipped with an omnidirec-tional antenna that has radio transmission range of R. Twodistinct nodes are called neighbors if they are in transmis-sion range of each other (i.e., the euclidean distancebetween them is less than or equal to R). We assume thateach node is able to obtain its position using an existingpositioning technique, such as the Global PositioningSystem (GPS). Each node periodically broadcasts a shortHELLO message containing its unique ID and its position.Therefore, each node knows the position of its 1-hopneighbors as well.

To prove that a broadcast algorithm guarantees fulldelivery, we assume that nodes are static during thebroadcast and that the Medium Access Control (MAC)layer is ideal, i.e., there are no transmission errors such ascollision and contention. In addition, we assume that thenetwork is connected. A broadcast algorithm can beconsidered a delivery-guaranteed broadcast algorithm if itguarantees full delivery under these assumptions. Notethat, without these assumptions, even flooding may notachieve full delivery.

3 BROADCAST ALGORITHMS BASED ON 1-HOP

NEIGHBOR INFORMATION

In this section, we consider two general structurescommonly used in neighbor-designating and self-pruningalgorithms based on 1-hop neighbor information. Thesestructures require one round of information exchange andemploy only 1-hop neighbor information to make adecision. We show that, using these structures, we are notable to guarantee both full delivery and a good bound onthe number of forwarding nodes in the network.

3.1 Neighbor-Designating Algorithms

Algorithm 1 shows a general structure of neighbor-designating broadcast algorithms based on 1-hop neighborinformation. As shown in Algorithm 1, each node sche-dules a broadcast for a received message if the node isselected by the sender and if it has not received the samemessage before. Let us represent a node NA with a point Alocated at its position and use PQ to denote the euclideandistance between two points P and Q. Suppose Algorithm 1is used as the basic structure in a delivery-guaranteedbroadcast algorithm based on 1-hop neighbor information.Let P be a point such that AP � R and, for every node,NB 6¼ NA in the network BP > R. Recall that each nodeuses 1-hop neighbor information, thus only NA knowswhether or not there is a node at the position of point P .Consequently, NA must be selected to forward the messagein order to guarantee full delivery.

Algorithm 1: A general structure of neighbor-designating

algorithms

1. Extract the required information from the receivedmessage M ;

2. if M has been received before or does not contain

node’s ID

3. drop the message;

4. else

5. set a defer timer;6. When defer timer expires

7. Based on 1-hop neighbor information:

Select a subset of neighbors to forward the message;8. Attach the list of forwarding nodes to the message;

9. Broadcast the message.

For example, suppose NA is a node on the boundary ofthe network. As shown in Fig. 1, let L be a line containing Asuch that all the nodes are either on the line or on one sideof it. This situation occurs, for instance, when A is a vertexof the network convex hull [15]. Consider a point P on theother side of the line L such that AP ¼ R and AP ? L. It isclear that BP > R for every node NB 6¼ NA. Therefore, NA

has to be selected to forward the message. This implies thatmost of the nodes around the boundary of the network(including all the nodes on the convex hull [15] of thenetwork) will broadcast the message if Algorithm 1 is usedas the basic structure in a delivery-guaranteed broadcastalgorithm based on 1-hop neighbor information. Conse-quently, neighbor-designating algorithms based on 1-hop


Fig. 1. An example of a node on the network boundary.

neighbor information may not be efficient if a large numberof nodes (compared to the total number of nodes) arelocated around the boundary of the network. The followingtheorem shows that a broadcast algorithm based on 1-hopneighbor information cannot guarantee both full deliveryand a good bound on the number of forwarding nodes if ituses Algorithm 1 as its basic structure.

Theorem 1. Suppose Algorithm 1 is used as the basic structure

in a delivery-guaranteed broadcast algorithm based on 1-hop

neighbor information. The ratio of the number of broadcasting

nodes over the minimum number of required broadcasts can be

as large as N , where N is the number of nodes in the network.

Proof. As shown in Fig. 2, suppose that all the nodes are

located on a line segmentAB of lengthR or on a circleCO;R2with a radius R

2 , where R is the radio transmission range.

For every node NA, we can find a point P outside the

network boundary such that AP ¼ R and AP is orthogo-

nal to the line segment AB=Circle CO;R2 . It is easy to

show that BP > R for every node NB 6¼ NA. Therefore,

all of the nodes will broadcast the message. However,

one broadcast is enough to transmit a message to all of

the nodes in the network for each scenario. Conse-

quently, the ratio of broadcasting nodes over the

minimum number of required broadcasts is N . tu

3.2 Self-Pruning Algorithms

Algorithm 2 shows a general structure of self-pruningbroadcast algorithms based on 1-hop neighbor information.Using this structure, a node schedules a broadcast for thefirst received copy of the message. When the defer timerexpires, the node may refrain from broadcasting themessage if a certain self-pruning condition is satisfied.Theorem 2 shows that this structure cannot guarantee bothfull delivery and a good bound on the number offorwarding nodes. Note that, in Algorithm 2, no informa-tion is piggybacked with the broadcast packet. An exten-sion of Algorithm 2 is to allow nodes to piggyback someinformation in the broadcast packet. Clearly, this extensionof Algorithm 2 is stronger than both Algorithm 1 andAlgorithm 2. In the next section, we show that a broadcastalgorithm based on 1-hop neighbor information canguarantee both full delivery and a good bound on the

number of forwarding nodes (a constant approximationratio to MCDS) if it employs this extension of Algorithm 2.

Algorithm 2: A general structure of self-pruning algorithms1. if the message M has been received before

2. Drop the message;

3. else4. Set a defer timer;

5. when defer timer expires

6. Decide whether or not to broadcast M .

Theorem 2. Suppose Algorithm 2 is used as the basic structurein a delivery-guaranteed broadcast algorithm based on 1-hopneighbor information. The ratio of the number of broadcastingnodes over the minimum number of required broadcasts is�ðNÞ in the worst case.

Proof. Consider the case where all the nodes are on two

circles CO;R2 and CO;3R2 centered at O with radii R2 and 3R2 ,

respectively. As shown in Fig. 3, suppose that, for every

node NAion CO;R2 , there is a corresponding node NBi

on

C0;3R2(and vice versa) such that AiBi ¼ R. By contra-

diction, suppose that neither NAinor NBi

broadcast the

message. Note that NAiand NBi

do not share any

neighbor. Since they have only 1-hop neighbor informa-

tion and there is no information piggybacked in the

broadcast packet, NAi(or NBi

) cannot be sure whether its

corresponding node has received the message. There-

fore, in order to guarantee full delivery, either NAior NBi

has to broadcast the message. However, only a constant

number of transmissions is enough to send the message

to all of the nodes. Therefore, the ratio of broadcasting

nodes over the minimum number of required broadcasts

is �ðNÞ. tu

4 AN EFFICIENT SELF-PRUNING BROADCAST

ALGORITHM

In the previous section, we considered two structurestypically used in the broadcast algorithms based on 1-hop


Fig. 2. Two worst-case examples for Theorem 1.

