On Construction of Virtual Backbone in Wireless Ad Hoc Networks with Unidirectional Links
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Transcript of On Construction of Virtual Backbone in Wireless Ad Hoc Networks with Unidirectional Links
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
Abstract—Since 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
undirected graphs, 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 Terms—Strongly Connected Dominating Set, Strongly Connected Dominating and Absorbing Set, disk graph, wireless ad hoc
network, virtual backbone, directed graph.
Ç
1 INTRODUCTION
IN wireless ad hoc networks, there is no fixed orpredefined infrastructure. Nodes in wireless ad hoc
networks 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 goodspatial 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 unidirectionallinks. 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) and
its 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 2
presents the preliminaries and problem definitions. Sec-
tion 3 describes related work on the virtual backbone
construction. Two approximation algorithms to SCDS and
their theoretical analyses are presented in Sections 4 and 5,
respectively. Section 6 extends the solutions of SCDS to
solve the SCDAS problem. Section 7 shows simulation
results and their performance comparison. Finally, Section 8
ends this paper with conclusion and future work.
2 PRELIMINARIES AND PROBLEM DEFINITIONS
2.1 Preliminaries
Let a directed graph G ¼ ðV ;EÞ represent a network,
where V consists of all nodes in a network and E represents
all the communication links.For any vertex v 2 V , the incoming neighborhood of
v is defined as N�ðvÞ ¼ fu 2 V jðu; vÞ 2 Eg, and the
outgoing neighborhood of v is defined as NþðvÞ ¼fu 2 V jðv; uÞ 2 Eg.
Likewise, for any vertex v 2 V , the closed incoming
neighborhood of v is defined as N�½v� ¼ N�ðvÞ [ fvg, and
the closed outgoing neighborhood of v is defined as
Nþ½v� ¼ NþðvÞ [ fvg.A subset S � V is called a dominating set (DS) of G iff
S [NþðSÞ ¼ V , where NþðSÞ ¼Su2S N
þðuÞ.Given a subset S � V , an induced subgraph of S,
denoted as G½S�, obtained by deleting all vertices in the set
V n S from 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 � V is called a strongly connected set
if G½S� 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 network
can be modeled using a DG G ¼ ðV ; EÞ. The nodes in V are
located in a Euclidean plane, and each node vi 2 V has
transmission range ri 2 ½rmin; rmax�, where rmin is the
minimum transmission range and rmax is the maximum
transmission range of a network. A directed edge ðvi; vjÞ 2 Eiff dðvi; vjÞ � ri, where dðvi; vjÞ denotes the Euclidean
distance between vi and vj. Such graphs are called DG. An
edge ðvi; vjÞ is unidirectional if ðvi; vjÞ 2 E and ðvj; viÞ =2 E.
An edge ðvi; vjÞ is bidirectional if both ðvi; vjÞ and ðvj; viÞ are
in E, i.e., dðvi; vjÞ � minfri; rjg. Fig. 1 gives an example of a
DG representing a network. In Fig. 1, dotted circles
represent transmission ranges, directed edges represent
unidirectional links, while undirected edges represent the
bidirectional links, and black nodes represent the virtual
backbone.Under such a model, we formulate the virtual backbone
as the following two problems:
Definition 1: SCDS problem. Given a directed DGG ¼ ðV ;EÞ, find a minimum size subset C � V such that1) C is a DS and 2) G½C� is strongly connected.
Definition 2: SCDAS problem. Given a directed DGG ¼ ðV ;EÞ, find a minimum size subset C � V such that1) C is an SCDS and 2) for all nodes u =2 C, NþðuÞ \ C 6¼ ;.
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 S and 2) connect all nodes in S to forman SCDS C by using the Breadth First Search (BFS) tree. Inthe first stage, we find a DS S of G 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 of u 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 ¼ V nNþ½u�. The algorithm terminates when V ¼ ;.Clearly, the set of black nodes S forms a DS of G.
Algorithm 1 Find a DS.
