A New Algorithm for Backbone Formation in Ad

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A New Algorithm for Backbone Formation in AdHoc Wireless Networks of Nodes with Different

Transmission RangesH K

Dept. of Comp ter Science, Concordia University Montreal, Q ebec, Canada H3G M8

Email: h_ [email protected]

Abs act-W consider the problem of backbone formationin ad hoc wireless networks composed of heterogeneous nodes. virtual backbone in an ad hoc wireless network provides ahierarchical infrastructure that can be used to address importantchallenges such as e cient routing, mu ticasting/broadcasting,

activi y-scheduling, and energy e ciency. We model a wirelessnetwork in which nodes have di erent transmission ranges by a d k graph. virtual backbone in such a network can be modeledby a S ongly Connected Dom ating and Absorbent Set (SCDAS)in the associated disk graph. For practical reasons, it is desirableto minimize the size of this backbone. In this paper, we propose ane cient distributed algorithm for the construction of anin ad hoc networks modeled by disk graphs. Extensive simulationresults show that the constructed by our algorithm issigni cantly smaller than those generated by the algorithms prior to our work.

Index Te s-W ireless ad hoc networks, dominating set, absorbent set, virtual backbone, directed graph.

I. NTRODUCTION

A wireless ad hoc networ is an in as ct reless, peer to-peer networ of wireless nodes comm nicating with eachot er in a m ltihop fashion. S ch a networ comes togetheras the need arises and achieves a goal witho t relying onany established infrast ct re. The lifespan of an ad hocnetwor depends on the application and may vary om a few ho rs to a possibly long period of time. Applications of s chnetwor s incl de conferencing, home networ ing, personalrea networ s, emergency services, and sensor networ s. The

niq e ch racteristics of ad hoc networ ing along with itsstringent req irements bring abo t many challenges that need to be addressed before it can emerge as a ll- edged technology. One s ch gro p of cr cial problems are the intertwinediss es of e cient ro ting, broadcasting, scalability and energy conservation. The se of a hierarchical in ast ct re in whicha small s bset of nodes forms a virt al bac bone in anotherwise inherently at networ has been s ggested in theliterat re to ad ress these problems. It is desirable to minimize the size of this bac bone beca se the smaller the size of the bac bone, the fewer the n mber of the nodes that need to eep their t ansceivers on at all times and the smaller the size of t e ro ting tables that these nodes need to maintain.

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L N Dept. of Comp ter Science, Concordia University

Mon eal, Q ebec, Canada H3G M8Email: [email protected]

Unit Dis Graphs (UDGs) are typically sed to modwireless ad hoc networ s with symmet ic lin s. In s ch model, the virt al bac bone described earlier corresponds to connected dominating set. In a UDGG E) a Connected

Dominating Set (CDS) is a s bset s ch that for everynode is either in or has a neighbor in and thes bgraph ind ced by is connected. Minimizing the sizeof the virt al bac bone corresponds to nding the smalles possible CDS in the nderlying graph however nding Minimum CDS (MCDS) in UDGs is nown to be NP-h rd[4].

In reality, nodes in a networ may not necessarily have same transmission range. This might simply occ r when tnetwor consists of vario s inds of wireless devices with di erent powers and di erent f nctionalities. Even when thnetwor consists of similar nodes, these nodes may ne to adj st their transmission ranges for many reasons. F

example, in many power cont ol schemes, nodes adj st th transmission power to save energy, red ce collisions and on. All of these scenarios res lt in int od cing asymmetrilin s in the networ . In the presence of nidirectional lin sin the networ , UDGs can no longer be sed to model thnetwor . Instead, Disk G phs (DGs) e sed to model ad hoc networ s of nodes with di erent transmission ranges. In DGG E) nding a bac bone t anslates to constr ctinga St ngly Connected Dominating and Absorbent Set (SCDAS)in G. An SCDAS is a s bset of nodes s ch that everynode is either in or has an o tgoing and an incomingneighbor in , and the s bgraph ind ced by is st ongly connected. A directed graph is strongly connected if for two vertices and there exists a directed path betweenand For the same practical reasons, we see to minimize tsize of the SCDAS in DGs, however, that remains an NP-h problem.

