Int. J. of Networking and Virtual Organisations, Vol. x, No. x, xxxx 1

Secure Local Algorithm for Establishing aVirtual Backbone in 3D Ad Hoc Network

Emad E. Abdallah*,Alaa E. Abdallah,Ahmad Al-Khasawneh,Mohammad Bsoul,Ayoub Alsarhan

Faculty of Information Technology,Hashemite University, Zarqa 13115, JordanEmail:{emad,aabdallah,akhasawneh,mbsoul,ayoubm}@hu.edu.jo*Corresponding author

Abstract: Due to the limited lifetime of the nodes in ad hoc

and sensor networks, energy efficiency needs to be an important

design consideration in any routing algorithm. It is known that by

employing a virtual backbone in a wireless network, the efficiency of

any routing scheme for the network can be improved. Most of the

current algorithms for electing the virtual backbone mainly focused

on the energy balance among the network nodes without considering

the nodes distribution in the real environment. In this paper, we use

the node’s geometric locations to introduce a first secure algorithm

that can construct the virtual backbone structure locally in 3D

environment; we have proofed that our new algorithm construction

time is constant.

Keywords: Virtual bakcbone; Ad hoc network; Domonating Set.

Reference to this paper should be made as follows: EE. Abdallah*,

AE. Abdallah, A. Al-Khasawneh, M. Bsoul, and A. Alsarhan(xxxx)

‘Secure Local Algorithm for Establishing a Virtual Backbone in 3D

Ad Hoc Network’, International Journal of Networking and Virtual

Organisations, Vol. x, No. x, pp.xxx–xxx.

Biographical notes: Emad E. Abdallah received his PhD in

Computer Science from Concordia University in 2008, where he worked

on multimedia security, pattern recognition and 3D object recognition.

He received his BS in Computer Science from Yarmouk University,

Jordan, and his MS in Computer Science from the University of Jordan

in 2000 and 2004, respectively. He is currently an Assistant Professor

in the Department of Computer Information Systems at the Hashemite

University (HU), Jordan. Prior to joining HU, he was a Software

Developer at SAP Labs Montreal. His current research interests

include computer graphics, multimedia security, pattern recognition,

and computer networks.

Copyright c© 2009 Inderscience Enterprises Ltd.

2 Emad E. Abdallah

Alaa E. Abdallah is an Assistant Professor in the Department ofComputer Science of Hashemite University, having joined in 2011.He obtained his B.Sc. in Computer Science (Yarmouk University2000), M.Sc. in Computer Science University of Jordan 2003), Ph.D.in Computer Science (Concordia University 2008) Montreal-Canada.Prior to joining Hashemite University, he was a Network Researcherat consulting private company in Montreal (2008 - 2011). His researchinterest includes the Routing Protocols for Ad Hoc Networks, Paralleland Distributed Systems, and Multimedia Security.

Ahmad Al-Khasawneh is an associate professor at HashemiteUniversity and currently acting dean of Prince Al-Hussein binAbdullah II Faculty of Information Technology. Dr. Khasawneh holdsa Ph.D. of Information Systems, M.S in Information Technology andComputer Engineering both from Newcastle University, Australia andB.S in Computer and Automatic Control Engineering, Jordan. Dr.Khasawneh has more than 40 published refereed articles in scholarlyinternational journals and proceedings of international conferences.He also served on the Editorial Board of some international journalsand as publicity chair and technical program committee member ofseveral International conferences and workshops. Among his previouspositions are assistance vice president for scientific affairs and Directorof the eLearning Center at Hashemite University. Prior to joiningthe Hashemite University of Jordan, he has been managing GalileoInternational and Royal Jordanian Airlines R&D project since 1994.

Mohammad Bsoul is an Assistant Professor in the Computer Sciencedepartment of Hashemite University. He received his BSc in ComputerScience from Jordan University of Science and Technology, Jordan, hisMaster from University of Western Sydney, Australia and his PhD fromLoughborough University, UK. His research interests include wirelesssensor networks, grid computing, distributed systems, and performanceevaluation.

Ayoub Alsarhan received his Ph.D. degree in Electrical and ComputerEngineering from Concordia University, Canada, in 2011, the M.Sc. in Computer Science from Al al-Bayt University, Jordan, in2001, and B.E. degree in Computer Science from the YarmoukUniversity, Jordan, in 1997. He is currently an assistant professor at theComputer Information System at Hashemite University, Zarqa, Jordan.His research interests include Cognitive Network, Parallel Processing,Machine Learning, and Real Time Multimedia Communication overInternet.