1: INPUT: A directed DG G ¼ ðV ;EÞ2: OUTPUT: A DS S
3: S ¼ ;4: while V 6¼ ; do
5: Find a node u 2 V with the largest radius ru and
color u black6: S ¼ S [ fug7: V ¼ V nNþ½u�8: 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 TfðsÞ ¼ðV f; EfÞ denote a forward tree rooted at s such that there
exists a directed path from s to all vi, i ¼ 1 . . . p. Also, let
TbðsÞ ¼ ðV b; EbÞ denote a backward tree rooted at s such
that for any node vi, there exists a directed path from vi to s.
Let H be the union of TfðsÞ and TbðsÞ. Thus, the vertex set
of H is a feasible solution to SCDS.
The detail of the second stage is as follows: First,
construct a BFS tree T1 of G rooted at s. Let Lj, j ¼ 1 . . . l be
the set of nodes at level j in T1, where l is the depth of T1.
Note that L0 ¼ fsg. At each level j, let Sj be the black nodes
in Lj, i.e., Sj ¼ Lj \ S, and �Sj be the nonblack nodes in Lj,
i.e., �Sj ¼ Lj n Sj. We construct TfðsÞ as follows: Initially,
TfðsÞ has only one node s. At each iteration j, for each
node u 2 Sj, we find a node v such that v 2 N�ðuÞ \ Lj�1. If
v is not black, color it blue. In other words, we need to find
a node v such that v is an incoming neighbor of u in G, and
v is in the previous level of u in T1. Add v to TfðsÞ, where v
is the parent of u. This process stops when j ¼ l. Next, we
need to identify the parents of all the blue nodes. Similarly,
at each iteration j, for each blue node u 2 �Sj, find a node
v 2 N�ðuÞ \ Sj�1 and set v as the parent of u in TfðsÞ. If no
such black v exists, select a blue node in N�ðuÞ \ �Sj�1.
Thus, TfðsÞ consists of all the black and blue nodes, and
there is a directed path from s to all other nodes in S.
Now, we need to find the TbðsÞ. First, construct a graph
G0 ¼ ðV ;E0Þ, where E0 ¼ fðu; vÞjðv; uÞ 2 Eg, i.e., reverse all
the edges in G to obtain G0. Next, we build the second BFS
tree T2 of G0 rooted at s. Then, follow the above procedure
to find a Tf0 ðsÞ such that there exists a directed path from
s to all the other nodes in S. Then, reverse all the edges in
Tf0 ðsÞ back to their original directions, we have TbðsÞ.
Hence, H ¼ TfðsÞ [ TbðsÞ is the strongly connected sub-
graph where all the nodes in H form an SCDS. The
construction of our proposed BFS_SCDS algorithm is
described in Algorithm 2.
Algorithm 2 BFS_SCDS.
1: INPUT: A directed DG G ¼ ðV ;EÞ2: OUTPUT: An SCDS C
3: Find a DS S using Algorithm 1
4: Choose node s 2 S such that rs is maximum
5: Construct a BFS tree T1 of G rooted at s
6: Construct a tree TfðsÞ such that there exists a directedpath in TfðsÞ from s to all other nodes in S as follows:
7: for j ¼ 1 to l do
8: Lj is a set of nodes in T1 at level j
9: Sj ¼ Lj \ S; �Sj ¼ Lj � Sj; TfðsÞ ¼ fsg10: for each node u 2 Sj do
11: select v 2 ðN�ðuÞ \ Lj�1Þ and set v as a parent of
u. If v is not black, color v blue
12: end for
13: end for
14: for j ¼ l to 1 do
15: for each blue node u 2 �Sj do
16: if N�ðuÞ \ Sj�1 6¼ ; then
17: selectv 2 ðN�ðuÞ \ Sj�1Þand setvas a parent ofu.
18: else
19: selectv 2 ðN�ðuÞ \ �Sj�1Þand setvas a parent ofu.