Fig. 3. A worst-case example for Theorem 2.

neighbor information. We showed that these structures

cannot guarantee the following properties simultaneously:

. full network delivery and

. good bound on the number of broadcasting nodes.

An open question is whether a localized broadcast

algorithm (e.g., a neighbor-knowledge-based broadcast

algorithm) is able to guarantee these properties. In this

section, we propose a novel self-pruning broadcast algo-

rithm based on one round of information exchange that not

only guarantees full delivery but also a constant approx-

imation ratio to the optimal solution (MCDS). As its basic

structure, the proposed broadcast algorithm (Algorithm 3)

uses a simple extension of Algorithm 2 in which each node

is allowed to piggyback some information within the

broadcast packet. In Algorithm 3, each broadcasting node

adds a list of its 1-hop neighbor IDs and positions into the

broadcast packet. Upon receiving a packet, this information

is extracted and used to determine the node’s status

(forwarding/nonforwarding) based on the following self-

pruning condition.

Definition 1 (Responsibility condition). Node NA is pruned

(has nonforwarding status) if it is not responsible for any of its

neighbors. Node NA is not responsible for its neighbor NB if

NB has received the message or if there is another node NC

such that NC has received the message and NB is closer to NC

than NA.

Algorithm 3: The proposed broadcast algorithm

1. Extract the required information from the received

packet;2. Add the broadcasting node and its 1-hop neighbor

information

(except the node’s own information) to the listListRecID .

3. if the message has been received before

4. drop the message;5. else

6. set a defer timer;

7. When defer timer expires8. if the responsibility condition is satisfied

9. Remove the previous information attached to the

message;10. Attach the list of 1-hop neighbors to the message;

11. Broadcast the packet.

Suppose NA computes a list of nodes that have received

the message ðListRecNAÞ by collecting the information piggy-

backed in the broadcast packets. To check the responsibility

condition, NA can first determine which neighbors have not

received the message. The responsibility condition is then

satisfied if and only if there is a neighbor NB that has not

received the message and

8NC 2 ListRecNA: AB � CB:

In Section 4.2, we describe efficient algorithms for comput-ing the responsibility condition.

Example 1. As shown in Fig. 4, NA has six neighbors.Suppose that NA has received the message from NH .Recall that NA extracts the list of NH ’s neighbors fromthe received packet. Therefore, it knows that NE , NF ,and NG have received the message and NB, NC , and ND

have not. According to the responsibility condition, NA

is not required to broadcast because

BE < BA; CF < CA and DG < DAðor DF < DAÞ:

4.1 Analysis of the Proposed Broadcast Algorithm

In this section, we prove that Algorithm 3 can achieve fulldelivery and a constant approximation ratio to MCDS. Inorder to prove these properties, we assume that nodes arestatic during the broadcast, the network is connected, andthe MAC layer is ideal, i.e., there is no communicationerror. As mentioned earlier in Section 2, even floodingcannot guarantee full delivery without these assumptions.

Theorem 3. Algorithm 3 guarantees that all of the nodes in thenetwork will receive the message.

Proof. Using Algorithm 3, each node broadcasts themessage at most once. Therefore, broadcasting willeventually terminate. By contradiction, suppose thereis at least one node that has not received the messageafter the broadcast termination. Let us consider the set

� ¼�ðNX;NY ÞjNX and NY are neighbors;

NX has received the message and

NY has not received the message

�:

Suppose NS is the source node (the node that initiatedthe broadcast) and NT is a node that has not received themessage. The network is connected, thus there is a pathbetween NS and NT . Clearly, we can find two neighbornodes NB and NA along the path from NT to NS suchthat NB has not received the message, while NA has.Consequently, ðNA;NBÞ 2 �, thus � 6¼ �. As a result,

9ðNA0 ; NB0 Þ 2 � s:t: 8ðNX;NY Þ 2 � : A0B0 � XY : ð1Þ

Clearly, NA0 has not broadcast since NB0 has not receivedthe message. Therefore, there must be a node NC0 such


Fig. 4. An example of self-pruning based on the responsibility condition.

that NC0 has received the message and C0B0 < A0B0 � R.

This result contradicts (1) since ðNC0 ; NB0 Þ 2 �. tu

Lemma 1. Using Algorithm 3, the number of broadcasting nodes

inside a disk DO;R4with a radius R

4 is bounded by a constant.

Proof. As shown in Fig. 5, consider three disks, DO;R4, DO;3R4

,

and DO;5R4, centered at O with radii R

4 , 3R4 , and 5R

4 ,

respectively. Let us define

R1 �R2 ¼ fP jP 2 R1 and P =2 R2g;

where R1 and R2 are two regions and P is a point.

Suppose k is the minimum number such that, for every

set of k points Pi 2 DO;5R4�DO;3R4

, 1 � i � k, we have

9Pi; Pj : i 6¼ j and PiPj �R

2:

Note that the area DO;5R4�DO;3R4

can be covered with a

constant number of disks with radius R4 . If the number of

points inside DO;5R4�DO;3R4

is greater than the number of

covering disks, at least one covering disk will contain

more than one point. For two points P1 and P2 inside a

disk with a radius R4 , we have P1P2 � R

2 . Therefore, k is

bounded by a constant (the number of covering disks

plus one).

We prove that the number of broadcasting nodes inside

DO;R4is less than or equal to k. By contradiction, suppose

that there are more than k broadcasting nodes insideDO;R4.

Let NAi2 DO;R4

, 0 � i � k, be the first kþ 1 broadcasting

nodes ordered chronologically based on their broadcast

time. Based on the responsibility condition, for every

node NAi, there is a corresponding neighbor NBi

such

that NBi62 ListRecNAi

and AiBi � CBi for every node

NC 2 ListRecNAi, where ListRecNAi

is the list of nodes that have

received the message based on NAi’s collected informa-

tion. Note that every node inside DO;3R4will receive the

message after NA0’s broadcast. Therefore, NBi

62 DO;3R4for

1 � i � k. On the other hand, NBiis a neighbor of NAi

,

thus NBi2 DO;5R4

. Consequently,

81 � i � k : NBi2 DO;5R4

�DO;3R4:

It follows that

81 � i � k : AiBi >R

2: ð2Þ

Suppose 1 � i < j � k. The nodes NAiand NAj

areneighbors because AiAj � R

2 . Recall that NAipiggybacks

the list of its neighbors within the broadcast packet.Therefore, NBi

2 ListRecNAj, thus NBi

6¼ NBj. It follows that

NB1. . .NBk

are k different points inside DO;5R4�DO;3R4

.Therefore,

9NBi;NBj

: i < j and BiBj �R

2: ð3Þ

This is a contradiction because NBi2 ListRecNAj

and, basedon (2) and (3),

AjBj >R

2¼)AjBj > BiBj:

Thus, according to the responsibility condition, NAjis

not responsible for NBj. tu

Theorem 4. Algorithm 3 gives a constant approximation ratio tothe optimal solution (MCDS).

