base.on construction of virtual backbone in wireless ad hoc networks with unidirectional links

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On Construction of Virtual Backbone in WirelessAd Hoc Networks with Unidirectional Links

My T. Thai, Member, IEEE, Ravi Tiwari, and Ding-Zhu Du, Member, IEEE

AbstractSince there is no fixed infrastructure in wireless ad hoc networks, virtual backbone has been proposed as the routing

infrastructure to alleviate the broadcasting storm problem. The virtual backbone construction has been studied extensively in

undirectedgraphs, especially in unit disk graphs, in which each node has the same transmission range. In practice, however,

transmission ranges of all nodes are not necessarily equal. In this paper, we model such a network as a disk graph (DG), where

unidirectional links are considered. To study the virtual backbone construction in DGs, we consider two problems: Strongly Connected

Dominating Set (SCDS) and Strongly Connected Dominating and Absorbing Set (SCDAS). We propose a constant approximation

algorithm and present its improvements for the SCDS problem. Based on the solutions of SCDS, we discuss how to maintain its

constant approximation ratio for SCDAS and also propose an efficient heuristic. Through extensive simulations, we verify our

theoretical analysis and demonstrate that the SCDS can be extended to form an SCDAS with a marginal extra overhead.

Index TermsStrongly Connected Dominating Set, Strongly Connected Dominating and Absorbing Set, disk graph, wireless ad hoc

network, virtual backbone, directed graph.

1 INTRODUCTION

IN wireless ad hoc networks, there is no fixed orpredefined infrastructure. Nodes in wireless ad hocnetworks communicate via shared medium, either througha single-hop communication or multihop relays. In order toenable data transfer in such networks, all the wireless nodesneed to frequently flooding control messages, thus causinga lot of redundancy, contentions, and collisions [20]. As aresult, a virtual backbone has been proposed as the routinginfrastructure of a network for efficient routing, broad-casting, and collision avoidance protocols [22]. With virtualbackbones, routing messages are only exchanged betweenthe backbone nodes, instead of being broadcasted to all thenodes; therefore, routing is easier and can adapt quickly tonetwork topology changes. It has also shown that virtualbackbones could dramatically reduce routing overhead [21].Furthermore, using virtual backbone as forwarding nodescan efficiently reduce the energy consumption, which isalso one of the critical issues in wireless ad hoc networks.

The virtual backbone construction has been studiedintensively in a network where all nodes have the sametransmission ranges. Under such an assumption, a wirelessad hoc network can be modeled as a Unit Disk Graph(UDG) G [1]. Note that, in this case, G is undirected.

However, transmission ranges of all nodes in a networkare not necessarily equal. Nodes in a network may havedifferent powers due to differences in functionalities, power

control to alleviate collisions, topology control to achieve acertain level of connectivity, and so on. For example, in aclustered network, the cluster head or gateway nodes mighthave higher power than other nodes. On the other hand, ina certain power control scheme, a node enlarges or shrinksits transmission range according to a measured frequency incollisions. Likewise, in some topology-control networks,each node may adjust its transmission range to maintain acertain number of neighbors in order to make use of a good

spatial reuse. Such an adjustment of transmission rangedepends on node distribution in a network. All thesescenarios result in a wireless ad hoc network with differenttransmission ranges. Such a network can be modeled as aDisk Graph (DG) G. Note that G is directed, consisting ofboth bidirectional and unidirectional links.

While the study of virtual backbone in UDGs has drawna lot of attentions, the study of virtual backbone in wirelessad hoc networks with different transmission ranges hasbeen insufficient. To the best of our knowledge, the onlytwo works that have addressed this problem are in [18] and[19]. In [18], Wu extended their color marking scheme toobtain a virtual backbone in networks with unidirectional

links. Later, Dai and Wu generalized their pruning rulesused in [18] for any number of neighbors in directed graphs[19]. Although the algorithms are simple, they do notguarantee a performance bound.

Since finding a virtual backbone in UDG is NP-hard [2]and UDG is a special case of DG, we expect that finding avirtual backbone in DG is also NP-hard. Due to thehardness of these problems, it is important to devise andanalyze an approximation algorithm with a guaranteedapproximation ratio. To study this problem, we formulatethe virtual backbone in DG as a Strongly ConnectedDominating Set (SCDS) and Strongly Connected Dominat-ing and Absorbing Set (SCDAS) (see Section 2 for defini-

tions). In this paper, we propose a constant approximation

1098 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 7, NO. 9, SEPTEMBER 2008

. M.T. Thai and R. Tiwari are with the Department of Computer andInformation Science and Engineering, University of Florida, Gainesville,FL 32611-6120. E-mail: {mythai, rtiwari}@cise.ufl.edu.

. D.-Z. Du is with the Department of Computer Science, Erik Jonsson Schoolof Engineering and Computer Science, University of Texas at Dallas,Richardson, TX 75083-0688. E-mail: [email protected].

Manuscript received 18 Apr. 2007; revised 5 Oct. 2007; accepted 23 Jan. 2008;published online 6 Feb. 2008.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TMC-2007-04-0109.

Digital Object Identifier no. 10.1109/TMC.2008.22.1536-1233/08/$25.00 2008 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS


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algorithm to the SCDS problem, called Strongly Connected

Dominating Set using Breadth First Search (BFS_SCDS) andits improvements, called Strongly Connected Dominating

Set using Minimum number of Steiner Nodes (MSN_SCDS).We later extend these results to solve the SCDAS problem.

The rest of this paper is organized as follows: Section 2presents the preliminaries and problem definitions. Sec-

tion 3 describes related work on the virtual backboneconstruction. Two approximation algorithms to SCDS and

their theoretical analyses are presented in Sections 4 and 5,

respectively. Section 6 extends the solutions of SCDS tosolve the SCDAS problem. Section 7 shows simulation

results and their performance comparison. Finally, Section 8ends this paper with conclusion and future work.