1 Introduction

The recent advances in technologies have enabled a new kind of networks, so calledwireless ad hoc networks, which do not require any pre-existing infrastructurefor establishing connectivity and routing messages. Two nodes can communicate

Virtual Backbone in 3D Ad Hoc Network 3

in a bidirectional manner if and only if the distance between them is at mostthe minimum of their transmission ranges. If one node wishes to communicatewith another node outside its transmission range, the set of nodes between thetwo endpoints should forward their packets so they can communicate. This meansthat in ad hoc network, any node must be able to play the role of a router in aconventional network.

A crucial problem in multihop routing is to find an efficient and correct routebetween a source and a destination; however for many networks, a more importantproblem is to provide an energy efficient routing protocol because of the limitedbattery life of the wireless nodes. One way to decrease the power consumption(communication overhead) of the routing protocols is to narrow down the searchspace for a route to the node in the virtual backbone. A connected dominating set(CDS) Alzoubi et al. (2002b), Das & Bharghavan (1997), Wu et al. (2002), Wu &Li (2001) can form an interesting virtual backbone. A connected dominating set ofa graph is a connected subset of nodes such that each node in the graph is eitherin the subset or adjacent to at least one node in that subset.

The routing algorithms that use virtual backbone only allow nodes ofthe connected dominating set (dominators) act as routers; all other nodescommunicate via a neighbor in the dominating set. Clearly, the efficiency of thisapproach depends largely on the process of finding the dominating sets and thesize of the corresponding virtual backbone.

Due to the nature of ad hoc network, Algorithms to construct a virtualbackbone should be local, where each node of the network only uses informationobtained uniquely from the nodes located no more than a constant (independent ofthe size of the network) number of hops from it. In this paper we propose a securelocal algorithm to construct a virtual backbone for 3d wireless ad hoc network.The new algorithm has a constant time complexity.

The rest of the paper is organized as follows: In Section 2, we briefly present thenetwork model and some related work. A Space Partition System of the 3D spaceneeded in our algorithms is described in Section 3. In Section 4, we introduce ourlocal algorithm to construct a virtual backbone. We conclude our paper in Section5.

2 Preliminaries

2.1 The Network Model

We assume that the set of n wireless nodes is represented as a point set S in 3Dspace; each mobile host knows the coordinates of its position. All network nodeshave the same communication range r. Two nodes are connected by an edge if theEuclidean distance between them is at most r. The resulting graph is called a unitdisk graph (UDG)Bose et al. (2001), Mauve et al. (2001). A dominating set fora graph is a set of vertices whose neighbors, along with themselves, constitute allthe vertices in the graph.

4 Emad E. Abdallah

2.2 Ad hoc network security threats

The authors in Kuhn & Wattenhofer (2003) showed several threats against ad hocand sensor network. We summarize two of them as follows:

• Spoofed or altered information, in this attack, the information thatexchanged between nodes is targeted by the attackers; they try to change itto false information. This could lead to a wrong assessment and an incorrectconstruction of the virtual backbone.

• Selective forwarding: in most virtual backbone construction algorithms itis assumed that all participating nodes will faithfully forward and receivedmessages. In this attack, nodes may refuse to forward the coming packetsand simply drop them. This would definitely affect the whole constructionalgorithm and may lead to a disconnected backbone.

There were many attempts to secure ad hoc routing algorithms by usingpublic key cryptography Zhou & Haas (1999), Hubaux et al. (2001), Binkley &Trost (2001). It is known that ad hoc network nodes usually have limited powerresource Aslam et al. (2003), Chang & Tassiulas (2004), it’s also known thatpublic key cryptography algorithms are resource expensive Kuhn & Wattenhofer(2003), thus using public key cryptography is not really applicable for virtualbackbone construction algorithms because these algorithms need a fair amount ofcommunication between nodes.