Proof. Clearly, a disk with radius R can be covered with aconstant number of disks with radius R

4 . Therefore,based on Lemma 1, the number of broadcasting nodesinside a disk with radius R is bounded by a constantCmin. Let jMCDSj be the number of nodes in the MCDS.Each broadcasting node is inside the transmission rangeof at least one node in the MCDS. On the other hand, thenumber of broadcasting nodes inside the transmissionrange of each node in the MCDS is bounded by Cmin.Therefore, the total number of broadcasting nodes is notmore than Cmin � jMCDSj. tu

The following theorem shows that the message complex-ity of the proposed broadcast algorithm is OðNÞ, where Nis the total number of nodes in the network and eachmessage contains a node’s ID and its position. In Section 5,we will show that the message size can be reduced to aconstant number of bits by removing the node ID from themessage and using a constant number of bits to representeach node’s position.

Theorem 5. The message complexity of Algorithm 3 is OðNÞ,where N is the total number of nodes in the network and eachmessage contains a node’s ID and its position.

Proof. There are two types of messages used in theproposed broadcast algorithm. First, each node broad-casts its ID and position in a short HELLO message.Second, each broadcasting node piggybacks the list of its1-hop neighbor IDs and positions into the broadcastpacket. We will consider this packet as n separatemessages, where n is the number of 1-hop neighbors ofthe broadcasting node. Therefore, each message contains


Fig. 5. Three cocentered disks; NAi2 DO;R4

and NBi2 DO;5R4

�DO;3R4.

exactly one node’s ID, together with its position. Asshown in the proof of Theorem 4, the number ofbroadcasting neighbors of each node is bounded by aconstant Cmin. Therefore, each node’s ID and positionappears in at most Cmin þ 1 messages, including theHELLO message. Consequently, the total number ofmessages is not more than ðCmin þ 1Þ �N . tu

4.2 Efficient Algorithms for Computing theResponsibility Condition

The problem of computing the responsibility condition canbe expressed by the following geometry problem:

Definition 2 (Closest-point problem (CPP)). Let P ¼fP0; P1; . . . ; Pmg and Q ¼ fQ1; Q2; . . . ; Qng be two disjointsets of points. Is there a point Q 2 Q such that

8P 2 P : QP0 � QP?

The CPP attempts to establish whether or not P0 is theclosest point in P to at least one point in Q. One approachto solving CPP is to find the closest point in P to everypoint in Q. The trivial algorithm for finding the closestpoint in P to a query point Q 2 Q is to compute the distancefrom Q to all of the points in P, keeping track of the “closestpoint so far.” Clearly, using this algorithm, the computa-tional complexity of solving the CPP would be Oðm� nÞ.Finding the closest point inP to a query pointQ 2 Q is a well-known problem called the Nearest Neighbor Search (NNS)problem. To solve the NNS problem, several space-partition-ing methods, including the kd-tree and the Voronoi diagram[15], can be used. A kd-tree is a data structure used fororganizing points in k-dimensional space. A kd-tree for the setof 2D points P can be constructed in Oðm logmÞ. Moreover,the computational complexity of a query is OðlogmÞ [15].Therefore, using kd-tree, the CPP can be solved inOðm logmþ n logmÞ. Similarly, we can use the Voronoidiagram to solve CPP. The Voronoi diagram is a partitioningof a plane with n generating points into convex polygons,called Voronoi cells, such that each cell contains exactly onegenerating point and every node in the cell is closer to thegenerating point of the cell than to other generating points. Itis known that the Voronoi diagram can be constructed inOðn lognÞ and that each cell hasOðnÞ edges in the worst case.Therefore, using the set of points P as the generating points,the Voronoi diagram can be generated in Oðm logmÞ. ThenodeP0 is the closest node inP to a query pointQ if and only ifQ is in the Voronoi cell of P0. We can check whether a querypoint Q is in the Voronoi cell of P0 in OðlogmÞ since theVoronoi cell of P0 is a convex polygon with at mostOðmÞ edges. Therefore, the CPP can be solved inOðm logmþn logmÞ using the Voronoi diagram. Note that we only needto compute the Voronoi cell of P0. The Voronoi cell of P0 canbe computed inOðm log hÞ, whereh is the number of edges ofthe cell. The average number of vertices of a Voronoi cell isless than six [15], thus h is a constant on average. Therefore,the average case computational complexity of this approachis Oðmþ nÞ.

Suppose P0 is not the closest point in P to any node in Q.In this case, �ðnÞ is a lower bound for every algorithm tosolve the CPP for sets P and Q because the algorithmcannot ignore any node in the set Q. On the other hand,every algorithm to solve the CPP has to consider all of the

points in P if P0 is the closest point in P to a node in Q.Thus, �ðmÞ is also a lower bound for such algorithms.Considering both cases, it follows that �ðmþ nÞ is a lowerbound for every algorithm to solve the CPP. We describedalgorithms that compute the CPP in OððnþmÞ logmÞ in theworst case. These algorithms are nearly optimal in terms ofcomputational complexity because logm is a small factor inpractice. Moreover, by constructing the Voronoi cell of P0,we showed that the CPP can be solved in OðnþmÞ in theaverage case.

Theorem 6. The worst case complexity of computing theresponsibility condition is Oð�G log �GÞ, where �G is themaximum node degree of the network.

Proof. Based on Algorithm 3, a node NA stores thebroadcasting neighbors’ information and their 1-hopneighbors’ information (except NA’s information) inListRecNA

. The node NA can compute ListNotRecNA, the list of

its neighbors that have not received the message, inOðl� nÞ, where 1 � l � n is the number of broadcastingneighbors and n is the total number of neighbors.Consider the set of points

P ¼ fP0; P1; . . . ; Pmg;

where P0 is located at the position of NA andP1; P2; . . . ; Pm are located at the positions of the nodesin ListRecNA

. Suppose

Q ¼ fQ1; Q2; . . . ; Qkg

is the set of points located at the position of the nodes inListNotRecNA

. As mentioned earlier, by constructing theVoronoi cell of P0, we can compute the CPP in OððmþkÞ logmÞ for sets P and Q. Clearly, the responsibilitycondition is satisfied if and only if there exists a pointQ 2 Q such that

8P 2 P : QP0 � QP:

As shown in the proof of Theorem 4, l (the number ofbroadcasting neighbors of NA) is bounded by a constant.Moreover, we have k � n, m � l��G, and n � �G,where �G is the maximum node degree of the network.Therefore, the complexity of computing the responsi-bility condition is

Oðl� nÞ þO ðmþ kÞ logmÞ ¼ Oð�G log �Gð Þ:

Note that the CPP can be solved in OðkþmÞ in theaverage case. Therefore, the average complexity ofcomputing the responsibility condition is

Oðl� nÞ þOðmþ kÞ ¼ Oð�GÞ:ut

4.3 Reducing Bandwidth Requirements

In Algorithm 3, each forwarding node piggybacks the list ofits 1-hop neighbors in the broadcast packet. In this section,we show how a forwarding node can reduce bandwidthoverhead by piggybacking a list of a subset of its 1-hopneighbors.