2 PRELIMINARIES AND PROBLEM DEFINITIONS

2.1 Preliminaries

Let a directed graph G V ; E represent a network,where Vconsists of all nodes in a network and Erepresents

all the communication links.For any vertex v 2 V, the incoming neighborhood of

v is defined as Nv fu 2 Vju; v 2 Eg, and theoutgoing neighborhood ofv is defined as Nv fu 2 Vjv; u 2 Eg.

Likewise, for any vertex v 2 V, the closed incomingneighborhood ofv is defined as Nv Nv [ fvg, andthe closed outgoing neighborhood ofv is defined as

Nv Nv [ fvg.A subset S Vis called a dominating set (DS) ofG iff

S[ NS V, where NS Su2SNu.Given a subset S V, an induced subgraph of S,

denoted as GS, obtained by deleting all vertices in the set

Vn Sfrom G.A graph G is said to be strongly connected if for any pair

of nodes u, v 2 V, there exists a directed path between them.Likewise, a subset S Vis called a strongly connected setifGS is strongly connected.2.2 Network Model and Problem Definitions

In this paper, we study the virtual backbone in a network

with different transmission ranges. In this case, a networkcan be modeled using a DG G V ; E. The nodes in Varelocated in a Euclidean plane, and each node vi 2 Vhastransmission range ri 2 rmin; rmax, where rmin is theminimum transmission range and rmax is the maximum

transmission range of a network. A directed edge vi; vj 2 Eiffdvi; vj ri, where dvi; vj denotes the Euclideandistance between vi and vj. Such graphs are called DG. Anedge vi; vj is unidirectional ifvi; vj 2 Eand vj; vi =2 E.An edge vi; vj is bidirectional if both vi; vj and vj; vi arein E, i.e., dvi; vj minfri; rjg. Fig. 1 gives an example of aDG representing a network. In Fig. 1, dotted circles

represent transmission ranges, directed edges representunidirectional links, while undirected edges represent the

bidirectional links, and black nodes represent the virtualbackbone.

Under such a model, we formulate the virtual backbone

as the following two problems:

Definition 1: SCDS problem. Given a directed DGG V ; E, find a minimum size subset C Vsuch that1) Cis a DS and 2) GC is strongly connected.

Definition 2: SCDAS problem. Given a directed DGG V ; E, find a minimum size subset C Vsuch that1) Cis an SCDS and 2) for all nodes u =2 C, Nu \ C6 ;.

To study the SCDS and SCDAS problems, we assumethat the input graph G is strongly connected.

3 RELATED WORK

Although the virtual backbone problem has been exten-sively studied in general undirected graphs and UDGs [4],[5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17],little research has been done on directed DGs.

In directed graphs, the only two works that haveaddressed this problem are in [18] and [19]. In [18], Wuextended their color marking scheme to obtain an SCDASin networks with unidirectional links. No approximationratio has been presented. Later, Dai and Wu generalizedtheir pruning rules used in [18] for any number ofneighbors in directed graphs [19]. This generalized pruningrule, called Rule k, does not guarantee a constant approx-imation ratio. Instead, the authors showed a probabilisticperformance ratio. In UDG, the average size of the DSderived from Rule k was proved to be upper bounded by aconstant. In DG, this claim is held if the nonrestrictedRule k, which requires a global information, is applied.Thus, in the DG context, the proposed pruning rule is nolonger localized.

4 THE BFS_SCDS ALGORITHM

In this section, we introduce the BFS_SCDS algorithm toconstruct an SCDS for a directed DG. We then analyze itsapproximation ratio based on the geometric characteristicsof DGs.

4.1 Algorithm Description

The BFS_SCDS algorithm has two stages as follows:1) construct a DS Sand 2) connect all nodes in Sto forman SCDS Cby using the Breadth First Search (BFS) tree. Inthe first stage, we find a DS SofG using a greedy methodshown in Algorithm 1. Specifically, as described inAlgorithm 1, at each iteration, we find a node u, whichhas the largest transmission range in V, and color it black.

Remove closed outgoing neighbors ofu from V,

THAI ET AL.: ON CONSTRUCTION OF VIRTUAL BACKBONE IN WIRELESS AD HOC NETWORKS WITH UNIDIRECTIONAL LINKS 1099

Fig. 1. A DG representing a wireless ad hoc network.


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i.e., V Vn Nu. The algorithm terminates when V ;.Clearly, the set of black nodes Sforms a DS ofG.

Algorithm 1 Find a DS.1: INPUT: A directed DG G V ; E2: OUTPUT: A DS S3: S ;4: while V6 ; do5: Find a node u 2 Vwith the largest radius ru and

color u black6: S S[ fug7: V Vn Nu8: end while

9: Return S

In the second stage, two BFS trees are constructed to

connect S. Let s denote a node with the largest transmission

range in S, and vi be all other nodes in S. Let Tfs Vf; Ef denote a forward tree rooted at s such that thereexists a directed path from s to all vi, i 1 . . . p. Also, letTb

s

Vb; Eb

denote a backward tree rooted at s such

that for any node vi, there exists a directed path from vi to s.Let Hbe the union ofTfs and Tbs. Thus, the vertex setofHis a feasible solution to SCDS.

The detail of the second stage is as follows: First,

construct a BFS tree T1 ofG rooted at s. Let Lj, j 1 . . . l bethe set of nodes at level j in T1, where l is the depth ofT1.

Note that L0 fsg. At each level j, let Sj be the black nodesin Lj, i.e., Sj Lj \ S, and Sj be the nonblack nodes in Lj,i.e., Sj Lj n Sj. We construct Tfs as follows: Initially,Tfs has only one node s. At each iteration j, for eachnode u 2 Sj, we find a node v such that v 2 Nu \ Lj1. Ifv is not black, color it blue. In other words, we need to find

a node v such that v is an incoming neighbor ofu in G, andv is in the previous level ofu in T1. Add v to Tfs, where v

is the parent ofu. This process stops when j l. Next, weneed to identify the parents of all the blue nodes. Similarly,

at each iteration j, for each blue node u 2 Sj, find a nodev 2 Nu \ Sj1 and set v as the parent ofu in Tfs. If nosuch black v exists, select a blue node in Nu \ Sj1.Thus, Tfs consists of all the black and blue nodes, andthere is a directed path from s to all other nodes in S.

