fault-tolerant spanners for ad hoc networks

INTERNATIONAL JOURNAL OF NETWORK MANAGEMENTInt. J. Network Mgmt (2011)Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/nem.807

Fault-tolerant spanners for ad hoc networks

D. Satyanarayana1,*† and S. V. Rao2

1School of Electronics and Electrical Engineering, University of Leeds, Leeds, UK2Department of Computer Science and Engineering, Indian Institute of Technology, Guwahati, India

SUMMARY

Spanners for ad hoc networks provide several benefits such as low communication cost and resourceconsumption. These spanners need to be fault tolerant in resource-constrained ad hoc networks. In thispaper, we have proposed three spanners, called fault-tolerant local Delaunay triangulation (FTLDel),fault-tolerant relative neighborhood graph (FTRNG), and fault-tolerant Gabriel graph (FTGG). The fault-tolerant spanners provide reliability to the network by avoiding heavy packet loss and retaining usefulgeometric properties. The performance of fault-tolerant spanners FTLDel, FTRNG, and FTGG areevaluated using the network simulator ns2.28. Copyright © 2011 John Wiley & Sons, Ltd.

Received 13 December 2009; Revised 24 July 2011; Accepted 31 July 2011

1. INTRODUCTION

A wireless ad hoc network is a collection of wireless nodes communicating via radio without anypre-existing infrastructure. A generic model of an ad hoc network is the unit disk graph (UDG) [1],where all nodes in the network have the same transmission range of one unit. The communication costover UDG is costly owing to the large number of edges, O(n2), where n is the number of nodes in thenetwork. In order to reduce the communication cost, researchers have derived a special type ofproximity subgraph of UDG called a spanner, which contains a linear number of edges. Some ofthe spanners include Gabriel graph (GG) [2], relative neighborhood graph (RNG) [2], Delaunaytriangulation (Del) [3], Yao graph [4,5], local Delaunay triangulation (LDel) [6,7], planarized localDelaunay triangulation (PLDel) [6,7], partial Delaunay triangulation (PDT) [8], restricted Delaunaygraph [9,10] and its variants [11,12].In general, various resource limitations and environmental constraints create frequent link and node

failures in ad hoc networks, which make the network unreliable. For example, edge disconnectionsoccur because of buildings, walls, mountains, and obstacles between the wireless nodes. Similarly, nodefailures occur as a result of exhausted battery power, accidents, landslides, debris, eruption of volcano,and cyclones. Thus the network topology should be fault tolerant to handle these failures. Hence thespanners should take care of the network topology even in the event of k-node or k-link failure, for someconstant k. In the literature, two types of fault-tolerant spanners are considered [13–15], namely k-nodefault-tolerant spanners (k-NFTS) and k-link fault-tolerant spanners (k-LFTS).A graph G= (V,E) is called a k-node fault-tolerant t-spanner for the given set of nodes V, if for any

subset V′ of V of size at most k, the graph G \V′ is a t-spanner for the node set V \V′. Similarly, thegraph G= (V,E) is called k-link fault tolerant t-spanner for V, if for any subset E′ of E of size at mostk and for any pair of nodes a and b in V, the shortest path length between a and b in the graph G \E′

*Correspondence to: Degala Satyanarayana, Information Systems, University of Nizwa, Oman. PC:616.†E-mail: [email protected]

Copyright © 2011 John Wiley & Sons, Ltd.

D. SATYANARAYANA AND S.V. RAO

is at most t times the length of the shortest path between a and b in graph KV \E′. Here, KV denotes thecomplete graph with the node set V. The notation G \V′ denotes the graph with the vertex set V \V′,and edge set E that have both end points in V \V′. In ad hoc networks, one can use UDG(V) instead of KV.The computational complexity for constructing these spanners is in O(n log n). But these are

centralized algorithms, which need massive communications. In this paper, we have given localizedalgorithms for constructing the fault-tolerant spanners in ad hoc networks. These spanners considerstable nodes for forming the network graph. If we are aware of the nodes which are likely to be failedor unstable in advance, we can avoid such nodes as a part of the network graph to overcome theundesirable problems. The node stability depends on various parameters such as power level, mobility,its environment, and usability. In this paper, we consider more stable nodes in the network forconstructing the spanners, which are called fault-tolerant spanners.The proposed fault-tolerant spanners are fault-tolerant local Delaunay triangulation (FTLDel),