Definition 3 (Representative set). Let P be a set of pointsfP1; P2; . . . ; Png inside a circle CO;R. The set S � P is a


representative set for P if, for every point Q outside CO;R (i.e.,

OQ > R),

9P 0 2 S s:t: 8P 2 P : P 0Q � PQ:

A representative set Smin of P is called the minimum

representative set of P if Smin has the minimum cardinality

among all the representative sets of P.

Definition 4 (Representative neighbor set). We say P is the

corresponding point of the node NX if P is located at the

position of NX . Let P be the set of corresponding points of

NA’s neighbors. A subset of NA’s neighbors is called a

representative neighbor set if their corresponding point set is a

representative set of P. The set of NA’s representative

neighbors with the minimum number of nodes is called the

minimum representative neighbor set.

Lemma 2. In Algorithm 3, a forwarding node can piggyback a

list of its representative neighbors instead of the list of all its

neighbors without affecting the forwarding status of any other

node in the network.

Proof. Let ListRepNAbe a list of representative nodes of NA.

Suppose NA piggybacks ListRepNAinstead of the list of all

its neighbors, and NB 62 ListRepNAis a neighbor of NA. The

node NB is required in computing the responsibility

condition of NA’s neighbor NC if NC has a neighbor ND

that has not received the message and BD < CD. Note

that ND is not a neighbor of NA since all of NA’s

neighbors have received the message. Therefore, ND is

outside CA;R, where CA;R is the transmission range of

NA. Based on the definition of representative neighbor

set, there exists a node NE 2 ListRepNAsuch that

ED � BD ¼) ED < CD:

Therefore, for each neighbor of NA required in comput-ing the responsibility condition of NC , we can find anode in ListRepNA

that leads to the same result. tuLemma 2 shows that forwarding nodes can save

bandwidth by piggybacking the list of their representativeneighbors instead of the list of all of their neighbors.Clearly, finding the minimum representative neighbor setis equivalent to computing the minimum representative setof P, where P is the set of points corresponding to thenode’s neighbors. Let us denote the Voronoi diagram of Pby VorðPÞ. Assume that CellðP Þ is the set of vertices of thecell of VorðPÞ that corresponds to a point P 2 P. Note thatCellðP Þ includes a vertex v1 “at infinity” if the Voronoi cellof P has an infinite edge. Theorem 7 shows that theminimum representative set of P is unique and can becomputed in OðjPj log jPjÞ.Lemma 3. Suppose P is a set of points inside the circle CO;R and

Smin is the minimum representative set of P. We have

P 2 Smin () 9V 2 CellðP Þ : OV > R:

Proof. Suppose that

8V 2 CellðP Þ : OV � R:

In this case, the entire Voronoi cell of P will be inside thecircle CO;R. Therefore, for every point Q outside thecircle, we have

9P 0 2 P : P 0Q < PQ

because Q is outside the Voronoi cell of P . Conse-quently, P is not required to be in a representative set ofP, thus it is not in the minimum representative set of P.Now, suppose that

9V 2 CellðP Þ : OV > R:

Since the point P is inside the circle, the line segment PVcrosses the circle in a point U . Consider a point Q on theline segment UV such that Q 6¼ U and Q 6¼ V . Clearly, Qis outside the circle CO;R, i.e., OQ > R. Since the Voronoicell of P is a convex polygon, the point Q is inside thecell and is not located on any edge of it. Therefore, wehave

8P 0 2 P n fPg : PQ < P 0Q:

Consequently, P must be in every representative set ofP, including its minimum representative set. tu

Theorem 7. Suppose P is a set of points inside a circle CO;R. The

minimum representative set of P is unique and can becomputed in OðjPj log jPjÞ.

Proof. Based on Lemma 3, a point P is in the minimumrepresentative set of P if and only if

9V 2 CellðP Þ : OV > R:

There is a single Voronoi diagram associated with eachset of generating points. Therefore, the minimumrepresentative set is unique. From Lemma 3, it alsofollows that the minimum representative set of P is asubset of every representative set of P. To compute theminimum representative set of P, we first compute theVoronoi diagram VorðPÞ. For each Voronoi cell, we thencheck whether it has a vertex outside the circle CO;R. TheVoronoi diagram of P can be computed in OðjPj log jPjÞ.Moreover, the average number of vertices of a cell is lessthan six [15]. In other words, the total number of vertex-checking operations is less than 6� jPj. Therefore, thecomplexity of computing the representative set is

O jPj log jPjð Þ þO jPjð Þ ¼ O jPj log jPjð Þ:ut

In the worst case, we have jSminj ¼ jPj, where Smin is theminimum representative set of P. This situation occurs, forexample, when all the points in P are located in a linesegment or on a circle. However, it can be shown that

limjPj!1

E jSminjð ÞjPj ¼ 0;

where EðjSminjÞ is the expected cardinality of the minimumrepresentative set of P. In order to avoid complex analyticalanalysis, we used simulation to analyze EðjSminjÞ. Thesimulation results are presented in Section 6.


Suppose that the forwarding nodes can piggyback a listof their representative neighbors instead of the list of alltheir neighbors. In this case, Theorem 8 shows that a nodecan simply avoid broadcasting if it is not in the list of thenodes piggybacked in at least one received packet.

Theorem 8. A node NA can set its status to nonforwarding for amessage M if it receives M in a packet that does not list NA asa representative node.

Proof. Suppose NA receives the message from node NB andthat NA is not in the list of NB’s representative nodespiggybacked in the packet. We prove that NA’s status isnonforwarding based on the responsibility condition.NA’s status is nonforwarding if all of its neighbors havereceived the message, so we assume that this is not thecase. Suppose NC is a neighbor of NA that has notreceived the message. Therefore, NC is not a neighbor ofNB, hence it is outside CB;R, where CB;R is thetransmission range of NB. Note that NA is not in theminimum representative neighbor set of NB becausethere exists a representative neighbor set of NB that doesnot contain NA. Therefore, based on Lemma 3, theVoronoi cell of the point A is inside CB;R and, thus, C isoutside the Voronoi cell of A. Consequently, there existsa node ND 6¼ NA in the representative neighbor set of NB

such that

DC < AC:

It follows that NA is not responsible for any of itsneighbors and, therefore, has a nonforwarding status. tu

5 RELAXING SOME OF THE SYSTEM-MODEL

ASSUMPTIONS

The assumptions made in Section 2 are often used in theexisting broadcast algorithms in order to model thenetwork. Some of these assumptions are not practical. Forexample, a node may obtain its position using the GPS.However, the position is not accurate due to several sourcesof error, including positioning error and the limitednumber of bits used to represent the position. In thissection, we discuss how to relax some of the system-modelassumptions or replace them with more practical ones inorder to improve the practicality of our proposed broadcastalgorithm. We show that all of the extensions of theproposed algorithm can provide both full delivery and aconstant approximation ratio to the optimal solution underthe new assumptions.