Now, we need to find the Tbs. First, construct a graphG0 V ; E0, where E0 fu; vjv; u 2 Eg, i.e., reverse allthe edges in G to obtain G0. Next, we build the second BFStree T2 ofG

0 rooted at s. Then, follow the above procedure

to find a Tf0

s such that there exists a directed path froms to all the other nodes in S. Then, reverse all the edges in

Tf0s back to their original directions, we have Tbs.

Hence, H Tfs [ Tbs is the strongly connected sub-graph where all the nodes in Hform an SCDS. Theconstruction of our proposed BFS_SCDS algorithm is

described in Algorithm 2.

Algorithm 2 BFS_SCDS.1: INPUT: A directed DG G V ; E2: OUTPUT: An SCDS C3: Find a DS Susing Algorithm 14: Choose node s 2 Ssuch that rs is maximum

5: Construct a BFS tree T1 ofG rooted at s

6: Construct a tree Tfs such that there exists a directedpath in Tfs from s to all other nodes in Sas follows:

7: for j 1 to l do8: Lj is a set of nodes in T1 at level j9: Sj Lj \ S; Sj Lj Sj; Tfs fsg

10: for each node u 2 Sj do11: select v

2 N

u

\Lj

1

and set v as a parent of

u. Ifv is not black, color v blue12: end for13: end for14: for j l to 1 do15: for each blue node u 2 Sj do16: if Nu \ Sj1 6 ; then17: select v 2 Nu \ Sj1andset v asaparentofu.18: else

19: select v 2 Nu \ Sj1andset v asaparentofu.20: Color v blue21: end if

22: end for23: end for

24: Reverse all edges in G to obtain G0

25: Construct a BFS tree T2 ofG0 rooted at s

26: Construct a tree Tf0s such that there exists a directed

path in Tf0s from s to all other nodes in S

27: Reverse all edges back to their original directions,then Tf

0s become Tbs, where there exists a directedpath from all other nodes in Sto s

28: H Tfs [ Tbs29: Return all nodes in H

4.2 Theoretical Analysis

Lemma 1. For any two black nodes u and v in a DS Sobtained by

Algorithm 1, du; v > rmin.Proof. This is trivial. Without loss of generality, assume

that ru > rv ! rmin. Algorithm 1 would mark u as ablack node before v. Assume that du; v rmin, thenv 2 Nu. Hence, v cannot be black, contradicting toour assumption. tu

Lemma 2. In a directed DG G V ; E, the size of any DS Sobtained by Algorithm 1 is upper bounded by

2:4 k 12

2 opt 3:7 k 1

2

2;

where k rmaxrmin , and opt is the size of the optimal solution of theSCDS problem.

Proof. From Lemma 1, the set of all the disks centered at

nodes in Swith radius rmin=2 are disjoint. Thus, the size

ofSis bounded by the maximum number of disks with

radius rmin=2 packing in the area covered by an optimal

SCDS OP T. Similar to [23], we will prove this by two

main steps: 1) calculate an area A covered by OP Tand

2) compute how many disks with radius rmin=2 can be

packed in A.

1. Calculate the area A covered by OP T. Let vi, 1 i

opt be the nodes in OP Tand vl be all the



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dominated nodes, i.e., nodes in G but not in OP T.

Let D0i be a disk centered at vi with radius rmax,andD0l be a disk centered at vl with radius

rmin2

. Clearly,all disks D0i are intersected. Let L bethesetof disks

Li with radius rmax rmin2 centered at vi. Hence,all disks D0i and D

0l must be contained in the union

of the disks Li. Each disk Li is added as follows:At each iteration i, add a disk Li centered at visuch that there exists a node vj 2 fv1; . . . ; vi1g anddvi; vj rmax. This node vj exists since all nodesin OP Tare connected. The newly covered area Aiis bounded by two arcs of disks Li and Lj,as shown in Fig. 2, where dvi; vj rmax. Notethat in Fig. 2, the disk Lj was added beforethe disk Li, i .e. , j < i. L et ffXvjvi an dc rmax rmin2 , we have

Ai area ofLi 2 area of the ffXvjYsector ofLj area of the diamond XvjY vi

c2 2c2 rmaxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 rmax

2

2r

c2 23

crmax

c2 3

1

:

Hence, the total area A covered by OP Tis, at

most, c2

3

1

opt

c2.

Note that dvi; vj ! rmin, where vi, vj are inDS S. Hence, all disks with radius rmin

2centered at

nodes in Sare disjoint. Now, we proceed to the

second step:2. Compute how many such disks are in A. We know

that the densest packing of unit disks in the plane

is attained by a hexagonal lattice. For each disk diwith radius rmin

2centered at a vertex vi, where vi is

in the DS S, place a regular hexagon ofwidth rmin,

as shown in Fig. 3. Each hexagon has an area offfiffi3

p2

r2min. For example, in Fig. 3, the disk d1 uses an

area of at least

ffiffi3

p2

r2min. Notice that the disks

nearby the boundary might not use all that area.

For example, in Fig. 3, the hexagon of the disk d2

centered at v2 has one part outside of disk D,which is the biggest disk in Fig. 3. That part has

an area offfiffi3

p2

r2min rmin2 2=6. Hence, each unitdisk can use an area of

ffiffiffi3

p

2r2min

ffiffiffi3

p

2r2min

r2min4

06

! :85r2min:

Therefore, the size ofSis bounded by

jSj Total Area A:85r2min

opt c2 3 1 c2:85r2min

opt 1 =3:85

rmaxrmin

12

2

:85

rmaxrmin

12

2

2:4 rmaxrmin

12

2opt 3:7 rmax

rmin 1

2

2:

tuTheorem 1. The BFS_SCDS algorithm has an approximation

ratio of12k 122, where k rmaxrmin .