2.3 Related Work

The general problem of finding the smallest virtual backbone (smallest connecteddominating set) for a graph is known to be NP-Hard Clark et al. (1990), Garey& Johnson (1979), Hochbaum (1995), Karp (1972). Several algorithms for findingan approximation for a small size virtual backbone have been proposed. TheGreedy algorithm Chvatal (1979), Johnson (1974), Slavik (1996) for constructing avirtual backbone is a global algorithm where the run time depends on the numberof nodes. The greedy algorithm grows a tree rooted at the node that has themaximum number of neighbors. The root is colored black and all its neighbors arecolored gray. Then, the algorithm scans the gray nodes and their white neighborsiteratively, and selects the gray node or the pair of gray and white node with themaximum number of white neighbors. The selected node(s) are marked black andtheir neighbors are marked gray. The algorithm terminates when all the nodes havebeen marked either black or gray.

Alzoubi et al. (2002a,b) proposed a distributed algorithm to construct virtualbackbone; in this algorithm if the node unique ID is minimum among its neighbors,it adds itself to the dominating set and removes all its neighbors from theconsideration of the set members. This process is repeated at each node, such thatthe resulting set is a non-connected dominating set. The nodes in the resultingset use local topology information for a node, up to 3 hops away, to add gatewaynodes to the set until the set becomes a connected dominating set. The maindisadvantage of this algorithm is the construction time of the independent setwhich can be proportional to the number of nodes, thus it is a non-local algorithm.

Virtual Backbone in 3D Ad Hoc Network 5

Figure 1 Unit diameter cube. top face is orange, right side face is light green

It has been proved in Abdallah et al. (2010) that this algorithm is not a localalgorithm.

None of the algorithms mentioned above has both constant approximationbound and constant worst case time bound. One approach to achieve these boundsis to use the underlying geographic information. The first algorithm to determine avirtual backbone in 2D within a constant approximation of the optimal dominatingset in a constant time was proposed by Czyzowicz et al. (2008). A 3D version ofthe algorithm has been proposed in Abdallah et al. (2010).

3 Space Partition System

Our space partition system uses identically shaped tiles that fill the entire space.Each tile used in our tiling system consists of 27 cube of diameter equal to 1 unit,see Fig. 1. Each cube in the tile represents one class which has a unique integer.Assume that the first cube is centered at the coordinates (x1, y1, z1), i.e. the z-axispasses through the center of face 1 (top face), the x-axis passes through the centerof face 2 (right side face) and y-axis passes through the center of face 3 (front sideface). We will call this orientation as the centering orientation; the coordinates ofthe centers of the classes from 2 to 27 are shown in table 1. They all have thesame orientation as class 1. See Fig. 2 for an example of the tile used, showing theplacement of the cubes in the tile with the associated classes labels.

Let the tiling starts by placing the center of one tile, TLCN , at the coordinate(x1, y1, z1), with orientation equal to the centering orientation. To cover all thefaces of TLCN we need 26 other adjacent tiles that are in contact with TLCN

in the positions summarized on Table 2. Each tile has the same orientation asTLCN . Fig. 3 shows the space tiling process used in our algorithm. It is clearthat any point can calculate locally its class number by determining to which tileand corresponding cube it belongs. If a node located exactly on the shared facebetween two cubes C1 and C2, the node is considered of class 1 (the class with thesmaller ID).

6 Emad E. Abdallah

Figure 2 The tile used in the space partition system divided into 27 cube ofdiameter 1 and the class numbering associated with the cube (a) shows thecubes class number. (b) shows the whole tile.

Virtual Backbone in 3D Ad Hoc Network 7

Table 1 Coordinates of the 27 cubes that forms the center tile, first cube is centeredat (x1, y1, z1).

C 1 (x1, y1, z1) C 2 (x1 + 1/√

3, y1, z1)

C 3 (x1 − 1/√

3, y1, z1) C 4 (x1, y1 − 1/√

3, z1)

C 5 (x1, y1, z1 + 1/√

3) C 6 (x1, y1 + 1/√

3, z1)

C 7 (x1, y1 − 1/√

3, z1) C 8 (x1 + 1/√

3, y1, z1 + 1/√

3)

C 9 (x1 − 1/√

3, y1, z1 + 1/√

3) C 10 (x1 − 1/√

3, y1, z1 − 1/√

3)

C 11 (x1 + 1/√

3, y1, z1 − 1/√

3) C 12 (x1 + 1/√

3, y1 − 1/√

3, z1)

C 13 (x1 − 1/√

3, y1 − 1/√

3, z1) C 14 (x1, y1 − 1/√

3, z1 − 1/√

3)

C 15 (x1, y1 − 1/√

3, z1 + 1/√

3) C 16 (x1, y1 + 1/√

3, z1 − 1/√

3)