CDS const ction in ad hoc networ s wit symmet ic lin s has been extensively st died over the past decade. Howevnot many algorit ms have been proposed for networ s wiasymmetric lin s and those few at have been proposed const ct relatively l rge SCDASs or have high complexiti that ma e them impractical in the context of ad hoc networ



In this paper, we propose a distrib ted algorithm for the constr ction of SCDAS in ad hoc networ s with asymmetriclin s. We ass me t ere is no topology change ind ced bynode mobility in the networ and that in the face of mobility, the recalc lation of a new SCDAS is initiated once thenetwor stabilizes. Extensive sim lation res lts show that o ralgorithm o tperforms the existing algorithms in the literat reand const cts SCDASs that are signi cantly smaller in size.

The remainder of this paper is organized as follows. Insection 2, we give an overview of the relevant wor donein this area. We t en present o r algorithm in Section 3.The sim lations cond cted to eval ate the performance of o r algorithm re detailed in Section 4. Finally, Section 5 concl des the paper and points o t possible directions fort re wor .

II. ELATED WORK

CDS formation in homogeneo s wireless ad hoc networ s modeled by UDGs is a well-st died problem [1, 2, 3, 7, 10].However, the co nterp t of this problem in networ s of

heterogeneo s nodes is relatively new. W [9] extended the concept of a connected dominating set (CDS) in ndirectedgraphs to a connected dominating and absorbent set (SCDAS)in directed graphs and gave a simple local algorithm for its constr ction. Initially all the nodes are nm ed. A nodeuis mar ed if there exists a nodev in its dominating set anda node in its absorbent set, b tv does not dominate Inother words, a node is mar ed only if it lies on the shortest path om one neighbor to another. J st as in the m r ingalgorithm of [10], this m ing r le generates a l rge SCDASwith a lot of red ndant nodes. Therefore, W proposed two r les to red ce the size of the constr cted SCDAS. These two r les are indeed the extended forms of the r les proposed

in [10], which are applied to the nodes in the SCDAS. Therst r le allows a node u to be removed om the SCDASif its dominating (absorbent) neighbor set is covered by the dominating (absorbent) neighbor set of nodev with a higherID. The second r le allows a nodeu to be removed om theSCDAS if its dominating (absorbent) neighbor set is covered by the nion of the dominating (absorbent) sets of the two connected nodesv and provided that t e ID of nodeuis the smallest among the three nodes. This algorithm doesnot provide a constant approximation ratio and its time and message complexities are9(m) and O( 3) respectively.

In [8], Par et al proposed a cent alized constant ap proximation algorithm for the constr ction of a MinimumSCDAS (MSCDAS) in wireless ad hoc networ s with di erent transmission ranges. In their wor , it is ass med t at the ratio of the maxim m to the minim m t ansmission range is bo nded. They also present two he ristics and eval ate their performance thro gh sim lations. The he ristics re indeed the co nterp t of G ha and Kh ller's algorithms [5] for ndirected graphs.

In the constant approximation algorithm, an o tgoing spanning t ee and an incoming spanning tree rooted at an ar bitrary nodeu are constr cted. The non-leaf nodes of the

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two trees form an SCDAS. Par et al. also presented t cent alized he ristics for the const ction of the SCDAS i directed graphs. Both of these he ristics rely on a Dominatand Absorbent Set (DAS) as t e inp t. The algorithm th constr cts t e DAS consists of two stages: constr ction of Dominating Set (DS) and an Absorbent Set (AS). The nof the two sets then forms the nal DAS. In nding the DAinitially all nodes are m r ed as ncolored. Then, at eaciteration, a nodeu with the highest degree is colored blacand all its ncolored neighbors with an incoming edge omu(its dominatees) become gray. This process terminates when ncolored node is left At the end of this stage, the set of blnodes forms a DS. Before proceeding to the constr ction AS, a preprocessing step is performed whose goal is to red the n mber of nodes selected as AS by chec ing if any grnode (a dominated node) is also absorbed by a blac node there exists s ch node(s), they are colored white.