5.1 Broadcasting under Uncertain PositionInformation

As mentioned earlier, the assumption of having the preciseposition is not practical. In practice, there are severalsources of error. For example, the position informationobtained by GPS typically includes some errors, which varybetween different GPS devices. Another source of error isroundoff error or representation error, which is associatedwith the fact that a finite number of bits is used to representthe position information. Typically, each node in thenetwork is able to reduce the representation error byassigning a fairly large number of memory bits to representa number. However, to reduce the bandwidth and power

overhead, it is desirable to use as few bits as possible torepresent the position information added in the HELLOmessages or the broadcast packets. In this section, we showthat a constant number of bits suffices to maintain the mainproperties of the proposed algorithm.

Let � be an upper bound on the maximum position errorand E � � an upper bound for maximum error in comput-ing the distance between two points. Suppose that E isknown by all the nodes in the network. We have

AB� E � �ðABÞ � ABþ E; ð4Þ

where �ðABÞ is the approximated distance between the

points A and B. Algorithm 4 shows how to compute the

responsibility condition under uncertain position informa-

tion. Let ListNeighNAbe the list of NA’s neighbors, ListBrdNA

be

the list of NA’s neighbor that have broadcast the message,

and ListRecNAbe the list of nodes that have received the

message, i.e., all of the nodes in ListBrdNAtogether with their

1-hop neighbors. Algorithm 4 first computes ListNotRecNA, the

list of its neighbors that have not received the message

(lines 1-8). Node NA assumes that its neighbor NB has

received the message if and only if

9NC 2 ListBrdNA: NB 2 ListNeighNC

: ð5Þ

Algorithm 4 returns true if and only if NA has a neighbor

NB 2 ListNotRecNAsuch that

8NC 2 ListRecNA: �ðABÞ � E � �ðBCÞ þ E: ð6Þ

The output of Algorithm 4 determines whether theresponsibility condition is satisfied.

Algorithm 4: Computing the responsibility condition underuncertain position information

Input: ListNeighID , ListBrdID , ListRecID and the max error E.

output: true or false.

1. emptyðListNotRecID Þ; //make the list empty

2. for (i ¼ 1; i � lengthðListNeighID Þ; ++i)3. chk true;

4. for (j ¼ 1; j � lengthðListBrdID Þ; ++j)

5. if ðListNeighID ½i� 2 ListNeighListBrd

ID½j�Þ

6. chk false; break;

7. if ðchkÞ8. AddðListNeighID ½i�; ListNotRecID Þ

9. for (i ¼ 1; i � lengthðListNotRecID Þ; ++i)

10. d distðListNotRecID ½i�; IDÞ;11. chk true;

12. for (j ¼ 1; j � lengthðListRecID Þ; ++j)

13. if ðdistðListNotRecID ½i�; ListRecID ½j�Þ < d� 2EÞ14. chk false; break;

15. if ðchkÞ16. return true;17. return false.

Theorem 9. Algorithm 3 guarantees full delivery under

uncertain position information if it uses Algorithm 4 to

compute the responsibility condition.


Proof. The proof is similar to that of Theorem 3. Thebroadcasting will eventually terminate since each nodetransmits the message only once. By contradiction,suppose there is a node that has not received themessage after termination of broadcasting. Let usconsider the set

� ¼�ðNX;NY Þ j NX and NY are neighbors;

NX has received the message and

NY has not received the message

�:

Suppose NS is the source node and NT is a node that hasnot received the message. The network is connected;thus there is a path between NS and NT . Clearly, we canfind two neighbor nodes NB and NA along the path fromNT to NS such that NB has not received the message,while NA has. Consequently, ðNA;NBÞ 2 �, thus � 6¼ �.As a result,

9ðNA0 ; NB0 Þ 2 � s:t: 8ðNX;NY Þ 2 � : A0B0 � XY : ð7Þ

Node NB0 has not received the message. Therefore, for

every broadcasting node NF , we have

NB0 62 ListNeighNF:

Consequently, based on (5), NA0 does not consider NB0 asa node that has received the message. Clearly, NA0 hasnot broadcast since NB0 has not received the message.Considering the responsibility condition, it follows thatthere is a node NC0 such that NC 0 has received themessage and

�ðB0C0Þ þ E < �ðA0B0Þ � E:

Using (4), we get

B0C0 < A0B0:

This result contradicts (7), since ðNC0 ; NB0 Þ 2 �. tu

Theorem 10. Using Algorithm 4 to compute the responsibility

condition, the proposed broadcast algorithm (Algorithm 3)

guarantees a constant approximation ratio to the optimum

solution under uncertain position information if E ¼ R8 � �R,

where 0 < � � 18 and 1

� is bounded by a constant number.

Proof. We show that the number of broadcasting nodes

inside a disk DO;R4with a radius R

4 is bounded by a

constant. Using the same approach used in the proof of

Theorem 4, it then follows that the total number of

broadcasting nodes is bounded by a constant factor of

that of the optimum solution. Let DO;R4, DO;3R4

, and DO;5R4

be three disks centered at O with radii R4 , 3R

4 , and 5R4 ,

respectively. Suppose k is the minimum number such

that, for every set of k points, Pi 2 DO;5R4�DO;3R4

,

1 � i � k, we have

9Pi; Pj : i 6¼ j and PiPj � �R:

Note that 1� is bounded by a constant, so the area DO;5R4

�DO;3R4

can be covered with a constant number of disks

with radius �2R. If the number of points inside DO;5R4

�DO;3R4

is greater than the number of covering disks, at

least one covering disk will contain more than one point.

For two points P1 and P2 inside a disk with radius �2R,

we have P1P2 � �R. Therefore, k is bounded by a

constant (the number of covering disks plus one). We

prove that the number of broadcasting nodes inside DO;R4

is less than or equal to k. By contradiction, suppose that

there are more than k broadcasting nodes inside DO;R4.

Let NAi2 DO;R4

, 0 � i � k, be the first kþ 1 broadcasting

nodes ordered chronologically based on their broadcast

time. Based on the responsibility condition, for each

node NAi, there is a corresponding neighbor NBi

such

that NBi62 ListRecNAi

and

8NC 2 ListRecNAi: �ðAiBiÞ � E � �ðBiCÞ þ E;

where ListRecNAiis the list of nodes that have received the

message based on NAi’s collected information. Suppose

1 � i < j � k. Nodes NAiand NAj

are neighbors because

AiAj � R2 . Recall that NAi

piggybacks the list of all of its

neighbors within the broadcast packet. Therefore,

NBi2 ListRecNAj

, thus NBi6¼ NBj

. Note that every node

NC 2 DO;3R4will receive the message after NA0

’s broad-

cast because A0C � R. Therefore, NC is in the list of

nodes piggybacked by NA0in the broadcast packet.