fault-tolerant relative neighborhood graph (FTRNG), and fault-tolerant Gabriel graph (FTGG). Thespanners FTLDel, FTRNG, and FTGG are simulated using a network simulator, ns-2.28 [16], and theyhave shown better performance than their counterparts PLDel, RNG, and GG, respectively.The rest of this paper is organized as follows. In Section 2 we discuss the definitions and notations

used in this paper, which is followed by the survey. Section 3 describes the new contributions FTLDel,FTRNG, and FTGG. In Section 4 the simulation details and results are given. Section 5 concludes.

2. RELATED WORK

The unit disk graph (UDG) [1] of a given set of nodes can be defined as a network graph where anedge between any two nodes in the graph exists if and only if the Euclidean distance between themis not more than one unit. In the UDG model, the nodes contain same transmission range which isnormalized to one unit. The position information of a node is obtained using any positioningtechniques such as Global Positioning System (GPS) [17]. The number of edges in UDG can be aslarge as O(n2) [6]; hence higher communication cost. Thus researchers have proposed lower-costnetwork graphs called spanners.A graph G is a t-spanner of UDG if and only if for any two nodes u and v, |ΠG (u, v) |≤ t � |ΠUDG (u, v) |,

where ΠG(u, v) represents the shortest path between nodes u and v in the graph G and |ΠG(u, v)| representsthe length of the path. The value t is called the stretch factor of the spanner. In other words, a spannerG′ is a spanning subgraph of the given graph G such that the length of the shortest path between anytwo nodes in G′ is bounded in the graph G. The following are some of the important criteria to evaluatespanners [11,12]:

1. Spanning ratio. The length of the shortest path between two nodes in a spanner is bounded. Itis desirable that the spanning ratio of the graph is as small as possible, which is useful fordelay-bound applications.

2. Sparseness. Since the medium is shared, the sparseness of the topology reduces the channelcontention at the MAC level. Sparseness enables most routing protocols to run efficiently.

3. Planarity. The planarity of the spanner is required for many geographic routing protocols toguarantee the delivery of packets to their destination.

4. Localized construction. Distributed local algorithms are desirable for construction andmaintenance of the spanners in ad hoc networks, because it is easy to maintain the informationof nodes within a constant number of hops.

5. Bounded degree. The bounded node degree of the spanner will make the network to consumefewer resources. This is crucial for ad hoc networks because nodes have limited memory, power,and other resources.

Several spanners have been studied recently both by computational geometry scientists and networkengineers. Here, we review the definitions of some that are used frequently in wireless networks.The minimum spanning tree (MST) [18] is a tree that connects all the nodes in the network and whose

total edge length is minimal. MST is obviously one of the sparsest possible connected subgraphs, where

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Network Mgmt (2011)DOI: 10.1002/nem

FAULT TOLERANT SPANNERS FOR AD HOC NETWORKS

the stretch factor can be as large as (n-1). The other geometric graph is called the nearest neighborhoodgraph (NNG) [19], where the edges are determined by minimum distance. More precisely, for any nodep there exists an edge�pq in NNG if and only if the node q is the nearest neighbor of p. Although NNGincorporates a useful notation, it has the disadvantage of disconnectedness.The relative neighborhood graph (RNG) [2] is a spanner of UDG, where the edges are formed

between the nodes based on the lune. The lune L(u,v) is the intersection of two circles with radius�uvj j and centered at the nodes u and v, respectively. The edge�uv exists in RNG if and only if the luneL(u,v) does not contain any other node inside, as shown in 1. The spanning ratio of RNG is in O(n) [2].The algorithm to construct RNG for ad hoc networks can be found in Karp and Kung [20]. Anotherimportant spanner frequently used in wireless networks is the Gabriel graph [2,21]. As shown inFigure 1, there exists an edge �uv in GG between two nodes u and v if and only if the circle withdiameter �uvj j centered at uþv