Once t e preprocessing phase is nished, nodes in the grapare either blac , gray or white. Since white nodes are alreaabsorbed, the constr ction of AS is eq ivalent to nding a

absorbent node for every gray node. In doing so, a greapproach is adopted: at each iteration, a gray node that abso the highest n mber of gray nodes is mar ed blac and itabsorbed gray nodes e m r ed white. This stage terminateswhen no gray node is le , at which time the set of blac nodforms a DAS.

Then two he ristics are proposed to ma e the above DA connected: i greedy spider cont action algorithm (G-SCAand (ii) greedy strongly connected component merging alg rithm (G-CMA). The rst one ses a greedy approach to nan approximation for the directed Steiner t ee with minimSteiner nodes problem to minimize the n mber of white nod req ired to connect the blac nodes. The second he risti

iteratively nds two st ongly connected components whic can be merged at minim m cost among all the pairs of s components, and merges them by coloring the white noon the two directed paths between the two components blaThey cond ct sim lations that showG-CMA consistently o t performsG-SCA in terms of the size of he res lting set.

III. UR LGORITHM

In this section, we give an algorithm to const ct an SCDAin wireless ad hoc networ s wit asymmetric lin s. O ralgorithm ses a pr ning-based approach proposed in [7adapted to asymmetric networ s modeled by directed graph

A. De nitions and preliminariesIn o r algorithm, we seNd(U) to denote the dominating

neighbor set of nodeu i.e. Nd(U) {v (v, u)EE}. A nodev E Nd(U) is also referred to as anincoming or ingressneighbor of nodeUin the literat re. Li ewise,N (u) is sed to denote the absorbent neighbor set of nodeu; i.e N (u){v (u, v)E E}. A nodev E N (u) is also referred to as anoutgoing or egress neighbor of u. Fig re 1 ill strates the dominating and absorbent neighbor sets of a node. Note t these two sets may overlap. In other words, a neighbor of n



u can be both a dominator and an absorbent of nodeu. We de ne the degree of a nodeu to be Nd(U)1+ N (u) .

v

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w

A

Fig. Dominating and abso bent neighbor sets of node

Every nodeu has a ran ( (u), i (u)) which is an ordered pair of itse ective degree and i where the e ective degree of node u is the n mber of u's neighbors in SCDAS, i.e(u){v v E N (u) v E S DAS} + {v v E Nd(U) v ES DAS} . Since the membership of nodes in the SCDAS changes d ring the algorithm, so does the e ective degreeof a node. Assigning a niq ei to every node provides a mechanism to brea ties.

The ran of a node in o r algorithm can be considered ageneral weight that is de ned based on the goal f nction. Sincewe intend to minimize the size of the cons cted SCDAS,we se a node's effective degree in the de nition of its ran . Alternatively, if we see to prolong the node's lifetime, we can de ne its weight as the node's remaining energy. In thefollowing section, we present the centralized description of o r algorithm for ease of exposition, and then we will disc ss the dist ib ted implementation.

Centralized DescriptionWe adopt a pr ning-based approach in this algorithm.

Initially, every node is in the SCDAS and has pending stat s.

At each step, we select the nodeu with the smallest ranom the set P of the nodes with pending stat s. We removenode u om P and apply two local tests to see if it can be removed from the nal SCDAS. The two tests are called(i) the domination and absorbency test and (ii) the st ngconnectivi test. If u passes both tests, we change its stat s toout; otherwise, its stat s becomesin and we will eep it in thenal SCDAS. Nodeu passes thedomination and absorb ncyt st DAT if:a all the nodes that re dominated byu have at least one

other dominator.b all the nodes that are absorbed byu have at least one other

absorbent.Nodeu passes hestrong conn c vity t stif the s bgraph in d ced by its dominators and absorbents is strongly connected.The formal description of this extended algorithm is given in Algorithm 1.