Thus,

81 � i � k : NC 2 ListRecNAi:

It follows that

81 � i � k : NBi=2DO;3R4

:

On the other hand, NBiis a neighbor of NAi

, thusNBi2 DO;5R4

. Consequently,

81 � i � k : NBi2 DO;5R4

�DO;3R4:

Thus,

81 � i � k : AiBi >R

2: ð8Þ

There are k different nodes NB1. . .NBk

insideDO;5R4

�DO;3R4. Therefore,

9NBi;NBj

: i < j and BiBj � �R: ð9Þ

From (4), we have

�ðBiBjÞ þ E � BiBj þ 2E:

Thus, using (9), we get

�ðBiBjÞ þ E �R

8þ E: ð10Þ


On the other hand, we have AjBj >R2 . Hence,

�ðAjBjÞ � E >R

2� 2E: ð11Þ

Since E � R8 , we have

R

8þ E � R

2� 2E:

Thus, based on (10) and (11), we get

�ðBiBjÞ þ E < �ðAjBjÞ � E:

This result is a contradiction because, based on (6), NAjis

not responsible for NBj. tu

Example 2. Suppose that E � R9 . Therefore, we have

E ¼ R8� �R;

where � � 172 . Note that 1

� ð� 72Þ is bounded by a

constant. Thus, based on Theorem 10, the proposed

broadcast algorithm can guarantee a constant approx-

imation ratio to the optimal solution if the maximum

error E is less than or equal to R9 .

As shown in Theorem 10, Algorithm 3 can guarantee aconstant approximation ratio to MCDS if the maximumerror is bounded by a factor of the transmission range. As aresult, employing the same approach used in the proof ofTheorem 5, we can show that the message complexity ofAlgorithm 3 under uncertain position information is stillOðNÞ, where N is the total number of nodes in the networkand each message contains a node ID and its position. Notethat node IDs can be simply removed from the messagewithout affecting the functionality of the proposed broad-cast algorithm. This is because the responsibility conditionis based solely on the approximate/precise positioninformation of the nodes.

Suppose that all of the nodes in the network are in an

L� L square area. Clearly, each node requires an infinite

number of bits in general to precisely represent its position.

However, allowing errors in position information, node

position can be represented by a limited number of bits,

provided that the size of square area is finite. For example,

assume that we allow the maximum position error of �R.

As shown in Fig. 6, the L� L square area can be partitioned

into d Lffiffi2p

�Re � d Lffiffi

2p

�Re grid cells of size

ffiffiffi2p

�R�ffiffiffi2p

�R. Let

the tuple ðI; JÞ denote the coordinate of the cell in the grid.

For instance, (1, 1) indicates the bottom left cell of the gird.

Suppose each node uses the coordinate of the cell in which

it is located to indicate its position. In this case, the position

of the node can be estimated by the position of the center of

the cell. Therefore, the number of bits required to represent

the approximate position is at most

lg2

Lffiffiffi2p

�R

� �� þ lg2

Lffiffiffi2p

�R

� �� 2 lg2

L

�R

� �

and the maximum position error is one-half of the cell

diagonal, i.e.,

ffiffiffi2pðffiffiffi2p

�RÞ2

¼ �R:

Note that the number of bits required to represent the

approximate position increases as LR increases.

Lemma 4. Suppose that the node NA is required to send its

position with the maximum error �R to its neighbor NB. In

this case, NA requires only a constant number of bits to

represent its position if 1� is bounded by a constant.

Proof. Suppose ði; jÞ is the coordinate of the cell in which

NA is located and that the side of each grid cell isffiffiffi2p

�R.

Node NA can transmit the tuple

imod

ffiffiffi2p

�

þ 2

� �; jmod

ffiffiffi2p

�

þ 2

� ��

to node NB. Suppose ði; jÞ and ði0; j0Þ are the coordinates

of two different cells such that

i i0 mod

ffiffiffi2p

�

þ 2

� �;

and

j j0 mod

ffiffiffi2p

�

þ 2

� �:

Let P1 and P2 be two points inside the cells ði; jÞ and

ði0; j0Þ, respectively. It is easy to show that

P1P2 �ffiffiffi2p

�

þ 1

� �� ð

ffiffiffi2p

�RÞ ¼) P1P2 > 2R:

It follows that a circle with radius R cannot intersect

both cells. Therefore, NB can uniquely identify the cell

using the tuple

imod

ffiffiffi2p

�

þ 2

� �; jmod

ffiffiffi2p

�

þ 2

� �� :

Consequently, NA is required to transmit only

2 lg2

ffiffiffi2p

�

þ 2

� �� 2 lg2

1

�

� �þ 2


Fig. 6. Partitioning the network into square cells.

bits, which is constant if 1� is bounded by a constant.

Node NB can approximate the position of NA by takingthe position of the center of the cell. tuAs mentioned earlier, the message complexity of Algo-

rithm 3 is OðNÞ under uncertain position information if themaximum position error is bounded. Moreover, usingLemma 4, it follows that the size of each message can bereduced to a constant number of bits by removing the node IDand using a constant number of bits to represent theapproximate position. In [3], the authors proved that everydistributed algorithm for constructing a nontrivial CDS hasthe lower message complexity bound of �ðN logNÞ, whereNis the number of nodes and the message size is a constantmultiple of the number of bits representing the node IDs.Although finding a nontrivial CDS and designing a non-trivial broadcast algorithm are closely related problems, ourresults show that the same message complexity lower bounddoes not apply for the nontrivial broadcast algorithms.

5.2 Relaxing the Homogeneous NetworkAssumption

In practice, devices may have different transmission ranges.Suppose G0 is a graph for which there is a link from NA toNB if NB is in transmission range of NA. Let G be anundirected graph obtained by removing unidirectionallinks of the graph G0. We assume that G is connected anddefine two nodes as neighbors if there is a link betweenthem in G (i.e., they are in transmission range of eachother). Note that many wireless MAC protocols, such asIEEE 802.11, require bidirectional links.

Let us use RNAto denote a lower bound of

NA’s transmission range. Suppose each node includes thelower bound of its transmission range in the HELLOmessage. Moreover, assume that the forwarding nodesinclude the lower bounds of their neighbors’ transmissionranges in the broadcast packets. The responsibility condi-tion of node NA is satisfied if NA has a neighbor NB suchthat NB has not received the message and, for every nodeNC that has received the message,

AB � CB or CB > RNC:

Using a similar approach as that used in the proof ofTheorem 3, we can show that Algorithm 3 guarantees fulldelivery in a heterogeneous network if it employs the newresponsibility condition.

Suppose Rmin is a lower bound for the transmissionranges of all the nodes in the network and Rmax is an upperbound. The following theorem shows that the proposedbroadcast algorithm can still guarantee a good bound onthe number of forwarding nodes in a heterogeneousnetwork if Rmax

Rminis bounded by a constant.

Theorem 11. The proposed broadcast algorithm can guarantee aconstant approximation ratio to the optimal solution in aheterogeneous network if Rmax

Rminis bounded by a constant.