20: Color v blue21: end if
22: end for
23: end for
24: Reverse all edges in G to obtain G0
25: Construct a BFS tree T2 of G0 rooted at s
26: Construct a tree Tf0 ðsÞ such that there exists a directed
path in Tf0 ðsÞ from s to all other nodes in S
27: Reverse all edges back to their original directions,then Tf
0 ðsÞ become TbðsÞ, where there exists a directed
path from all other nodes in S to s
28: H ¼ TfðsÞ [ TbðsÞ29: Return all nodes in H
4.2 Theoretical Analysis
Lemma 1. For any two black nodes u and v in a DS S obtained by
Algorithm 1, dðu; vÞ > rmin.
Proof. This is trivial. Without loss of generality, assume
that ru > rv � rmin. Algorithm 1 would mark u as a
black node before v. Assume that dðu; vÞ � rmin, then
v 2 NþðuÞ. Hence, v cannot be black, contradicting to
our assumption. tuLemma 2. In a directed DG G ¼ ðV ;EÞ, the size of any DS S
obtained by Algorithm 1 is upper bounded by
2:4 kþ 1
2
� �2
� optþ 3:7 kþ 1
2
� �2
;
where k ¼ rmaxrmin
, and opt is the size of the optimal solution of the
SCDS problem.
Proof. From Lemma 1, the set of all the disks centered at
nodes in S with radius rmin=2 are disjoint. Thus, the size
of S is bounded by the maximum number of disks with
radius rmin=2 packing in the area covered by an optimal
SCDS OPT . Similar to [23], we will prove this by two
main steps: 1) calculate an area A covered by OPT and
2) compute how many disks with radius rmin=2 can be
packed in A.
1. Calculate the area A covered by OPT . Let vi, 1 � i �opt be the nodes in OPT and vl be all the
1100 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 7, NO. 9, SEPTEMBER 2008
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dominated nodes, i.e., nodes in G but not in OPT .
LetD0i be a disk centered at vi with radius rmax, and
D0l be a disk centered at vl with radius rmin2 . Clearly,
all disksD0i are intersected. LetL be the set of disks
Li with radius rmax þ rmin2
� �centered at vi. Hence,
all disks D0i and D0l 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; . . . ; vi�1g and
dðvi; vjÞ � rmax. This node vj exists since all nodes
in OPT are connected. The newly covered area Ai
is bounded by two arcs of disks Li and Lj,
as shown in Fig. 2, where dðvi; vjÞ ¼ rmax. Notethat in Fig. 2, the disk Lj was added before
the disk Li, i.e., j < i. Let � ¼ ffXvjvi and
c ¼ rmax þ rmin2 , we have
Ai � area of Li � 2 area of the ffXvjY sector of Lj
þ area of the diamond XvjY vi
��c2 � 2�c2 þ rmaxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 � rmax
2
� �2r
� c2 �� 2�
3
� �þ crmax
� c2 �
3þ 1
� �:
Hence, the total area A covered by OPT is, at
most, c2ð�3 þ 1Þ � optþ c2�.
Note that dðvi; vjÞ � rmin, where vi, vj are in
DS S. Hence, all disks with radius rmin2 centered at
nodes in S are 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
2 centered at a vertex vi, where vi is
in the DS S, place a regular hexagon of width rmin,
as shown in Fig. 3. Each hexagon has an area offfiffi3p
2 r2min. For example, in Fig. 3, the disk d1 uses an
area of at leastffiffi3p
2 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 of ðffiffi3p
2 r2min � ðrmin2 Þ
2�Þ=6. Hence, each unit
disk can use an area of
ffiffiffi3p
2r2min �
ffiffiffi3p
2r2min �
�r2min
4
� �6
� �� :85r2
min:
Therefore, the size of S is bounded by
jSj � Total Area A
:85r2min
�opt c2 �
3 þ 1� �� �
þ c2�
:85r2min
� opt 1þ �=3:85
rmaxrmin
þ 1
2
� �2
þ �
:85
rmaxrmin
þ 1
2
� �2
� 2:4rmaxrmin
þ 1
2
� �2
optþ 3:7rmaxrmin
þ 1
2
� �2
:
tuTheorem 1. The BFS_SCDS algorithm has an approximation
ratio of 12ðkþ 12Þ
2, where k ¼ rmaxrmin
.