It is easy to see that the removal of a nodeu that passes both tests om the nal SCDAS neither leaves any of its neighborswitho t a dominator/absorbent, nor disconnects the SCDAS,and that both tests are needed to g arantee this. Therefore, the set of the nodes that form the nal sol tion is indeed anSCDAS of the original graph. Also, since at each ro nd one

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Algorithm Cent alized Strongly Connected Dominating an Absorbent Set (SCDAS) Algorithm

S DAS VP Vwhil P = 0 do

u argmin{( (v), i (v)) v EP}- {u}

DAT true For eve node v dominated by u, check there is someother node that dominates v for allv EN (u) do

if ((Nd( ) S DAS) - {u} 0) th nDAT false

nd if nd for For eve node v absorbed by u, check there is someother node that absorbs v for allv ENd(U)do

if ((N (v) S DAS) - {u} 0) th nDAT false

nd if nd for if DAT th n

G Graph[((N (v) UNd( )) S DAS) - {u}]if G is st ongly connectedth n

S DAS S DAS - {u}for allv E(N (u) S DAS) do

(v) (v) - 1nd for for allv E(Nd(U) S DAS) do

(v) (v) - 1nd for nd if

nd if nd whilRet SCDAS

node is removed om the setP the algorithm terminates inn steps.

Distributed Implementation

In order to r n the above algorithm in a dist ib ted fashio every node needs to perform a preliminary set p whiinvolves obtaining some information from its neighbors asetting some variables that will be sed later d ring th exec tion of the algorithm.

Initially, every nodeu sets its stat s to pen ing and ex changes its ran with all its neighbors (incoming and o tgoinand stores the set of its neighbors inNu SCDASa v iable holding the set of neighbors in the SCDAS. Also, it maint the list of its lower ran neighbors inLower _Rank . Note that in this section, whenever we tal abo t neighbors general, we mean both incoming and o tgoing neighboWhen a node nishes r nning the algorithm, it sends Finishe _M sg( Status) to all its neighbors with theStatus



Algorithm Dist ib ted Connected Dominating and Absorbent Set Algorithm, exec ted by nodeu

when Lower _Rank 0DAT trueSendDominator Query to all nodes inN (u)Send AbsorbenCQuery to all nodes inNd(U)for all(v E(N (u) UNd(U)))do

Wait for Dominator _Reply(v, D ) andAbsorbenCReply(v, A )

if ((D - {u} 0)or(A - {u} 0)) th nDAT false

nd if nd for if DAT th n

if G[NuSCDAS- {u}] is st ongly connectedth nStatus outSend Finishe _Msg(out) to (N (u) UNd(U))

lsStatus inSend Finishe _Msg(in) to (N (u) UNd(U))

nd if ls

Status inSendFinishe Msg(in) to (N (u) UNd(U))

nd if

Upon iving ( o inato I bso bentLQ eT_Msgfr mv:

if ! Replyjn_Transit th nReplyjn_Transit trueSend(Dominator/Absorbent) Reply(u, to v

ls Enq e e (Dominator/ Absorbent)_Query(v) inDQQnd if

Upon iving Fin hed_Msg(s t sfromv:

if status out th nif ((v EN (u))&(v ENd(U))) n

8(u) 8(u) - 2ls8(u) 8(u) 1nd if

Nu SCDASNu SCDAS- {v}

nd if if (Rank(v) < Rank(u)) th nLower Rank Lower Rank - {v}

nd if if DQQ 0 th n

v Deq e e DQQSendDominator/AbsorbenCReply(u, D /A ) to v

ls Reply_In_Transit falsend if

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being in or out depending on the decision it has made a e r nning the tests.

A er the initial set p, a nodeu with the lowest ran amongits neighbors becomes the initiator and r ns Algorithm 2. Sin the ran s assigned to nodes e niq e, here exists at leastone s ch initiator. In fact, the experimental res lts show thin random scenarios, on the average, t e algorithm exhibit desirable degree of parallelism.