C 17 (x1, y1 + 1/√

3, z1 + 1/√

3) C 18 (x1 + 1/√

3, y1 + 1/√

3, z1)

C 19 (x1 − 1/√

3, y1 + 1/√

3, z1) C 20 (x1 + 1/√

3, y1 + 1/√

3, z1 + 1/√

3)

C 21 (x1 − 1/√

3, y1 + 1/√

3, z1 + 1/√

3) C 22 (x1 + 1/√

3, y1 + 1/√

3, z1 − 1/√

3)

C 23 (x1 − 1/√

3, y1 + 1/√

3, z1 − 1/√

3) C 24 (x1 + 1/√

3, y1 − 1/√

3, z1 + 1/√

3)

C 25 (x1 + 1/√

3, y1 − 1/√

3, z1 − 1/√

3) C 26 (x1 − 1/√

3, y1 − 1/√

3, z1 − 1/√

3)

C 27 (x1 − 1/√

3, y1 − 1/√

3, z1 + 1/√

3)

4 A LOCAL ALGORITHM FOR 3D VIRTUAL BACKBONE(3DVBP)

Our local algorithm to construct a virtual backbone consists of two phases. Inthe first phase, a dominating set 3DDOM is constructed using Algorithm 2. Inthe second phase, each node from 3DDOM creates paths connecting dominatorsthat are at most three hops apart. Using the space partition described in theprevious section, each node can determine its class number locally using a constantnumber of arithmetic operations. The nodes are aware of the locations of all theirneighbors, using the periodic hello messages, so they can also calculate the classnumber of each neighbor. It is clear that the nodes that are in same cube areneighbors because the diameter of the cube is 1 Our local construction of thedominating sets is based on a similar algorithm proposed by Czyzowicz Czyzowiczet al. (2008) for 2D. To calculate a dominating set of a unit disk graph G, eachnode of G executes Algorithm 2.

In the following we will discuss some properties of Algorithm 2. To begin with,we show that Algorithm2 is local algorithm by showing that it will terminate in aconstant number of steps. Let 3DDOM be the set of dominator nodes that resultsfrom applying Algorithm2 on each node in from UDG.

1. Algorithm 2 is a local algorithm.

If the selection of a dominator in a cube depends only on the nodes thatare at most constant hops away from the nodes in the given cube, then the

8 Emad E. Abdallah

Table 2 Coordinates of the centers of the 26 neighbors.

center tile 1 (x1, y1, z1)

neighbor 1 (x1 + 3/√

3, y1, z1)

neighbor 2 (x1 − 3/√

3, y1, z1)

neighbor 3 (x1, y1 − 3/√

3, z1)

neighbor 4 (x1, y1, z1 + 3/√

3)

neighbor 5 (x1, y1 + 3/√

3, z1)

neighbor 6 (x1, y1 − 3/√

3, z1)

neighbor 7 (x1 + 3/√

3, y1, z1 + 3/√

3)

neighbor 8 (x1 − 3/√

3, y1, z1 + 3/√

3)

neighbor 9 (x1 − 3/√

3, y1, z1 − 3/√

3)

neighbor 10 (x1 + 3/√

3, y1, z1 − 3/√

3)

neighbor 11 (x1 + 3/√

3, y1 − 3/√

3, z1)

neighbor 12 (x1 − 3/√

3, y1 − 3/√

3, z1)

neighbor 13 (x1, y1 − 3/√

3, z1 − 3/√

3)

neighbor 14 (x1, y1 − 3/√

3, z1 + 3/√

3)

neighbor 15 (x1, y1 + 3/√

3, z1 − 3/√

3)

neighbor 16 (x1, y1 + 3/√

3, z1 + 3/√

3)

neighbor 17 (x1 + 3/√

3, y1 + 3/√

3, z1)

neighbor 18 (x1 − 3/√

3, y1 + 3/√

3, z1)

neighbor 19 (x1 + 3/√

3, y1 + 3/√

3, z1 + 3/√

3)

neighbor 20 (x1 − 3/√

3, y1 + 3/√

3, z1 + 3/√

3)

neighbor 21 (x1 + 3/√

3, y1 + 3/√

3, z1 − 3/√

3)

neighbor 22 (x1 − 3/√

3, y1 + 3/√

3, z1 − 3/√

3)

neighbor 23 (x1 + 3/√

3, y1 − 3/√

3, z1 + 3/√

3)

neighbor 24 (x1 + 3/√

3, y1 − 3/√

3, z1 − 3/√

3)

neighbor 25 (x1 − 3/√

3, y1 − 3/√

3, z1 − 3/√

3)

neighbor 26 (x1 − 3/√

3, y1 − 3/√

3, z1 + 3/√

3)