Proof. Let C denote the SCDS obtained from the BFS_SCDS
algorithm. Let BTf and BTb be the blue nodes in TfðsÞand TbðsÞ, respectively. We have
jCj ¼ jBTf j þ jBTb j þ jSj
� 5jSj
� 5 2:4 kþ 1
2
� �2
� optþ 3:7 kþ 1
2
� �2" #
� 12 kþ 1
2
� �2
optþ 18:5 kþ 1
2
� �2
:
tu
THAI ET AL.: ON CONSTRUCTION OF VIRTUAL BACKBONE IN WIRELESS AD HOC NETWORKS WITH UNIDIRECTIONAL LINKS 1101
Fig. 2. On the proof of the size relationship between an S and 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 factorof Oð1Þ.
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 SCDS’s 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 ¼ ðV ;EÞ anda set of nodes S � V called terminals, find a directedSteiner tree T rooted at r 2 V such that there exists a directedpath from r to all the terminals in T and the number of theSteiner nodes is minimum.
Note that a Steiner node is a node in T but 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 S are black and the other nodesin V are white. First, let us introduce the followingdefinitions:
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.
A v-spider is a spider rooted at a white node v. 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 of G is
called a black-blue component if G0 is connected and consists of
only 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 a
v-spider performs in the following ways:
. Step 1: Start from level l ¼ 1.
. Step 2: For each undeleted node u at level l in thespider, do the following:
– Step 2.1: Add a unidirectional edge ðv; woÞ foreach w0 2 NþðuÞ such that ðv; woÞ =2 E.
– Step 2.2: If ðv; uÞ is bidirectional, add a unidirec-tional edge ðwi; vÞ for each wi 2 N�ðuÞ such thatðwi; vÞ =2 E.
– Step 2.3: If ðv; uÞ is unidirectional, add a unidirec-tional edge ðwi; woÞ for each wo 2 NþðuÞ and wi 2N�ðuÞ such that ðwi; woÞ =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 ¼ ðV ;EÞ, and a set of black nodes S
2: OUTPUT: A tree T ðrÞ rooted at any node r 2 V n Sspanning all nodes in S
3: T ;;4: while The number of black and blue nodes in G > 1 do
1102 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 7, NO. 9, SEPTEMBER 2008
Fig. 4. The spider rooted at V is 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 B is 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 operation
and update G
7: end while
8: Construct T ðrÞ from the set of black and blue nodes
rooted at node r, where r is the root of the last
contracted v-spider
9: Return T ðrÞSince the GSC algorithm is a solution of the DSMSN
problem, 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 S using a greedy methodshown in Algorithm 1. In the second stage, it calls theAlgorithm 3 twice to generate two trees: Tfðr1Þ andTbðr2Þ. Tfðr1Þ is a tree of blue and black nodes rooted atr1, in which there is a path from r1 to all the blue andblack nodes, and Tbðr2Þ 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 ¼ ðV ;EÞ2: OUTPUT: An SCDS C
3: Find a DS S using Algorithm 1
4: Tfðr1Þ ¼ GSCðG; SÞ5: Reverse all the edges in G to obtain G0
6: Tf0 ðr2Þ ¼ GSCðG0; SÞ
7: Reverse all edges in Tf0 ðr2Þ to obtain Tbðr2Þ
8: Find the path P ðr2; r1Þ from r2 to r1, having minimum
number of white nodes on to it.
9: Color all the white nodes in P ðr2; r1Þ blue
10: H ¼ Tfðr1Þ [ Tbðr2Þ [ P ðr2; r1Þ11: Let C be all nodes in H
12: 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 T ðrÞobtained 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, the
connectivity between A and B is still preserved. There
are four possible ways in which A and B can be
connected via u.