Fo r messages are sed in conj nction with the domination and absorbency test.Dominator _Query andDominator Reply messages are sed to verify if the node dominated byu have other dominators or not Li ewiseAbsorbenCQueryand AbsorbenCReply messages are sed to veri if the nodes absorbed byu have other absorbents. An o tgoing neighboring nodev incl des the list of its dominators,D in Dominator _Reply(v, D ) and an incoming neighborv incl des the list of its absorbents,A inAbsorbenCReply(v, A ). In order to r le o t the possibilityof the sim ltaneo s opo t of two nodes that dominate/absorb the same nodev node v also reserves two ags Domina

tor_Replyjn_Transit and Absorben Reply jn_Transit ags that are initially set to false. These ags e set to tr ewheneverv has sent a reply to aDominator _Replyor anAbsorbenCReply message, b t has not received the correspondingFinishe _M sg( Status) message om that node. As long as t e ags e t e, node enq e es any rther q eries that it might receive om other neighbors and repli to them only when it becomes aware of the decision made the nodeu to which it has a reply in transit.

Once all the replies are received om the neighbors, nou proceeds to the st ong connectivity test if all the neighb dominated byu have at least one other dominator and a the nodes absorbed byu have at least one other absorbent

Otherwise, nodeu changes its stat s toin and sends aFinishe M sg(in) to all its neighbors.The s ong connectivity test at nodeu examines if the

s bgraph ind ced byNuSCDASis st ongly connected. Note that this test does not req ire any message passing and c be done locally byu sing the local variables which it maintains. If it also passes the second test, nodeu drops o tof the SCDAS and sends aFinishe _M sg( out) to all itsneighbors. Otherwise, it stays in the nal SCDAS and sena Finishe Msg(out) to its neighbors.

Upon receiving aFinishe _M sg( Status) om a neighborv node u removesv om the list of its lower ran neighborsLower Rank . Additionally, if the stat s of the nodev isout it removes it om the set of its neighbors in the SCDANu SCDASThe formal description of this algorithm is givin Algorithm 2.

Before proceeding to the experimental res lts, we expla how a node's e ective degree is comp ted and pdated in thabove algorithm. Initially, every nodeu sets its e ective degree to 0 8(u) ). Then it increments8(u) for each incomingor o tgoing neighborv. If nodeu has a bidirectional lin tonode v; i.e. nodev is both inN (u) and Nd(u) then nodev ca ses 8 (u) to be incremented by 2. Correspondingly, w



decrease8(u by 2 d ring the exec tion of he algorithm if a bidirectional neighbor drops o t of SCDAS in Algorithm 2.

D. -Hop Extension

The s ong connectivity test in Algorithm 2 can be extended to a k-hop neighborhood. It is easy to see that a node can ma e more informed decisions abo t the role it playsin maintaining the strong connectivity of e SCDAS if it considers a larger neighborhood. By doing so, it is able to decide if it is red ndant and if so, removes itself om henal SCDAS, thereby res lting in red cing the size of thenal sol tion However, it sho ld be noted at this comes at

the expense of message overhead. In order to implement thek-hop extension, every node sho ld exchange its ran withits k-hop neighbors and adj st its local variables accordingly. Also, when a node nishes r nning the tests, it sho ld send the Finishe _Msg(Status) to its k-hop neighborhood.

Pe o ance Analysis

The analysis of time and message complexities of Algorithm

2 for the comp tation of an SCDAS is complicated by the existence of nidirectional lin s in the networ . If node u hasa nidirectional lin to node then can directly receive pac ets om u and is therefore aw re of the existence of its incoming neighbor (dominator); however, node u cannot directly hear om node and th s is not aware of its existence.In other words, the main iss e is that a node may not be able to directly identi its o tgoing (absorbent) neighbor(s). Onesol tion is to have each node in the networ emit a beacon,with its ID appended to it, at reg lar intervals. Any node that receives a beacon appends its ownID and forw ds it.Since we ass me the nderlying graph is st ongly connected, every node will event ally hear om its absorbent neighbor(s)

and can detect hem sing the chain of IDs appended byforwarding nodes. Using t is sol tion, the messages exchangeswhen a node wants to identify the set of its absorbents, orwhen it as s them abo t their other dominators d ring the domination and absorbency test, is not necess ily restricted to a xed-size neighborhood. The reply messages om anabsorbent node to a dominator u may be forwarded along a path containingO(n) nodes. With his complication in mind, he analysis of Algorithm 2's time and message complexity isas follows.