Proof. Let DO;

Rmin4

, DO;

3Rmin4

, and DO;

Rmin4 þRmax

be three disks

centered at O with radii Rmin

4 , 3Rmin

4 , and Rmin

4 þ Rmax,

respectively. Suppose k is the minimum number such

that, for every set of k points Pi 2 DO;Rmin

4 þRmax�D

O;3Rmin

4

,

1 � i � k, we have

9Pi; Pj : i 6¼ j and PiPj �Rmin

2:

Note that Rmax

Rminis bounded by a constant, so the area

DO;

Rmin4 þRmax

�DO;

3Rmin4

can be covered with a constant

number of disks with radius Rmin

4 . If the number of points

inside DO;

Rmin4 þRmax

�DO;

3Rmin4

is greater than the number

of covering disks, at least one covering disk will contain

more than one point. For two points P1 and P2 inside a

disk with radius Rmin

4 , we have P1P2 � Rmin

2 . Therefore, k

is bounded by a constant (the number of covering disks

plus one). Using a technique similar to that used in the

proof of Lemma 1, we can show that the number of

broadcasting nodes inside DO;

Rmin4

is less than or equal to

k. Since Rmax

Rminis bounded by a constant, a disk with radius

Rmax can be covered with a constant number of disks

with radii Rmin. It follows that the total number of

broadcasting nodes is bounded by a constant factor of

that of the optimum solution (refer to the proof of

Theorem 4). tu

5.3 Broadcasting in Three Dimensions

We assumed in Section 2 that the nodes are placed in a2D plane. In general, the nodes can be located in 3D spaceas the network area may not be perfectly flat. In this case,we assume that the nodes are provided with their positionin 3D. Interestingly, all of the properties of the proposedbroadcast algorithm are preserved in 3D. Assuming AB asthe euclidean distance of two points in 3D, we can use theproof of Theorem 3 to show that Algorithm 3 guaranteesfull delivery in 3D space. Note that the transmission rangeof a node in 3D is a sphere. Replacing circles by spheres inthe proofs of Lemma 1 and Theorem 4, we can similarlyargue that Algorithm 3 guarantees a constant approxima-tion ratio to the optimal solution in 3D. Finally, using atechnique similar to that used for 2D, we can show that, in3D, the proposed algorithm not only preserves its mainproperties under uncertain position information (providedthat the maximum error is bounded), but also can benefitfrom the proposed bandwidth-overhead reduction techni-que introduced in Section 4.3.

6 SIMULATION

6.1 Reducing Bandwidth Overhead

As shown in Section 4.3, a forwarding node can piggybackthe list of a subset of its neighbors (its representativeneighbor set) instead of all of them without violating thestatus (forwarding/nonforwarding) of any other node inthe network. To avoid the complexity of mathematicalanalysis, we used simulation to find the average cardinalityof the minimum representative set. For a given number ofneighbors 1 � n � 500, we randomly put n points inside aunit circle and computed the minimum representative setSmin. To get the average cardinality EðjSminjÞ, we ran thesimulation 104 times for each given n. Fig. 7 shows the ratioEðjSminjÞ

n � 100. As shown in Fig. 7, the ratio EðjSminjÞn decreases

as the number of neighbors increases. For instance, the


minimum number of representative nodes is, on average,less than 50 percent of total number of nodes when n � 33.

6.2 Performance of the Proposed BroadcastAlgorithm

To verify the theoretical results, we used the ns-2 simulator tocompute the average number of forwarding nodes. In ns-2,the total size of the data packet was fixed to 256 bytes andthe bandwidth of the wireless channel was set to 2 Mbps. Ineach simulation run, we uniformly distributed N nodes in a1;000� 1;000 m2 square area. A randomly generatedtopology was discarded if it led to a disconnected network.Only one broadcast was initiated in each simulation run bya randomly selected node. Table 1 summarizes some of theparameters used in ns-2.

Fig. 8 illustrates an instance of using the proposedalgorithm for the case where R ¼ 300 m, N ¼ 400, andnodes are placed in a square area of 1;000� 1;000 m2. Asshown in this figure, only 10 nodes (represented by stars)among 400 nodes broadcast the message. Figs. 9 and 10show the ratio of the number of broadcasting nodes overthe total number of nodes. To get the results shown inFig. 9, we varied the transmission range from 50 to 300 mand fixed the total number of nodes to 1,000. Fig. 10 showsthe result of the experiment for which the transmissionrange was fixed to 250 m and the total number of nodesvaried from 25 to 1,000. We compared the performance ofour proposed algorithm with that of the broadcast algo-rithm proposed in [5]. In [5], Liu et al. showed that thenumber of broadcasting nodes using their proposed flood-ing algorithm is significantly lower than that of previousnotable broadcast algorithms [16], [17]. We also considered

the ratio-8 approximation of MCDS [3] as a benchmark. As

shown in Figs. 9 and 10, our proposed broadcast

algorithm can significantly reduce the number of forward-

ing nodes. Moreover, it slightly outperforms the ratio-8

approximation. In [18], we proved that, using the

proposed broadcast algorithm, the probability of two


Fig. 7. The ratio EðjSmin jÞn � 100 against the total number of neighbors.

TABLE 1Simulation Parameters

Fig. 8. Broadcasting nodes in a 103 � 103 m2 square area with

400 nodes.

Fig. 9. Ratio of broadcasting nodes versus transmission range.

Fig. 10. Ratio of broadcasting nodes versus total number of nodes.

neighbor nodes transmitting the same message exponen-tially decreases when the distance between them decreases orwhen the node density increases. This property can furtherjustify why the proposed broadcast algorithm can signifi-cantly reduce the number of transmissions in the network.

Another metric evaluated using simulation was thealgorithm delay, which we define as the time between thefirst and the last transmission in a single message broad-cast. Fig. 11 shows the average delay of our proposedalgorithm and compares it with that of Liu et al.’s broadcastalgorithm. As shown in this figure, the average delay of ourbroadcast algorithm is 80 percent to 50 percent of that ofLiu et al.’s algorithm when the total number of nodes variesfrom 50 to 1,000.

We also evaluated the performance of the proposedalgorithm when there is uncertainty in position informa-tion. We set the transmission range to R ¼ 250 meters andfixed the maximum position error to �R, where0:01 � � � 0:15. Fig. 12 shows the average number ofbroadcasting nodes for a given � when the number ofnodes varies within 25-1,000. As shown in Fig. 12, theperformance of the broadcast algorithm slightly degradesas the maximum position error increases. Finally, weevaluated the performance of the proposed algorithm forthe case where nodes can have different transmissionranges. In the simulation, we randomly placed 25 � N �1;000 nodes in the network. The transmission range of eachnode was selected randomly from the interval ½Rmin; 250�,where Rmin < 250 is the minimum transmission range. Thevalue of Rmin varied within 150 to 250 m. As shown inFig. 13, the simulation results indicate that the performanceof the proposed broadcast algorithm slightly decreases asRmin decreases. Based on the simulation results shown inFig. 9, this effect is expected as the number of broadcastsincreases when the transmission range decreases.

7 CONCLUSION

We considered two general structures typically used inbroadcast algorithms based on 1-hop neighbor information.We showed that a broadcast algorithm cannot guarantee

both full delivery and a good bound on the number of

transmissions if it uses any of these structures. It is

commonly believed that a localized broadcast algorithm is

not able to guarantee a good bound on the number of

transmissions. We presented a localized broadcast algo-

rithm based on 1-hop information and proved that it

guarantees both full delivery and a constant approximation

ratio to the optimal solution. The proposed broadcast

algorithm has low message and computational complex-

ities. In fact, we proved that the message complexity of the

proposed algorithm is less than the lower message

complexity bound of finding a nontrivial CDS and that its

computational complexity is nearly optimal. Moreover, we

proposed a technique to further reduce the bandwidth

overhead. We relaxed several system-model assumptions,

or replaced them with practical ones, to improve the

practicality of the broadcast algorithm. Finally, we verified

the analytical results using simulation. Using the proposed

algorithm, a node decides “on the fly” whether or not to

broadcast and the set of broadcasting nodes can change

upon network changes. Therefore, we believe that our


Fig. 11. Average delay versus total number nodes.