Case 1: This case is shown in Part (a) of Fig. 5. Thereare paths from A to B and from v to B. When the v-spider
is contracted and u gets deleted, according to Step 2.1 of
the proposed contracting operation, there exists an edge
from v to B, and according to Step 2.3, there exists an
edge from A to B. Hence, after the deletion of u, the
connectivity from A to B and v to B are still preserved.
Case 2: This case is shown in Part (b) of Fig. 5. A and
B are 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 exists
an edge from v to A and from v to B, respectively, and
according to Step 2.3, there exists an edge from A to B.
Hence, after the deletion of u, the connectivity from A to
B, v to A, and v to B is still preserved.
Case 3: This case is shown in Part (c) of Fig. 5. There
is a path from B to A via u, and also there is a path fromv to A via u. When the v-spider contracts, according to
Step 2.1, there exists an edge from v to A, and according
to Step 2.3, there exists an edge from B to A. Hence,
after the deletion of u, the connectivity from v to A and
B to A is still preserved.
THAI ET AL.: ON CONSTRUCTION OF VIRTUAL BACKBONE IN WIRELESS AD HOC NETWORKS WITH UNIDIRECTIONAL LINKS 1103
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, B to A, and B to 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 B to v.Hence, the connectivity between v, A, and B ispreserved.
In conclusion, the connectivity of any two nonspidernodes A and B is 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 root
of 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 ¼ ðV ;EÞ 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 W be the set of allwhite nodes left, and D be the set of all nonwhite nodesleft. Since there does not exist any spider, we haveNþðWÞ \D ¼ ;. Thus, G½W [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 T of 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 T ðrÞ consists of only nonwhite nodes
and spans all the black nodes in S.Consider the last r-spider. From Lemma 3, all the
nodes 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. If u is black, then Lemma 6 holdsfor this black node u. If u 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 C is a set of nodes in T ðr1Þ [ T ðr2Þ [P ðr2; r1Þ. From Lemma 6, C contains a set S of all
black nodes. Hence, C is 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 exist
a directed path ðx;wi; r2; wj; r1; wk; yÞ, where wi are the
nodes in T ðr2Þ from x to r2, wj are the nodes in
P ðr2; r1Þ, and wk are the nodes in T ðr1Þ from r1 to y.tu
5.4 Theoretical Analysis
Lemma 7. Given a directed DG G ¼ ðV ;EÞ, for any arbitrary
node v 2 V n S, we have jNþðvÞ \ Sj � ð2kþ 1Þ2, where
k ¼ rmax=rmin.
Proof. Recall that NþðvÞ is a set of outgoing neighbors
of v, and S is a DS of G. Let v be a node with the
largest transmission range. From Lemma 1, we have
dðu; vÞ � rmin, where u, v 2 S. Hence, the size of
NþðvÞ \ S is bounded by the maximum number of
disjoint disks with radius rmin=2 packing in the disk
centered at v with radius of rmax þ rmin=2. We have
NþðvÞ \ Sj j � �ðrmax þ rmin=2Þ2
�ðrmin=2Þ2
�ð2kþ 1Þ2:
tu
Lemma 7 indicates that the maximum number of legs
in a spider is upper bounded by ð2kþ 1Þ2. Now, let T be
an optimal tree when connecting a given set S, and CðT Þis the number of the Steiner nodes in T . Also, let B be a
set of blue nodes in T , where T is the solution of the
DSMSN problem obtained from Algorithm 3. Then, we
have the following lemma:
Lemma 8. The size of B is, at most, ð2þ 2 lnð2kþ 1ÞÞCðT Þ.Proof. Let n ¼ jSj and p ¼ jBj. Let Gi be the graph G at the
iteration i after a spider contracting operation. Let vi,
i ¼ 1 . . . p be the blue nodes in the order of appearance in
Algorithm 3, and let ai be the number of the black and
blue components in Gi. Also, let CðT i Þ be the optimal
solution of Gi. If n ¼ 1, then the lemma is trivial. Assume