Ass ming is the degree of the nderlying graph that mod els the networ , the size of u's k-hop neighborhood is bo nded by . D ring the initial set p phase, every node sendsa constant n mber of messages to its k-hop neighborhoodinq iring abo t the ran s of the nodes in this neighborhood.If all the lin s were bidirectional, his wo ld translate toO(n ) messages, however, since a reply message might need to be forwarded along a path of O(n) nodes, the message complexity of this phase is( n2 ). D ring he exec tionof he algorithm, (n 2 ) messages e req ired for the domination and absorbency test and no message passing is req ired for the s ing connectivity test. Since every node sends(n ) Finishe _Msg(Status) messages once it completes

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the exec tion of the tests, the message complexity of this ro nd of the Algorithm is (n2 ). Th s, the overall message complexity isO(n2 ).

The exec tion time of Algorithm 2 depends on the longe chain of dependency in the nderlying graph where eanode has to wait for nodes of smaller ran in the chain nish r nning the algorithm before it can r n the algorithmSince this chain co ld be of lengthO(n) in the worst case,and each node may have to wait forO(n) time to receive the information abo t its absorbents, the time complexity Algorithm 2 isO(n2).

However, for practical reasons, we are interested in nwor s in which there is a bo nd on the maxim m length o the directed reverse path between any pair of nodes wit directed edge. We call a directed graph-reciprocal if for every directed edge u) E E there exists a directed pathom to u of length at most hops. Under t is ass mption,

t e bo nd for he time and message complexities of Algorith2 wo ld be O( n) and O( n ) respectively.

IV. XPERIMENTAL ESULTS

We cond cted extensive sim lations to eval ate the perfor mance of o r algorithm in networ s with asymmetric linin o r own sim lator developed sing Java SE To st dy t e impact of the pe entage of unidirectional links (PUL) aswell as di erent node densities on e size of the const ctedSCDAS, we experimented with varying n mber of nodes a transmission ranges. We considered a geographic area of 200m by200mand varied the n mber of nodes om 50 to 300 withincrements of 50. Also, each node was assigned a transmiss range randomly selected om the range[r _min, _max .

The only algorithms proposed in the literat re to const can SCDAS in networ s with di erent ansmission ranges

are the ones in [8] and [9], which were disc ssed in dein the related wor . Th s, we compared the performance o r proposed algorithm with the localized m ing algorithmin [9], herea er referred to asWu after the name of thea thor and the two centralized algorithms in [8], name Dominating-Absorbent Spanning Tree (DAST) and GreedyStrongly Connected Component Merging Algorithm (G_CMA)In o r performance comp ison, we foc sed on e impact of node density and the percentage of nidirectional lin s on thsize of the SCDAS constr cted by the fo r algorithms.

Cle rly, the[r _min, _max range a ects he percentage of nidirectional lin s in the networ Therefore, we created ve di erent scen ios in which we experimented with di erent ranges to generate di erent PULs. The t ansmission rangin the rst fo r scenarios were selected from[10m,50m[20m,50ml [30m,50mland [40m,50ml respectively. In thelast set, all the nodes were assigned the xed transmissi range of 50m to ma e it possible to also comp re di erences between UDGs and DGs as inp t graphs. For each val e n in each scenario, we generated as many random graphs req ired ntil we had 1000 strongly connected graphs. Tgraphs were stored and sed across di erent sim lations sing di erent algorithms.