Proof. Let Cdenote the SCDS obtained from the BFS_SCDS

algorithm. Let BTfand BTb be the blue nodes in Tfsand Tbs, respectively. We have

jCj jBTfj jBTb j jSj 5jSj

5 2:4 k 12

2 opt 3:7 k 1

2

2" #

12 k 12

2opt 18:5 k 1

2

2:

tu


Fig. 2. On the proof of the size relationship between an Sand an SCDS.

Fig. 3. The densest packing of unit disks.


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Corollary 1. If the transmission range ratio k is bounded,then the BFS_SCDS algorithm has an approximation factorofO1.

5 THE MSN_SCDS ALGORITHM

In this section, we propose an improved solution to theSCDS problem, namely the MSN_SCDS algorithm. Theimprovement (in terms of the SCDSs size) of MSN_SCDSover BFS_SCDS lies in the second stage. In BFS_SCDS, weuse the BFS tree to construct the forward and backwardtrees interconnecting all the black nodes in S. This schemeis simple and fast. However, we can reduce the size of theobtained SCDS further by reducing the number of theblue nodes that are used to connect all the black nodes. Inother words, we need to construct a tree with theminimum number of blue nodes to interconnect all theblack nodes.

Since the improvement of MSN_SCDS lies in thesecond stage, we begin introducing the Greedy SpiderContraction (GSC) algorithm that will be used as asubroutine of MSN_SCDS.

5.1 Greedy Spider Contraction Algorithm

As briefly mentioned, the objective of GSC is to construct atree with the minimum number of blue nodes to inter-connect all the black nodes. This problem can be formallydefined as follows:

Definition 3: Directed Steiner tree with Minimum SteinerNodes (DSMSN). Given a directed graph G V ; E anda set of nodes S

Vcalled terminals, find a directed

Steiner tree Trooted at r 2 Vsuch that there exists a directedpath from r to all the terminals in Tand the number of theSteiner nodes is minimum.

Note that a Steiner node is a node in Tbut not a terminal.In the SCDS problem context, Steiner nodes are theblue nodes where the terminals are the black nodes. It iswell known that the Steiner tree with minimum Steinernodes is NP-hard in undirected graphs [2]; thus, DSMSN isalso NP-hard. Therefore, we propose a greedy method tosolve the DSMSN problem, namely GSC algorithm.

Initially, all the nodes in Sare black and the other nodesin Vare white. First, let us introduce the following

definitions:

Definition 4. Spider. A spider is defined as a directed treehaving a white node as the root and all other nodes in the treeare either black or blue.

Av-spider

is a spider rooted at a white nodev

. Eachdirected path from v to a leaf is called a leg. Note that allthe nodes in each leg except v are either blue or black.

Definition 5: Black-blue component. A subgraph G0 ofG iscalled a black-blue component ifG0 is connected and consists ofonly black and blue nodes.

The main idea of GSC is that we repeatedly find a spidersuch that this spider has a maximum number of legs, i.e.,maximum number of black-blue components, and thencontract this spider. The detail of this algorithm isdescribed in Algorithm 3. The contracting operation isdefined as follows:

Contracting Operation: The contracting operation of av-spider performs in the following ways:

. Step 1: Start from level l 1.

. Step 2: For each undeleted node u at level l in thespider, do the following:

Step 2.1: Add a unidirectional edge v; wo foreach w0 2 Nu such that v; wo =2 E.

Step 2.2: Ifv; u is bidirectional, add a unidirec-tional edge wi; v for each wi 2 Nu such thatwi; v =2 E.

Step 2.3: Ifv; u is unidirectional, add a unidirec-tional edge wi; wo for each wo 2 Nu and wi 2Nu such that wi; wo =2 E.

. Step 3: Delete u.

. Step 4: Repeat the Step 2 for all the levels in thespider.

. Step 5: Color v blue.

Fig. 4 shows an example of a spider contracting

operation.

Algorithm 3 GSC(G, S, r).1: INPUT: Graph G V ; E, and a set of black nodes S2: OUTPUT: A tree Tr rooted at any node r 2 Vn S

spanning all nodes in S3: T ;;

4: while The number of black and blue nodes in G > 1 do


Fig. 4. The spider rooted at Vis getting contracted. The black node A is deleted according to steps 2.1, 2.3, and 3 of the Spider Contracting

Operation. The black node Bis deleted according to steps 2.1, 2.2, and 3 of the Spider Contracting Operation.


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5: Find a v-spider with the largest number of legs, i.e.,largest number black-blue components

6: Contract the v-spider using the contracting operationand update G

7: end while

8: Construct Tr from the set of black and blue nodesrooted at node r, where r is the root of the last

contracted v-spider9: Return Tr

Since the GSC algorithm is a solution of the DSMSNproblem, we are now ready to introduce the MSN_SCDSalgorithm.

5.2 MSN_SCDS Algorithm Description

The MSN_SCDS algorithm consists of three stages. In thefirst stage, it constructs a DS Susing a greedy methodshown in Algorithm 1. In the second stage, it calls theAlgorithm 3 twice to generate two trees: Tfr1 andTbr2. Tfr1 is a tree of blue and black nodes rooted atr1, in which there is a path from r1 to all the blue and

black nodes, and Tbr2 is another tree of blue andblack nodes rooted at r2, in which there is a path fromall the blue and black nodes to r2. Finally, in the thirdstage, the algorithm obtains the shortest path containinga minimum number of white from r2 to r1, and all thesewhite nodes on this path are colored blue. The set of allblack and blue nodes is an SCDS. The main steps ofMSN_SCDS is shown in Algorithm 4.