that n � 2, thus CðT Þ � 1. Since at each iteration i, we
pick 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 must
be at least aiCðT i Þ
. Thus, we have
aiþ1 � ai �ai
C T i� �þ 1
� ai �ai
CðT Þ þ 1:
This results to the recurrence
ai � a0 1� 1
CðT Þ
� �iþXi�1
j¼0
1� 1
CðT Þ
� �j
� a0e� iCðTÞ þ CðT Þ:
1104 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 7, NO. 9, SEPTEMBER 2008
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The last step uses the fact that �lnð1� xÞ � x andthe second term is a geometric series. Now, leti ¼ CðT Þ ln a0
CðT Þ , we have
ai �a0
elna0
CðTÞþ CðT Þ ¼ 2CðT Þ:
Thus, after CðT Þ ln a0
CðT Þ iterations, the number of
black-blue components left is less than or equal to
2CðT Þ. Using Lemma 7, we conclude that
jBj ¼ iþ 2CðT Þ
�CðT Þ ln a0
CðT Þ þ 2CðT Þ
� 2þ 2 lnð2kþ 1Þð ÞCðT Þ:
tu
6 THE EXTENDED SCDS (Ext_SCDS) ALGORITHM
In this section, we discuss the solutions to the SCDASproblem and present an Ext_SCDS algorithm, which is anextended result from the MSN_SCDS algorithm.
There are several ways to extend a solution of SCDSto be a solution of SCDAS. Note that a set C is calledan SCDAS iff C is SCDS and for any node u =2 C,NþðuÞ \ C 6¼ ;. Therefore, there are basically three mainmethods to extend these solutions:
1. In the first stage, instead of constructing a DS S, wefind a set S such that S is a DS and for any u =2 S,NþðuÞ \ S 6¼ ;. 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 C first,then iteratively add more nodes into C to 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 transmissionrange 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 ¼ ðV ;EÞ2: OUTPUT: An SCDAS C
3: Call the MSN_SCDS algorithm to construct an SCDS S.
Note that all the nodes in S are either black or blue and
the rest of the nodes are white
4: for All u 2 S do
5: All nodes v 2 N�ðuÞ 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 of
white nodes in N�ðvÞ, color v blue and color all the
white nodes in N�ðvÞ gray
9: end while
Basically, after constructing an SCDS S, all nodes not inS are white. Now, we need to check if these white nodesare also absorbed, that is, for each white node x, thereexist a directed edge ðx; uÞ 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 N�ðvÞ gray. This processterminates when there is no white node left. The union ofS and 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 ½rmin; rmax�, where rmin ¼ 200 m,and rmax ¼ 600 m. For each value of n, 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. Thedecrease 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 nodesadded in a path P ðr2; r1Þ.
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Varying the area size. In this experiment, we fixed the
number of nodes in a network to 100, and increased the
area size from 800 m 800 m to 1,600 m 1,600 m, with an
increment of 50. To evaluate the impact of network density
by varying the area, we randomly deployed 100 nodes in
the area with the size changing as described. Each node
randomly chose a transmission range in ½rmin; rmax�,where rmin ¼ 200 m and rmax ¼ 600 m. For each network
instance, we ran the simulations for 100 times and averaged
the results.Fig. 7 provides the performance comparison of
MSN_SCDS and BFS_SCDS, in terms of SCDS size. As
revealed in Fig. 7, the size of MSN_SCDS is smaller than
that of BFS_SCDS for all values of the area size. Another
observation is that, when the area size is small, there is a
small gap between the SCDS size obtained from the
MSN_SCDS and BFS_SCDS. However, as the area is larger,
the gap between them also increases. For example, when
the area is 1,000 m 1,000 m, the SCDS size obtained
from BFS_SCDS is 8, which is only three nodes more
than that of MSN_SCDS. However, when the area size is
1,500 m 1,500 m, the SCDS size obtained from BFS_SCDS
is 19, which is six more nodes than that of MSN_SCDS.