2 does not contain any other node inside. The spanning ratio of GG isin O

ffiffiffin

pð Þ [2]. The algorithm to construct GG for ad hoc networks can be found in Karp and Kung[20] and Bose et al. [22]. The RNG is a subgraph of GG and the two spanners are planar graphs.The Yao graph (Yao) [4,5] is a spanner especially designed for energy-efficient communication in ad

hoc networks. The Yao graph is defined as follows: the area around each node u is divided into k equalcones, where k≥ 6. In each cone, the edge�uv belongs to the Yao graph if v is the closest, as shown inFigure 1. Ties are broken arbitrarily. The spanning ratio of Yao is 1

1�2 sin pk, where k≥ 6. Unlike RNG and

GG, Yao is not a planar graph.Delaunay triangulation (Del) [3] has many attractive geometric properties useful for efficient

communication in ad hoc networks. A triangle 4 uvw of three nodes u, v, and w is called a Delaunaytriangle if the circumcircle of three nodes does not contain any other node inside, as shown in Figure 2.The spanning ratio of Del is 2.4 [23,24]. However, constructing the spanner Del for ad hoc networks iscostly, because the edges in Del can be larger than the transmission range. The unit Delaunay triangu-lation (UDel) [7] contains the edges that are present in both UDG and Del. The UDel is a t-spanner ofUDG, but how to construct UDel locally for ad hoc networks is not known. Xiang Li et al. [6,7] proposeda localized algorithm called planarized local Delaunay triangulation (PLDel) with useful Delaunayproperties for ad hoc networks. A triangle 4 uvw of three nodes u, v, and w is called a local Delaunaytriangle if the circumcircle of the three nodes does not contain any other node that is a neighbor of node

u v

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GG RNG Yao

Figure 1. Rules of GG, RNG, and Yao graphs

Del PLDel

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Figure 2. Rules of Del and PLDel



u, v, or w, as shown in Figure 2. The spanning ratio of PLDel is 2.4 [6,7]. Gao et al. [9,10] proposed aspanner called the restricted Delaunay graph (RDG), which combines the node clustering algorithmwiththe Delaunay triangulation graph. A similar type of spanner is proposed by Wang et al. [11,12] whichuses an efficient clustering algorithm.In wireless networks, many geometric routing protocols use these spanners as an underlying network

topology to solve local maxima and local loop problems. Some of these protocols include greedy perimeterstateless routing (GPSR) [20], face routing (FACE) [22], adaptive face routing (AFR) [25], greedy facegreedy (GFG) [22], greedy other adaptive face routing (GOAFR+) [26], and localized routing protocols [27].Even though the existing spanners contain many desirable geometric properties, they cannot

maintain the geometric properties for longer periods owing to various resource and environmentalconstraints in the network. Thus we are motivated to propose fault-tolerant spanners which not onlymaintain the properties of the spanner but also save the packet loss occurring due to path breaks.

3. FAULT-TOLERANT GRAPHS

The spanners for ad hoc networks require fault tolerance to retain the geometric properties of thespanners under various network conditions such as frequent node failures and edge disconnections.In addition, heavy packet loss occurs when an edge disconnection or failed node exists in routing path.The existing fault-tolerant spanners [13–15] cannot be used for ad hoc networks owing to theircentralized nature of algorithm. Satyanarayana et al. [28] proposed a fault-tolerant version of thespanners RNG and GG for ad hoc networks. The spanners assume that the nodes in the network arestatic and not suitable for mobile nodes. In this paper, we describe the fault-tolerant version of thespanners PLDel, RNG, and GG by introducing the concept of stability factor to the earlier work inorder to increase the fault tolerance in the spanners.The system model considered for fault-tolerant spanners is asynchronous wireless ad hoc networks.

Each node in the network has a unique identification number and is aware of its location informationthrough any positioning techniques. An omnidirectional antenna is used by the transceiver and all thenodes in the network have the same transmission range. The network has reliable communication,where a single packet transmitted is received by all the nodes in its vicinity. Each node has the facilityto measure the remaining energy in its battery.