Algorithm 1: SelectDominatorFromSetCandidates

// for a given node n, let nu be the number of times,where the node n hasbeen used before without problems.// for a given node n, let dc be the distance from the center of the nodeclass.// for a given node n, let rp be the remaining power of the node n.// rand is a random number between 0 and 1.begin

for i = 1 to j (the number of nodes in SX) do1

pi = nu∗rp∗1/dc∑i=1 nu∗rp∗1/dc ∗ rand2

return neighbor i with probability pi.3

end

Virtual Backbone in 3D Ad Hoc Network 9

Algorithm 2: A LOCAL ALGORITHM FOR 3D DOMINATING SETS

// Algorithm is executed independently by each node. Execution startseither when a node X needs to find its dominator, or if it receives a requestto find a dominator in its cube.begin

X determines its class number using its coordinates and the tiling1

information.X finds all its neighbors and obtains their coordinates and class2

numbers.Let CX be the cube that contains X3

if X of class number 1 then4

SelectDominatorFromSetCandidates(all nodes in the cube that5

contains X).

else6

X finds the set S1(X) that contains all nodes in CX that has no7

neighbor of lower class.if S1(X) is not empty then8

SelectDominatorFromSetCandidates(S1(X)).9

else10

X requests from every neighbor of lower class number to run the11

algorithm if not already running.When all nodes in CX finish their calculations.12

SelectDominatorFromSetCandidates(all nodes in the cube thatcontains CX that are not dominated).

X informs all its neighbors that a dominator selection in its cube is13

completed and gives them the results.end

10 Emad E. Abdallah

Figure 3 Tilling system used.

algorithm is local. The selection of a dominator of a node in a cube of class1 is done by checking only the nodes inside that cube, which is not morethan 1 hop away. If the node of class i <> 1, here the algorithm waits theresults that comes from neighbors nodes of lower classes, so eventually it willreaches nodes of class 1 after at most i− 1 steps. �

2. Every vertex of UDG is either in 3DDOM or adjacent to a vertex in3DDOM . Thus, set 3DDOM is a dominating set of the UDG.

For a node v in UDG that is not in 3DDOM . If v is of class 1, then oneof the nodes in the cube containing v is designated as a dominator in line 5.Else if v is of class i > 1 and at least one node of its cube is not dominated bya node of an adjacent lower class, one of the nodes in the cube is designatedas a dominator in line 12. Since the diameter of the cube is 1, node v isdominated by the designated node. �

3. The hop distance between a node u ∈ 3DDOM and it closest node v ∈3DDOM is at most 3.

Assume the shortest path between a node u ∈ 3DDOM and it closest nodev ∈ 3DDOM has at least 4 hops, i. e. u, x1, x2, x3, v. But x1 and x3 are not in

Virtual Backbone in 3D Ad Hoc Network 11

3DDOM because they are neighbors to other in 3DDOM . If x2 ∈ 3DDOM ,it makes the hop distance between u and x2 equal to 2, a contradiction. Ifx2 is a neighbor to a node w ∈ 3DDOM , it makes the hop distance betweenu and w equal to 3, a contradiction. �

4. For any node u, the number of dominators inside the sphere centered at uwith a radius of k units is bounded by a constant ηk.

It is known that the distance between any two dominators is greater thanone unit. Then the half-unit radius spheres centered at the dominators aredisjoint. So to find how many dominators can lie inside a sphere of a radiusK units centered at some node u, (see Figure 4) we have to find the ratioof the volume of the sphere of radius k + 0.5 to the volume of the disjointspheres of radius 0.5. Thus

ηk =43π(k + 0.5)3

43π(0.5)3

. (1)

When k = 2 or 3, we have ηk = 125 or 343, respectively.�

5. The backbone creation algorithm is more secure than any other availablealgorithm.

Most of the ad hoc network attacks depend on the idea of sending falseinformation from the attackers’ nodes to other nodes Kuhn & Wattenhofer(2003). When our virtual back bone construction algorithm chooses frommany possible candidates, it uses the previous information about thecandidates’ authentications; this is done by counting how many timeseach node has been used without problems. Moreover, our algorithm usesa randomize factor to reduce the chance of choosing the attackers falseinformation. Even with the little chance of constructing the backbone basedon the false information, our packets acknowledgement process will eliminatethat little chance. �

There are many algorithms proposed to connect a set of dominators Alzoubiet al. (2003, 2002a,b), Gao et al. (2004), most of them depend on using three hopsnode information. Our algorithm for finding the connectors can be described asfollows: each node X from 3DDOM independently runs Algorithm 3.