Algorithm 4 MSN_SCDS.1: INPUT: A directed strongly connected graph

G V ; E2: OUTPUT: An SCDS C

3: Find a DS Susing Algorithm 14: Tfr1 GSCG; S5: Reverse all the edges in G to obtain G0

6: Tf0r2 GSCG0; S

7: Reverse all edges in Tf0r2 to obtain Tbr2

8: Find the path Pr2; r1 from r2 to r1, having minimumnumber of white nodes on to it.

9: Color all the white nodes in Pr2; r1 blue10: H Tfr1 [ Tbr2 [ Pr2; r111: Let Cbe all nodes in H12: Return C

5.3 Correctness

The correctness of MSN_SCDS lies in the correctness of theGSC algorithm. Thus, in this section, we prove thecorrectness of GSC. First, we show that the proposedcontracting operation preserves the connectivity betweenany two nodes in G. Second, we show that the tree Trobtained from GSC is a tree rooted at r spanning all nodesin S. That is, for any node in S, there exists a directed pathfrom r to it.

Lemma 3. There will be always a path from v to all the nodes u inthe v-spider.

Proof. This is trivial by Definition 4. tuLemma 4. The contracting operation preserves connectivity for

any pair of nodes in G.

Proof. Consider any two nodes A and B. We will prove that

each time the contracting operation deletes a node u, theconnectivity between A and Bis still preserved. There

are four possible ways in which A and Bcan be

connected via u.

Case 1: This case is shown in Part (a) of Fig. 5. Thereare paths from A to Band from v to B. When the v-spideris contracted and u gets deleted, according to Step 2.1 of

the proposed contracting operation, there exists an edgefrom v to B, and according to Step 2.3, there exists anedge from A to B. Hence, after the deletion ofu, theconnectivity from A to Band v to Bare still preserved.

Case 2: This case is shown in Part (b) of Fig. 5. A and

Bare having a path to each other via nodes u, and alsothere is a path from v to A and B, respectively, via u.When u gets deleted, according to Step 2.1, there existsan edge from v to A and from v to B, respectively, andaccording to Step 2.3, there exists an edge from A to B.

Hence, after the deletion ofu, the connectivity from A toB, v to A, and v to Bis still preserved.

Case 3: This case is shown in Part (c) of Fig. 5. Thereis a path from Bto A via u, and also there is a path fromv to A via u. When the v-spider contracts, according toStep 2.1, there exists an edge from v to A, and accordingto Step 2.3, there exists an edge from Bto A. Hence,after the deletion ofu, the connectivity from v to A and

Bto A is still preserved.


Fig. 5. Contraction preserves connectivity.


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Case 4: This case is shown in Part (d) of Fig. 5.Similarly to the above three cases, however, in Case 4,the edge from v to u is bidirectional links. As shown inFig. 5d, there is a path from v to A, Bto A, and Bto v.When u is deleted from the contracting operation,according to Step 2.1, there exists an edge from v to A,and from Step 2.2, there exists an edge from Bto v.

Hence, the connectivity between v, A, and Bispreserved.

In conclusion, the connectivity of any two nonspidernodes A and Bis preserved after the contractingoperation. tu

Lemma 5. The GSC algorithm terminates only when a single

nonwhite node is left in the graph.

Proof. Assume that the GSC algorithm terminates (hangs)

and there are still more than one nonwhite nodes in the

graph. Because GSC hangs only when there does not

exist any v-spider for contracting anymore. Thus, we

consider the following two cases:

Case 1: There is no white node v left to be the rootof a v-spider. Thus, the contracted G consists of onlynonwhite nodes. Let D be the set of these nodes.Then, all nodes in D are not in any spider trees at theprevious contracting operations. It implies that for anyu 2 D, there does not exist any outgoing edge fromany previous v-spider to u. Since the contractingoperation preserves connectivity (Lemma 4), we con-clude that original graph G V ; E is not stronglyconnected, contradicting to the fact that G is stronglyconnected.

Case 2: There are white as well as nonwhite nodes left,but we cannot find a v-spider. Let Wbe the set of allwhite nodes left, and D be the set of all nonwhite nodesleft. Since there does not exist any spider, we haveNW \ D ;. Thus, GW[ D is not strongly con-nected. From Lemma 4, we conclude that G is notstrongly connected. Contradiction. tu

Lemma 6. The GSC algorithm produces a tree Tof all nonwhite

nodes rooted at r, and there is a path from r to all the black

nodes in S.

Proof. From Lemma 5, the GSC algorithm terminates

when there exists exactly one nonwhite node in the

contracted G. Let us call this node r. Now, we need to

prove that the tree Tr consists of only nonwhite nodesand spans all the black nodes in S.

Consider the last r-spider. From Lemma 3, all thenodes in the r-spider can be reached from its root. Hence,there is a path from r to all the (nonwhite) nodes inr-spider. Let u be the node in r-spider. Then, u iseither black or blue. Ifu is black, then Lemma 6 holdsfor this black node u. Ifu is blue, then u must be theroot of a u-spider in some previous iterations. Thus,there is a path from r to all nonwhite nodes in u-spidervia u. Eventually, there is a path from r to all theblack nodes in S. tu

Theorem 2 (Correctness). Algorithm 4 returns an SCDS C.

Proof. Note that Cis a set of nodes in Tr1 [ Tr2 [

Pr2; r1. From Lemma 6, Ccontains a set Sof all

black nodes. Hence, Cis a DS. Now, we need to prove

that there exists a path for any pair of black nodes via

nodes in C. For any pair of nodes x, y 2 S, there exista directed path x; wi; r2; wj; r1; wk; y, where wi are thenodes in Tr2 from x to r2, wj are the nodes inPr2; r1, and wk are the nodes in Tr1 from r1 to y.tu

5.4 Theoretical AnalysisLemma 7. Given a directed DG G V ; E, for any arbitrary

node v 2 Vn S, we have jNv \ Sj 2k 12, wherek rmax=rmin.