In addition, Fig. 7 shows the clear trend of increase inboth curves. This implies that the SCDS size is biggerwhen the network density decreases, due to the increasein network area size. This can be explained as when thenetwork density decreases, the number of neighbors of
each node decreases as well. Thus, SCDS size has to belarger to dominate all nodes in the network.
7.1.2 Impact of the Transmission Range Ratio
We also conducted simulations to compare the perfor-mance of MSN_SCDS and BFS_SCDS algorithms whenvarying the transmission range ratio k. To change k, wefixed rmin ¼ 200 m and changed rmax from 200 m to 800 mwith an increment of 10. In this experiment, we deployed100 nodes in a fixed area of size 1,000 m 1,000 m. Eachnode randomly chose a transmission range in ½rmin; rmax�.For each network instance, we ran the simulations for100 times and averaged the results.
As illustrated in Fig. 8, the SCDS size obtained fromMSN_SCDS is always smaller than that of BFS_SCDS. Fig. 8also shows the clear trend of decrease in both curves. Thisimplies that the size of SCDS gets smaller when thetransmission range ratio is higher. It can be explained thatwhen the transmission range ratio is higher, there is abigger gap in terms of transmission range between nodes.Therefore, the nodes with bigger transmission ranges candominate more nodes.
In addition, Fig. 8 reveals that when the transmission
range ratio is low, there is a huge gap between the SCDSsize obtained from MSN_SCDS and BFS_SCDS. Forinstance, when the transmission range ratio is 1, the SCDSsize obtained by BFS_SCDS is 53, which is 26 more nodesthan that of MSN_SCDS. However, when the transmissionrange ratio is high, the gap between them is small. Forexample, when the transmission range ratio is 2, theSCDS size obtained from BFS_SCDS is 10 nodes, which isonly two more nodes than that of MSN_SCDS. Addition-ally, when the transmission range ratio is 4, the SCDS sizeobtained from BFS_SCDS is almost equal to that of
MSN_SCDS. However, the SCDS size obtained fromBFS_SCDS is never lower than that of MSN_SCDS.
1106 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 7, NO. 9, SEPTEMBER 2008
Fig. 7. Impact of the area size.
Fig. 8. Impact of the transmission ratio “k”.Fig. 6. Impact of the number of nodes.
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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 are
averaged 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 nodesin 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 of
nodes 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.
THAI ET AL.: ON CONSTRUCTION OF VIRTUAL BACKBONE IN WIRELESS AD HOC NETWORKS WITH UNIDIRECTIONAL LINKS 1107
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 reveals
that when the transmission range ratio is small, the gap
between SCDS and SCDAS sizes is small, and when the
transmission range ratio is larger, this gap is also larger.As presented in Fig. 11b, when the transmission range
ratio is between 1.0 and 1.5, the size ratio of SCDAS and
SCDS is close to 1.0. As the transmission ratio increases
above 1.5, the size ratio of SCDS and SCDAS also increases.
This can explained as when the transmission range ratio is
low, there are more bidirectional links in the network.
Hence, there are more chances of a dominating node
having bidirectional link to the nodes it is dominating.Thus, less number of nodes are needed to extend an SCDS
to an SCDAS.
8 CONCLUSIONS
In this paper, we have studied the SCDS problem and the
SCDAS 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 ðm-SCDSÞ first, and then augment it based on
the connectivity to make it k-connected.
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1108 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 7, NO. 9, SEPTEMBER 2008
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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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, includingrouting protocols, coverage in sensor networks, broadcast tree, virtualbackbone, group testing, and nonunique probe selection. She serves onthe editorial board of the Journal of Combinatorial Optimization and 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 andseveral books. He is the editor in chief of the Journal of CombinatorialOptimization and is also on the editorial board for several other journals.Thirty PhD students have graduated under his supervision. He is amember of the IEEE.
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THAI ET AL.: ON CONSTRUCTION OF VIRTUAL BACKBONE IN WIRELESS AD HOC NETWORKS WITH UNIDIRECTIONAL LINKS 1109
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