Theorem 1 If UDG is connected, then the set of 3DDOM and connectorsconstructed by Algorithm 2 and Algorithm 3 is a virtual backbone.

We assume that the set 3DV BB (resulted virtual backbone) is disconnected;this means that there are at least two distinct components, R and L, and let R bethe closest component to L. We know that there is at least one path P between L

12 Emad E. Abdallah

Figure 4 Example on the number of dominator inside a sphere

Algorithm 3: A LOCAL ALGORITHM FOR 3D VIRTUAL BACKBONE

// Let 3DDOM be the set of dominating set calculated by applyingAlgorithm 2.// Each node X from 3DDOM run the rest of the algorithm.// Let H1(X) is the set of one hop neighbor of X, H2(X) is the set of thetwo hops neighbors of X, and H3(X) is the set of the three hops neighborsof X begin

for every node Y in H2(X) do1

if Y in 3DDOM and class number of Y is less than the class number2

of X thenNode X chooses from H1(X) a node U with the highest degree3

that creates a path (X,U, Y )

for every node Y in H3(X) doif Y in 3DDOM and class number of Y is less than the class number4

of X thenNode X chooses two nodes U from H1(X) and V from H2(X)5

that creates the path (X,U, V, Y ). Where U and V have thehighest degree.

The union of the selected nodes along with the nodes from 3DDOM6

describes the virtual backbone for UDG.end

Virtual Backbone in 3D Ad Hoc Network 13

Figure 5 Three cases example to prove 1.

and R because L and R are connected in the original UDG. Let P consist of thefollowing nodes A, x1, .., x2, ..x3, B, where A ∈ L, and B ∈ R.If the path length is greater than or equal to 4, (A, x1, x2, xi, x3, B), then x1, x2, xior x3 are not in 3DDOM otherwise there will be a closer component to R or L.But by point 2 of 3DDOM properties, if a node is not in 3DDOM , it has to bea neighbor to one from 3DDOM . This makes x2 has at least one neighbor from3DDOM , thus it belongs to some other component N . But this means that N iscloser to L from R and closer to R from N ; Contradiction.If the path length is equal to 3, (A, x1, x2, B), then x1 and x2 are not in 3DDOMotherwise there will be a closer component to R or L. By point 2 of 3DDOMproperties, if a node is not in 3DDOM , x1 and x2 have to be neighbors to nodesin 3DDOM which bring a new component closer to R or L; contradiction. ByAlgorithm 2, because the length of the path is equal to 3 nodes A or B will selectconnectors to connect R and L.If the path length is equal to 2, (A, x1, B) and if A,B are not in 3DDOM , then x1is not in 3DDOM otherwise there will be a closer component to R or L. By point2 of 3DDOM properties, if a node is not in 3DDOM , x1 has to be neighbors tonodes in 3DDOM which bring a new component closer to R or L; contradiction.If the path length is equal to 2, (A, x1, B), if A is in 3DDOM and B is not in3DDOM , then B is a neighbor to a node w from L that belong to 3DDOM . Thiswould make the path length between two nodes from 3DDOM is equal to 3. ByAlgorithm 2, because the length of the path is equal to 3 nodes A or W will selectconnectors to connect R and L.If the path length is equal to 2, (A, x1, B) and if A,B are in 3DDOM . Thiswould make the path length between two nodes from 3DDOM is equal to 2. ByAlgorithm 2, because the length of the path is equal to 2, nodes A or B will selectconnectors to connect R and L. �

14 Emad E. Abdallah

5 Conclusion

Existing virtual backbone construction algorithms for ad hoc network do notconsider the nodes distribution in real environment. Moreover, these algorithms aresubject to attacks due to lack of security in the design. In this paper, we proposeda fully local secure algorithm that constructs a virtual backbone of UDG in 3Denvironment with a constant construction time. The new algorithm is suitable forsituations, where the topology changes are frequent and unpredictable.

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