Proof. Recall that Nv is a set of outgoing neighborsofv, and Sis a DS ofG. Let v be a node with the

largest transmission range. From Lemma 1, we have

du; v ! rmin, where u, v 2 S. Hence, the size ofNv \ Sis bounded by the maximum number ofdisjoint disks with radius rmin=2 packing in the disk

centered at v with radius ofrmax rmin=2. We have

Nv \ Sj j rmax rmin=22rmin=22

2k 12:

tu

Lemma 7 indicates that the maximum number of legs

in a spider is upper bounded by 2k 12. Now, let T bean optimal tree when connecting a given set S, and CTis the number of the Steiner nodes in T. Also, let Bbe aset of blue nodes in T, where Tis the solution of the

DSMSN problem obtained from Algorithm 3. Then, we

have the following lemma:

Lemma 8. The size ofBis, at most, 2 2 ln2k 1CT.Proof. Let n jSj and p jBj. Let Gi be the graph G at the

iteration i after a spider contracting operation. Let vi,

i 1 . . . p be the blue nodes in the order of appearance inAlgorithm 3, and let ai be the number of the black and

blue components in Gi. Also, let CTi be the optimalsolution ofGi. Ifn 1, then the lemma is trivial. Assumethat n ! 2, thus CT ! 1. Since at each iteration i, wepick a white node v such that the v-spider has the

maximum number of black-blue components, the num-

ber of black and blue components (legs) in v-spider mustbe at least aiCTi . Thus, we have

ai1 ai aiC Ti 1

ai aiCT 1:

This results to the recurrence

ai a0 1 1CT

iXi1j0

1 1CT

j

a0e

iCT

C

T

:



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The last step uses the fact that ln1 x ! x andthe second term is a geometric series. Now, leti CT ln a0CT , we have

ai a0eln

a0CT

CT 2CT:

Thus, after C

T

ln a0C

T

iterations, the number of

black-blue components left is less than or equal to2CT. Using Lemma 7, we conclude that

jBj i 2CT CT ln a0

CT 2CT

2 2 ln2k 1 CT:tu

6 THE EXTENDED SCDS (Ext_SCDS) ALGORITHM

In this section, we discuss the solutions to the SCDAS

problem and present an Ext_SCDS algorithm, which is anextended result from the MSN_SCDS algorithm.There are several ways to extend a solution of SCDS

to be a solution of SCDAS. Note that a set Cis calledan SCDAS iffCis SCDS and for any node u =2 C,Nu \ C6 ;. Therefore, there are basically three mainmethods to extend these solutions:

1. In the first stage, instead of constructing a DS S, wefind a set Ssuch that Sis a DS and for any u =2 S,Nu \ S6 ;. We call such a set a Dominating andAbsorbent Set (DAS). Then, use the connectingmethod (second stage) to connect S. The obtainedset must be an SCDAS.

2. Keep the first stage the same, that is, finding a DS S.However, in the second stage, besides intercon-nected all nodes in S, we also make sure that this setis a DAS.

3. Use any algorithm of SCDS to find an SCDS Cfirst,then iteratively add more nodes into Cto make Cbecome an SCDAS.

Actually, we can modify Algorithm 2 to construct anoutgoing spanning tree and an incoming spanning treerooted at an arbitrary node r 2 V. Then, the nonleaf nodesof these two trees form an SCDAS. Using the similartechniques in Section 4, we can prove that this modifiedalgorithm can obtain a constant approximation ratio whenthe ratio of the maximum to the minimum in transmission

range is bounded. The detail of this algorithm togetherwith other proposed solutions specifically to SCDAS isreported in our other paper [3]. Here, we present how toextend the MSN_SCDS to Ext_SCDS using the thirdmethod. The simulation experiments show that thenumber of nodes added in is reasonable and expected.The details of Ext_SCDS are illustrated in Algorithm 5.

Algorithm 5 Ext_SCDS.1: INPUT: A directed DG G V ; E2: OUTPUT: An SCDAS C3: Call the MSN_SCDS algorithm to construct an SCDS S.

Note that all the nodes in Sare either black or blue and

the rest of the nodes are white

4: for All u 2 Sdo5: All nodes v 2 Nu that are white, color them gray.6: end for

7: while there exists a white node do

8: Find the gray node v having maximum number ofwhite nodes in Nv, color v blue and color all thewhite nodes in N

v

gray

9: end while

Basically, after constructing an SCDS S, all nodes not inSare white. Now, we need to check if these white nodesare also absorbed, that is, for each white node x, thereexist a directed edge x; u such that u 2 S. If yes, color ugray. Otherwise, we will choose a gray node v such that vabsorbs the most number of white nodes, color such nodev blue and all white nodes in Nv gray. This processterminates when there is no white node left. The union ofSand newly blue nodes form an SCDAS.

7 SIMULATION RESULTS

In this section, we conducted simulations to compare theperformance (in terms of the SCDS size) of the proposedalgorithms. We study two network parameters thatmay impact the performance of the proposed algorithms:1) network density and 2) transmission range ratio. Theperformance comparison of MSN_SCDS and BFS_SCDSalgorithms is presented in Section 7.1, while the perfor-mance comparison of MSN_SCDS and Ext_SCDS isevaluated in Section 7.2.

7.1 MSN_SCDS and BFS_SCDS

7.1.1 Impact of Network Density

To study the impact of network density, we varied thenetwork density in two ways: 1) varying the number ofnodes in a fixed area and 2) varying the area with a fixednumber of nodes.

Varying the number of nodes. We randomly deployedn nodes in a fixed area of 1,000 m 1,000 m. n changedfrom 10 to 200, with an increment of 5. Each node chose atransmission range in rmin; rmax, where rmin 200 m,and rmax 600 m. For each value ofn, we investigated100 network instances and averaged the results.