3.1. Fault-tolerant local Delaunay triangulation (FTLDel)

The algorithm for constructing FTLDel has two stages. In the first stage, we check the node’s stabilityfactor and select a subset of nodes which are stable. In the second stage of the algorithm, the planarizedlocal Delaunay triangulation is constructed with the nodes chosen in the first stage. The formalalgorithm is given in Algorithm 1.

3.2. Stability factor

In order to maintain a stable spanner under high topological changes, the stability factor is employed tomark the stability of each node in the network. The stability of a node represents the node’s robustness.The node stability is measured with various parameters depending on the application and environment.For example, the node’s stability can be computed with the following parameters: the node’s batterypower level, node mobility, the place of the deployment, and the node degree.In the literature, the stability factor is computed in various ways depending on the need. For exam-

ple, Yao Yu et al. [29] computed the stability factor using the parameters link quality and mobility of anode. Gandhi et al. [30] considers the parameters mobility and energy for calculating the stabilityfactor, whereas Chawla et al. [31] has considered only mobility. Liu [32] has taken the mobility forcalculating the stability factor, whereas Punde et al. [33] has considered both mobility and packetprocessing ratio.In this paper we have considered a node’s remaining battery power and node mobility for calculat-

ing the stability factor as they bring more topological changes into the network. The probability of anode failure is high if its battery power level is low. For a node with a high percentage of remaining


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Figure 3. Stability factor


power (PRP), the probability of node failure is low. Hence the stability factor increases exponentiallywith remaining battery power as shown in Figure 3. Similarly, the stability of a node increases expo-nentially with the decrease in node mobility as shown in Figure 4. For any node, the formula for nor-malized stability factor on a 10-point scale is given below:

Stability factor ðSFÞ ¼ 5 � log2ðPRPÞlog2100

þ log2 16-minðV ; 15Þð Þlog216

� �

where PRP is the percentage of remaining battery power and V is the velocity of the node.Each node finds its stability factor using the above formula and checks for its suitability to

participate in the spanner. Suitability depends on the threshold value. If the stability factor is more thanthe threshold value then the node participates in the spanner construction; otherwise it does notparticipate, by switching off its transmitter. However, the fixing of the threshold value depends onmany factors, such as communication pattern, application type, and user needs. For example, highthreshold values are preferable for large data transfers. On the other hand, if the threshold value is veryhigh, the number of nodes participating in the spanner may become fewer, which may lead to networkdisconnections. To avoid such problems, one can include the node degree as one of the parameters forcalculating the stability factor. However, these problems may not occur in the case of high-densitynetworks or nodes with a large communication range.In the second stage of the algorithm, after the selection of the stable nodes, each node u gathers one hop

neighborhood information n1(u) from the broadcasted hello_packets. Construct the PLDel with the stablenodes as follows: first each node u computes Del(n1(u)). For any node u the triangle4 uvw2Del(n1(u))for each v, w belong to n1(u) if and only if the circumcircle of the triangle 4 uvw does not contain anyother node p from n1(u), as shown in Figure 5. In order to find whether a node p is inside the circumcircleof three nodes u, v, and w, we can use the following determinant, where ux and uy represent x and y coor-dinates of node u:


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Figure 5. Delaunay triangle

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Figure 4. Stability factor vs. node mobility


ux uy u2x þ u2y 1vx vy v2x þ v2y 1wx wy w2

x þ w2y 1

px py p2x þ p2y 1

��

��Find all the Gabriel edges from Del(n1(u)) that will never be deleted. An edge�uv is called a Gabriel

edge, if the edge �uv is common edge for two triangles 4 uvw and 4 uvx, and the angles ∠ uwv and∠ uxvmust be less than or equal to 90�, as shown in Figure 6. Find all the consistent triangles. A triangle4 uvw is called consistent if the4 uvw is in Del(n1(u)), Del(n1(v)), and Del(n1(w)). The resulting graph iscalled LDel1, which is not a planar graph. For each consistent triangle4 uvw, compute the in-circle testwith all LDel1 nodes of its one-hop neighbors. If the circumcircle of the nodes is not empty, remove thetriangle 4 uvw from the graph and the resulting graph is PLDel.