Fig. 12. Ratio of broadcasting nodes of the proposed algorithm with

uncertain position information.

Fig. 13. Ratio of broadcasting nodes of the proposed algorithm in a

network with nodes having different transmission ranges.

proposed broadcast algorithm is very reliable and robustagainst node’s failure and packet losses.

REFERENCES

[1] M. Garey and D. Johnson, Computers and Intractability; A Guide tothe Theory of NP-Completeness. W.H. Freeman, 1990.

[2] B. Clark, C. Colbourn, and D. Johnson, “Unit Disk Graphs,”Discrete Math., vol. 86, pp. 165-177, 1990.

[3] P. Wan, K. Alzoubi, and O. Frieder, “Distributed Construction ofConnected Dominating Set in Wireless Ad Hoc Networks,” Proc.IEEE INFOCOM ’02, vol. 3, pp. 1597-1604, 2002.

[4] S. Funke, A. Kesselman, U. Meyer, and M. Segal, “A SimpleImproved Distributed Algorithm for Minimum CDS in Unit DiskGraphs,” ACM Trans. Sensor Networks, vol. 2, no. 3, pp. 444-453,2006.

[5] H. Liu, P. Wan, X. Jia, X. Liu, and F. Yao, “Efficient FloodingScheme Based on 1-hop Information in Mobile Ad Hoc Net-works,” Proc. IEEE INFOCOM, 2006.

[6] Y. Tseng, S. Ni, and E. Shih, “Adaptive Approaches to RelievingBroadcast Storms in a Wireless Multihop Mobile Ad HocNetworks,” Proc. 21st Int’l Conf. Distributed Computing Systems,pp. 481-488, 2001.

[7] Y. Sasson, D. Cavin, and A. Schiper, “Probabilistic Broadcast forFlooding in Wireless Mobile Ad Hoc Networks,” Proc. IEEEWireless Comm. and Networking Conf., pp. 1124-1130, 2003.

[8] P. Kyasanur, R. Choudhury, and I. Gupta, “Smart Gossip: AnAdaptive Gossip-Based Broadcasting Service for Sensor Net-works,” Proc. Third IEEE Int’l Conf. Mobile Adhoc and SensorSystems, pp. 91-100, 2006.

[9] J. Wu, W. Lou, and F. Dai, “Extended Multipoint Relays toDetermine Connected Dominating Sets in Manets,” IEEE Trans.Computers, vol. 55, no. 3, pp. 334-347, Mar. 2006.

[10] G. Calinescu, I. Mandoiu, P. Wan, and A. Zelikovsky, “SelectingForwarding Neighbors in Wireless Ad Hoc Networks,” ACMMobile Networks and Applications, vol. 9, no. 2, pp. 101-111, 2004.

[11] J. Wu and F. Dai, “Broadcasting in Ad Hoc Networks Based onSelf-Pruning,” Proc. IEEE INFOCOM ’03, pp. 2240-2250, 2003.

[12] W. Peng and X. Lu, “On the Reduction of Broadcast Redundancyin Mobile Ad Hoc Networks,” Proc. ACM MobiHoc ’00, pp. 129-130, 2000.

[13] I. Stojmenovic, M. Seddigh, and J. Zunic, “Dominating Sets andNeighbor Elimination-Based Broadcasting Algorithms in WirelessNetworks,” IEEE Trans. Parallel and Distributed Systems, vol. 13,no. 1, pp. 14-25, Jan. 2002.

[14] J. Wu and W. Lou, “Forward-Node-Set-Based Broadcast inClustered Mobile Ad Hoc Networks,” Wireless Comm. and MobileComputing, vol. 3, no. 2, pp. 155-173, 2003.

[15] M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf,Computational Geometry: Algorithms and Applications, second ed.Springer-Verlag, 2000.

[16] Y. Cai, K. Hua, and A. Phillips, “Leveraging 1-Hop NeighborhoodKnowledge for Efficient Flooding in Wireless Ad Hoc Networks,”Proc. 24th IEEE Int’l Performance, Computing, and Comm. Conf.,pp. 347-354, 2005.

[17] J. Wu and H. Li, “On Calculating Connected Dominating Set forEfficient Routing in Ad Hoc Wireless Networks,” Proc. Third Int’lWorkshop Discrete Algorithms and Methods for Mobile Computing andComm., pp. 7-14, 1999.

[18] M. Khabbazian and V.K. Bhargava, “Highly Efficient Flooding inMobile Ad Hoc Networks,” Technical Report TR-2006-28, Dept. ofComputer Science, Univ. of British Columbia, http://www.cs.ubc.ca/cgi-bin/tr/2006/TR-2006-28, 2006.

Majid Khabbazian received the BSc degree incomputer engineering from Sharif University ofTechnology, Tehran, Iran, in 2002 and theMASc degree in electrical and computer en-gineering from the University of Victoria, BritishColombia, Canada, in 2004. He is currently aPhD candidate in the Department of Electricaland Computer Engineering at the University ofBritish Columbia, where he is a UniversityGraduate Fellowship holder. He was a Univer-

sity Graduate Fellowship recipient at the University of Victoria. Hiscurrent research interests include cryptography and wireless networks.He is a student member of the IEEE.

Vijay K. Bhargava received the BSc, MSc, andPhD degrees from Queen’s University, King-ston, Ontario, Canada, in 1970, 1972, and 1974,respectively. He is currently a professor and thehead of the Department of Electrical andComputer Engineering at the University ofBritish Columbia, Vancouver, Canada. Pre-viously, he was with the University of Victoria(1984-2003) and with Concordia University,Montreal (1976-1984). He is a coauthor of the

book Digital Communications by Satellite (Wiley, 1981), coeditor ofReed-Solomon Codes and Their Applications (IEEE, 1994), andcoeditor of Communications, Information and Network Security (Kluwer,2003). His research interests include wireless communications. He is afellow of the Engineering Institute of Canada (EIC), the IEEE, theCanadian Academy of Engineering, and the Royal Society of Canada.He is a recipient of the IEEE Centennial Medal (1984), IEEE Canada’sMcNaughton Gold Medal (1995), the IEEE Haraden Pratt Award (1999),the IEEE Third Millennium Medal (2000), the IEEE Graduate TeachingAward (2002), and the Eadie Medal of the Royal Society of Canada(2004). He is very active in the IEEE and was nominated by the IEEEBoard of Directors for the Office of IEEE President-Elect. He has servedon the Board of the IEEE Information Theory Society and the IEEECommunications Society. He is a past president of the IEEE InformationTheory Society. He is the editor-in-chief of the IEEE Transactions onWireless Communications.

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