As shown in Fig. 6, the performance of MSN_SCDS isalways better than that of BFS_SCDS. The size of an SCDSconstructed by BFS_SCDS is mostly around 1.4 times that ofMSN_SCDS. As the number of nodes increases, the size ofSCDS for both algorithms decreases as predicted. The

decrease in the size of SCDS, with respect to the number ofnodes in a network, is not as large as expected. As thenumber of nodes increases, network density increases,nodes come closer to each other. Hence, it is expected that anode dominates more number of nodes. However, at thesame time, when number of nodes in the network is larger,more dominating nodes are required to dominate all thenodes in the network. We can notice in both the curves,the size of SCDS drops quickly at the beginning when thenumber of nodes increases from 10 to 80. However, it dropsslowly when the number of nodes increases from 135 to 200.In addition, the roots r1 and r2 constructed by MSN_SCDSare quite close, leading to a small number of white nodes

added in a path Pr2; r1.



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7.2 The Ext_SCDS and MSN_SCDS

In this section, we evaluate the number of nodes required to

extend an SCDS to an SCDAS through simulations. We

compare the SCDS size produced by MSN_SCDS and to the

SCDAS size produced by Ext_SCDS. To study the perfor-

mance of these two algorithms thoroughly, we also

investigated the impact of network density and transmis-

sion range ratio. For each parameter, we set up a network

instance the same as that in Section 7.1, and the results areaveraged as discussed before.

7.2.1 Impact of Network Density

Varying the number of nodes. As shown in Fig. 9a,Ext_SCDS needs only a few more nodes to extend an SCDSto an SCDAS. An interesting thing is that when the numberof nodes in a network increases, i.e., the network densityincreases, the difference between the size of SCDS andSCDAS also increases. As revealed in Fig. 9a, when thenumber of nodes in the network is between 10 and 55, thereis not much difference in the size of SCDS and SCDAS.However, as shown in Fig. 9b, when the number of nodes

in the network is between 10 and 55, the ratio of SCDS

and SCDAS sizes lies between 1.0 and 1.5. When the

number of nodes increases from 65 to 150, the ratio

fluctuates between 1.5 and 2.0. The reason for this is, in

the initial DS, the nodes with the large transmission ranges

are selected. With these criteria, the constructed DS is small.

However, the dominated nodes (which are large) may not

be absorbed. Therefore, when the SCDS is extended to the

SCDAS, less number of nodes are required if the number of

nodes in the network is less; otherwise, a larger number ofnodes are required to extend an SCDS to an SCDAS.

However, notice that throughout the simulations, the ratio

between the SCDAS and SCDS sizes never exceeds 2.0.

The average ratio is about 1.5.Varying the area size. The simulation results are shown in

Fig. 10. For both algorithms, as the area increases, i.e.,

network density decreases, the size of SCDS and SCDAS

also increases. As seen in Fig. 10a, the curves of both

algorithms increase. This is because as area increases, more

nodes are required to dominate all nodes in the network.

Fig. 10b shows that the ratio of the size of SCDAS and SCDS

mostly lies between 1.25 and 1.65.


Fig. 9. Impact of the number of nodes. (a) Performance comparison of

MSN_SCDS and Ext_SCDS. (b) The ratio of virtual backbone obtained

from MSN_SCDS and Ext_SCDS.

Fig. 10. Impact of the area size. (a) Performance comparison of

MSN_SCDS and Ext_SCDS. (b) The ratio of virtual backbone obtained

from MSN_SCDS and Ext_SCDS.


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7.2.2 Impact of the Transmission Range Ratio

The simulation results are shown in Fig. 11. Fig. 11a revealsthat when the transmission range ratio is small, the gapbetween SCDS and SCDAS sizes is small, and when thetransmission range ratio is larger, this gap is also larger.As presented in Fig. 11b, when the transmission rangeratio is between 1.0 and 1.5, the size ratio of SCDAS and

SCDS is close to 1.0. As the transmission ratio increasesabove 1.5, the size ratio of SCDS and SCDAS also increases.This can explained as when the transmission range ratio islow, there are more bidirectional links in the network.Hence, there are more chances of a dominating nodehaving bidirectional link to the nodes it is dominating.Thus, less number of nodes are needed to extend an SCDSto an SCDAS.

8 CONCLUSIONS

In this paper, we have studied the SCDS problem and theSCDAS problem in directed DGs, where both unidirectional

and bidirectional links are considered. The directed DGs

can be used to model wireless ad hoc networks, where

nodes have different transmission ranges. We have pro-

posed a constant approximation algorithm for the SCDS

problem and shown how to improve its performance

further. The main approach in our algorithms is to construct

a DS and connect them using the GSC technique. Through

the simulation experiments, we have shown that using a

Steiner tree with a minimum number of Steiner nodes to

interconnect nodes in DS can help to reduce the SCDS size.

We have also proposed an algorithm for SCDAS problem.

This algorithm works on an existing SCDS and extends it to

an SCDAS by adding a small number of nodes.Since the nodes in the virtual backbone need to carry

other nodes traffic, and node and link failures are inherent

in wireless ad hoc networks, it is desirable that the virtual

backbone is fault tolerant. Thus, we are interested in

studying the fault-tolerant virtual backbone problem in

directed DGs. One viable solution is to construct a

m-SCDS m-SCDS first, and then augment it based onthe connectivity to make it k-connected.

REFERENCES[1] B. Clack, C. Colbourn, and D. Johnson, Unit Disk Graphs,

Discrete Math., vol. 86, pp. 165-177, 1990.[2] M.R. Garey and D.S. Johnson, Computers and Intractability. A Guide

to the Theory of NP-Completeness. Freeman, New York, 1979.[3] M. Park, C. Wang, J. Willson, M.T. Thai, W. Wu, and A. Farago,

A Dominating and Absorbent Set in Wireless Ad-Hoc Networkswith Different Transmission Range, Proc. ACM MobiHoc, 2007.

[4] Y. Li, S. Zhu, M.T. Thai, and D.-Z. Du, Localized Construction ofConnected Dominating Set in Wireless Networks, NSF IntlWorkshop Theoretical Aspects of Wireless Ad Hoc, Sensor and Peer-to-Peer Networks (TAWN), 2004.