w

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Figure 6.�uv is Gabriel edge


Algorithm 1. FTLDel

1. Each node u computes the stability factor and switches off its transmitter if the stability factor is lessthan the threshold.2. Each node u broadcasts its ID and location information through hello_packet.3. Each node u collects one-hop neighborhood information, n1(u), through the hello_packets sent in
step 2.
4. Each node u computes Delaunay triangulation of n1(u), Del(n1(u)). Find all the Gabriel edges thatwill never be removed.

5. Each node u finds the consistent triangles. A triangle 4 abc is consistent if 4 abc2Del(n1(a)),4 abc2Del(n1(b)), and 4 abc2Del(n1(c)).

6. Each node u broadcasts all the edges of consistent triangles and Gabriel edges.7. Remove the consistent triangles which have the edge crossings with the edges sent in the previousstep.

The following are some of the properties of FTLDel.

Lemma 1. The time complexity for constructing FTLDel is in O(4 4), where 4 is the maximum nodedegree of UDG.

Proof. Each node requires O(1) time to compute the stability factor of the node. The time forcomputing the PLDel using one-hop neighbors is in O(4 4). Hence the time complexity of FTLDelis in O(4 4). □

Lemma 2. The FTLDel is a planar graph.

Proof. Let the graph FTLDel of the given set of nodes S be nonplanar. That is, the graph PLDel withthe set of nodes S′ is nonplanar, where S′ is a set of stable nodes and S′⊆ S. However, this contradictswith the statement that the PLDel is a planar graph. Hence the lemma is proved. □

Lemma 3. The message complexity for constructing FTLDel is in O(n).

Proof. The number of messages required for constructing PLDel is in O(n), which leads to the numberof messages required for constructing FTLDel is in O(n). □

Lemma 4. The spanning ratio of FTLDel is 2.5.



Proof. The length of the shortest path between two nodes p and q in FTLDel is t times the length of theshortest path between the same nodes in UDG with the same vertices, where t is the spanning ratio,which is 2.5 in the case of PLDel. Thus the spanning ratio of FTLDel becomes 2.5.

3.3. Fault-tolerant relative neighborhood graph (FTRNG)

The spanner FTRNG is constructed in two phases. In the first phase, a subset of the given nodes,called stable nodes, is selected. In the second phase of the algorithm, the relative neighborhoodgraph (RNG) [2] is constructed with the stable nodes computed in the first phase. The resulting graphis called a fault-tolerant relative neighborhood graph (FTRNG). The formal algorithm is given inAlgorithm 2.Initially, each node checks its stability factor. If the stability factor is below the critical threshold, it

switches off its transmitter and does not participate in the spanner. Hence only a subset of nodes existsin the spanner construction. Eachnode u gathers one-hop neighborhood information n1(u) from thebroadcasted hello_packets, computes the RNG edges, and adds it to FTRNG. For each node u, an edge�uai is added to FTRNG if the lune(u, ai) is empty, as shown in Figure 7.

Algorithm 2. FTRNG

1. Each node u switches off its transmitter if (stability factor ≤ threshold).2. Broadcast hello_packet with ID and location information.3. Node u receiving hello_packet stores the information in n1(u).4. Each node u follows the steps below:

for each node υ2 n1(u)if (lune(u, υ) does not contain any node in n1(u))add edge uv� to FTRNG.

The following are some of the properties of FTRNG.

Lemma 5. The time complexity of FTRNG is in O(4 2), where4 is the maximum node degree of UDG.

Proof. The time required for computing the stability factor is in O(1). The lune test needs O(4 2) time,which leads to the total time of RNG construction in O(4 2). Hence the time complexity of FTRNG isin O(4 2). □

Lemma 6. The FTRNG is a planar graph.

Proof. Let FTRNG be a nonplanar graph for the given set of nodes S. In other words, the RNG with theset of nodes S′⊆ S is not a planar graph, which is a contradiction. Hence the lemma is proved. □

a ui

Figure 7. RNG neighborhood relation



Lemma 7. The spanning ratio of FTRNG is in O(n).