[5] P.-J. Wan, K.M. Alzoubi, and O. Frieder, Distributed Con-

struction on Connected Dominating Set in Wireless Ad HocNetworks, Proc. IEEE INFOCOM, 2002.[6] Y. Li, M.T. Thai, F. Wang, C.-W. Yi, P.-J. Wang, and D.-Z. Du,

On Greedy Construction of Connected Dominating Sets inWireless Networks, Wireless Comm. and Mobile Computing(WCMC), special issue, 2005.

[7] M. Cardei, M.X. Cheng, X. Cheng, and D.-Z. Du, ConnectedDomination in Ad Hoc Wireless Networks, Proc. Sixth Intl Conf.Computer Science and Informatics (CSI), 2002.

[8] S. Guha and S. Khuller, Approximation Algorithms for Con-nected Dominating Sets, Algorithmica, vol. 20, pp. 374-387, 1998.

[9] L. Ruan, H. Du, X. Jia, W. Wu, Y. Li, and L.-I. Ko, A GreedyApproximation for Minimum Connected Dominating Sets,Theoretical Computer Science, vol. 329, nos. 1-3, pp. 325-330, Dec.2004.

[10] M.T. Thai, F. Wang, D. Liu, S. Zhu, and D.Z. Du, ConnectedDominating Sets in Wireless Networks with Different Transmis-

sion Ranges, IEEE Trans. Mobile Computing, vol. 6, no. 7, July2007.[11] M.T. Thai, N. Zhang, R. Tiwari, and X. Xu, On Approximation

Algorithms ofk-Connected m-Dominating Sets in Disk Graphs,J. Theoretical Computer Science, vol. 358, pp. 49-59, 2007.

[12] M.T. Thai and D.-Z. Du, Connected Dominating Sets in DiskGraphs with Bidirectional Links, IEEE Comm. Letters, vol. 10,no. 3, pp. 138-140, Mar. 2006.

[13] F. Wang, M.T. Thai, and D.Z. Du, On the Construction of 2-Connected Virtual Backbone in Wireless Network, IEEE Trans.Wireless Comm., accepted with revisions, 2006.

[14] X. Cheng, X. Huang, D. Li, W. Wu, and D.-Z. Du, Polynomial-Time Approximation Scheme for Minimum Connected Dominat-ing Set in Ad Hoc Wireless Networks, Networks, vol. 42, no. 4,pp. 202-208, 2003.

[15] X. Cheng, M. Ding, D.H. Du, and X. Jia, Virtual BackboneConstruction in Multi-hop Ad Hoc Wireless Networks, Wireless

Comm. and Mobile Computing, vol. 6, pp. 183-190, 2006.


Fig. 11. Impact of the transmission ratio k. (a) Performance

comparison of MSN_SCDS and Ext_SCDS. (b) The ratio of virtual

backbone obtained from MSN_SCDS and Ext_SCDS.


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[16] J. Blum, M. Ding, and X. Cheng, Applications of ConnectedDominating Sets in Wireless Networks, Handbook of CombinatorialOptimization, D.-Z. Du and P. Pardalos, eds., pp. 329-369, KluwerAcademic Publisher, 2004.

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Connected Dominating Set Formation in Ad Hoc WirelessNetworks, IEEE Trans. Parallel and Distributed Systems, vol. 15,no. 10, Oct. 2004.

[20] S.-Y. Ni, Y.-C. Tseng, Y.-S. Chen, and J.-P. Sheu, The BroadcastStorm Problem in a Mobile Ad Hoc Network, Proc. ACMMobiCom, 1999.

[21] P. Sinha, R. Sivakumar, and V. Bharghavan, Enhancing Ad HocRouting with Dynamic Virtual Infrastructures, Proc. IEEEINFOCOM, 2001.

[22] A. Ephremides, J. Wieselthier, and D. Baker, A Design Conceptfor Reliable Mobile Radio Networks with Frequency HoppingSignaling, Proc. IEEE, vol. 75, no. 1, pp. 56-73, 1987.

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Graphs, ACM Trans. Sensor Network, vol. 2, no. 3, pp. 444-454,Aug. 2006.

My T. Thai received the BS degree in computerscience and the BS degree in mathematicsfrom Iowa State University, in 1999 and thePhD degree in computer science from theUniversity of Minnesota, Twin Cities, in 2005.She is currently an assistant professor in theDepartment of Computer and InformationScience and Engineering, University of Florida,Gainesville. Her main research interests includecombinatorics, algorithms, wireless networks,

and computational biology. In particular, she is interested in developingand analyzing algorithms for many computationally hard problems incomputer networks and computational biology. Her work has coveredmany areas of wireless networks and computational biology, including

routing protocols, coverage in sensor networks, broadcast tree, virtualbackbone, group testing, and nonunique probe selection. She serves onthe editorial board of the Journal of Combinatorial Optimizationand theJournal of Optimization Letters. She is a member of the IEEE.

Ravi Tiwari is working toward the PhD degree inthe Department of Computer InformationScience and Engineering, University of Florida,Gainesville. His research interests include wire-less networks and community structures.

Ding-Zhu Du received the MS degree from theChinese Academy of Sciences, in 1982 and thePhD degree, under the supervision of ProfessorRonald V. Book, from the University of Califor-nia, Santa Barbara, in 1985. He was a professorin the Department of Computer Science andEngineering, University of Minnesota. He wasalso with the Mathematical Sciences ResearchInstitute, Berkeley, California, for one year, withthe Department of Mathematics, Massachusetts

Institute of Technology also for one year, and with the Department ofComputer Science, Princeton University for one and a half years. He iscurrently with the Department of Computer Science, Erik JonssonSchool of Engineering and Computer Science, University of Texas atDallas, Richardson. He has published about 140 journal papers and

several books. He is the editor in chief of the Journal of CombinatorialOptimizationand is also on the editorial board for several other journals.Thirty PhD students have graduated under his supervision. He is amember of the IEEE.

. For more information on this or any other computing topic,please visit our Digital Library at www.computer.org/publications/dlib.


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