Proof. The length of the shortest path between two nodes in FTRNG is t times the length of theshortest path between the same nodes in UDG, where t is the spanning ratio. In the worst case, itbecomes n. Thus the FTRNG spanning ratio is in O(n). □

Lemma 8. The message complexity for constructing the FTRNG is in O(n).

Proof. This follows from the fact that the number of messages required for constructing RNG with thegiven set of nodes is in O(n), as the computation of stability factor does not require any messagetransmission. □

3.4. Fault-tolerant Gabriel graph (FTGG)

A similar procedure is followed in FTGG construction. In the first phase of the FTGG spannerconstruction, a subset of the given nodes is selected based on their stability factor. In the second phaseof the algorithm, the Gabriel graph (GG) [2] is constructed on the subset of nodes. The resulting graphis called a fault-tolerant Gabriel graph (FTGG). The formal algorithm is given below. In this algorithm,initially the unstable nodes are switched off and will not participate in the network graph. The Gabrielgraph edges are calculated and added to FTGG. Step 4 of the FTGG algorithm describes the GGconstruction procedure with one-hop neighborhood information. Note that disk(u,v) represents thecircle with diameter �uvj j centered at uþv

2 , as shown in Figure 8.

Algorithm 3. FTGG

1. Each node u switches off its transmitter if (stability factor ≤ threshold).2. Broadcast hello_packet with ID and location information.3. Node u receiving hello_packet stores the information in n1(u).4. Each node u follows the steps below:

for each node υ2 n1(u)if (disk(u,v) does not contain any node in n1(u))add edge uv� to FTGG.

The following are some of the properties of FTGG.

Lemma 9. The time complexity of FTGG is in O(4 2).

Proof. The time complexity required to compute the stability factor is in O(1), whereas the timerequired to compute GG is in O(4 2). Thus the total time required to compute FTGG is in O(4 2).Hence the lemma is proved. □

Lemma 10. The FTGG is a planar graph.

u v

Figure 8. GG neighborhood relation



Proof. Let FTGG be a nonplanar graph. This leads to the statement that GG is not a planar graph,which is a contradiction. Hence FTGG is a planar graph. □

Lemma 11. The message complexity for FTGG is in O(n).

Proof. The message complexity of GG is in O(n). The message transmissions are not required forcomputing the stability factor. Hence the message complexity of FTGG is in O(n). □

Lemma 12. The spanning ratio of FTGG is in Offiffiffin

pð Þ.

Proof. The length of the shortest path between two nodes p and q in FTGG is t times the length of theshortest path between the same nodes in UDG of the same vertices, where t is called the spanning ratio,which is O

ffiffiffin

pð Þ in the case of GG. Thus the spanning ratio of FTGG is in Offiffiffin

pð Þ. □The fault-tolerant spanners FTLDel, FTRNG, and FTGG preserve the properties of the spanners

PLDel, RNG, and GG, respectively.

Lemma 13. FTRNG(V) ⊆ FTGG(V) ⊆ FTLDel(V) for a set of nodes V.

Proof. Let S be the set of stable nodes. We know that RNG is a subgraph of GG for any given set ofnodes. Thus RNG(S)⊆GG(S), which leads to FTRNG(V)⊆ FTGG(V). Similarly, it follows thatFTGG(V)⊆ FTLDel(V), as GG is a subgraph of PLDel. Hence the lemma is proved. □

Lemma 14.‘

FTRNG(V)≥‘

FTGG(V)≥‘

FTLDel(V).

Proof. This follows from Lemma 13 and the fact that |ΠFTRNG(u, v)|≥ |ΠFTGG(u, v)|≥ |ΠFTLDel(u, v)|,where |ΠG(u, v)| is the length of shortest path between two nodes u and v in the graph G. □

4. SIMULATION AND PERFORMANCE

We have used network simulator (ns-2.28) [16] in the simulation work. Apart from the fault-tolerantspanners FTLDel, FTRNG, and FTGG, we have simulated PLDel, RNG, and GG for comparisonpurpose. To evaluate the performance of the proposed spanners, we run six routing protocols on thesespanners. These routing protocols are greedy routing (Gdr) [22], compass routing (Cmp) [34,35],random compass routing (RandCmp) [35], most forward routing (MFR) [36], nearest neighbor routing(NNR) [37], and farthest neighbor routing (FNR) [7]. In the simulation, we have considered both thepower level and the node mobility for computing the stability factor, which has been described inSection 3.2.

4.1. Simulation model

In all the simulation experiments, we have taken five different node scenarios, with each containing 50nodes distributed randomly over a 600�600 m2 grid. The transmission range of each node is 150m.For each node, a random initial energy between 15 and 45 joules is given. Each node is given a randommobility of 0–20m/s velocity. The threshold value considered for selecting the stable nodes is 7. Thetotal simulation time is 600 seconds. For the connection patterns, we have chosen randomly threesource nodes on one side of the grid and three destination nodes on the other side. The energy isconsumed for each packet transmission with transmission power (txPower_) of 0.665 watts. For eachpacket reception, the energy is consumed with the receiving power (rxPower_) of 0.395 watts. Theidle power is 0.035 watts.

4.2. Simulation results

With the above simulation model, we run several simulations at different packet transfer rates. Toevaluate the performance of FTLDel, we run six geometric routing protocols Gdr [22], Cmp


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Figure 9. Greedy forward routing

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Figure 10. Compass routing



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Figure 11. Random compass routing

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Figure 12. Most forward routing



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Figure 13. Nearest neighbor routing

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Figure 14. Farthest neighbor routing



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Figure 15. Greedy routing

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[34,35], RandCmp [35], MFR [36], NNR [37], and FNR [7] on two spanners FTLDel and PLDel. Foreach spanner, we compute the delivery ratios for the above six routing protocols at different CBRintervals. Figures 9-14 show that the spanner FTRNG has better delivery ratios compared to PLDel.This happens because the spanner FTLDel considers stable nodes for fault tolerance and avoids heavypacket loss, whereas PLDel does not.Similarly, the next two experiments are performed to evaluate the performance of FTRNG and

FTGG. The delivery ratios are computed on each spanner for the above four routing protocols atdifferent CBR intervals. From Figures 15-22, we can say that the spanners FTRNG and FTGG showbetter delivery ratios than RNG and GG, respectively. This happens because the spanners FTRNG andFTGG avoid packet loss by considering stable nodes in the network graph, whereas their counterpartspanners suffer from packet loss.

5. CONCLUSIONS

In this paper we have proposed fault-tolerant spanners FTLDel, FTRNG, and FTGG for fault-tolerantrouting by choosing stable nodes in the network. These stable nodes provide network reliability byretaining the geometric properties of the spanners and avoid packet loss. The simulation resultsconsolidate the performance of the fault-tolerant spanners.

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AUTHORS’ BIOGRAPHIES

D. Satyanarayana received Doctorate from the Department of Computer Science and Engineering, IndianInstitute of Technology Guwahati in June-2010. He completed M.Tech and B.Tech degrees in Computer Scienceand Engineering at Visvesvaraya Technological University and Sri Venkateswara University in 2000 and 1998,respectively. He worked as a Postdoctoral Research Associate in the School of Electronics and ElectricalEngineering, University of Leeds, UK during Jan-2009 to Mar-2010. His areas of research include wirelessnetworks, ad hoc & sensor networks, mobility management, quality of service, energy efficient communication,and computational geometry applications.

S. V. Rao is currently working as an Associate Professor in Computer Science and Engineering department,Indian Institute of Technology Guwahati. He obtained Ph.D and M.Tech from the Department of ComputerScience and Engineering, Indian Institute of Technology, Kanpur in 1999 and 1993, respectively. He graduatedin Computer Engineering from Andhra University in 1990. His research interests include Ad hoc & SensorNetworks, Search Engines, Web Applications, Image Processing, Optical Character Recognition, ComputerGraphics, and Computational Geometry.


fault-tolerant spanners for ad hoc networks

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