' M -M x r m r ·

icB

4c - .

c

Numb r d

Fig. Relationship between axi u -to- ini u trans ission range ratio and percentage of unidirectional links

We present the res lts in the fo r following sections. In

Section IV-A, we loo at the relationship between the ratioof the maxim m to minim m t ansmission range on the per centage of nidirectional lin s in the inp t graphs. In SectionIV-B, we investigate the impact of the degree of locality withwhich nodes r n the connectivity test in Algorithm 2 on thesize of the cons cted SCDASs. Then, we will compare the performance of o r algorithm with that of its competitors inSections IV-C and IV-D nder varying node densities and percentages of nidirectional lin s.

A. Impact of T nsmission Range on The Percentage of Uni directional Links

As ill strated in Fig re 2, it can be generally seen that the

percentage of nidirectional lin s in the networ is a f nctionof the ratio of maxim m to minim m ansmission range and isalmost independent of the node density in the networ . For example, in o r sim lations, when the transmission range varies between40mand 50m;i.e. _ 25 the percentage of nidirectional lin s varies between 2% to 2%for different val es of n. When the ratio is increased to 67 2 5and5 the percentage of nidirectional lin s rises to25% 40%and 52% respectively. The only cases in which graphs exhibitslightly npredictable behavior is when the ratio of maxim m to minim m transmission range is high 5 and thenetwor is very sparsen 50, 00Fi ally, as expected,when all the nodes have the identical transmission range of

50m there are no nidirectional lin s in the networ .Impact of Locali on The Size of The SCDAS Constructed

by Our Algorithm

We also investigated the e ect of the degree of localityk with which the connectivity test is r n on the size of thegenerated SCDAS. O r goal was to experimentally determine the best tradeo between the degree of locality in this test and he n mber of nodes that can be pr ned. One sample of o r res lts is shown in Fig re 3. According to the res lts obtained

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S" 4 0c"

C"I"Q

_ _ : _ C : _ _ : t C

+ = :· ___.-___ 4____•

Locality of he connectiv y es (k

Fig. I pact of the locality of connectivity test on the size of the SCDA when ans ission ranges vary between and

om the experiments, on the average, k 4st i es a good compromise; while in sparser networ s the size of the set cstill be red ced by increasing k to 5 orin denser instances the gain is typically negligible for k's beyond 2 or 3. T can be seen in Fig re 3 where the c rves grad ally atten o beyond k 5in relatively sparse networ sn 50, 00and beyond k 3 in denser networ s. Therefore, in comparino r algorithm with its competitors, we selected the -hop an4-hop implementations of o r distrib ted algorithm; namel PInOu Unidirectional Distributed k ( Inou UD ) andnOu Unidirectional Distributed k 4 ( InOu UD4)

Impact of Node Densi

In this section we will investigate the impact of node denon the performance of the fo r selected SCDAS const ctioalgorithms. Fig res 4 thro gh show three di erent node densities, in ascending order, in the presence of different PUin the networ . Note that the graphs forn 50n 200andn 250are not shown beca se they exhibit the same trend As can be seen in Fig re 4, W performs better than DASin sp se networ s, especially when is higher. However,as drops below 2%and the nderlying graph tends toa UDG, DAST catches p and o tperforms W PInO UDand PInO UD4 consistently o tperform the other three algorithms while G_CMA is closer to PInO UD , standingin the middle. As the n mber of nodes increases to 1and 150 (moderate densities), DAST consistently o tperforW , and the gap between DAST and W as one gro p anG_CMA, PInO UD and PInO UD4 as the other gro pwidens, especially in the presence of PULs of 12% and hig Another noticeable end is that DAST almost maintains tsame distance from the algorithms in the second gro p p tPLUs of aro nd 12%, b t then considerably narrows down tgap as PUL tends to zero.



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M M t s ss o ge

oI pact of the node density nu ber of nodes

As the n mber of nodes increases to 200, 250, and 300(extremely dense networ s), PInO UD and PInO UD4 ta e the lead more conspic o sly and PInO UD increasesits distance from G CMA. On average, PInO UD andPInO UD4 const ct SCDASs which are 24% and 35%smaller than those constr cted by G_CMA respectively. The

89

e ciency of o r schemes becomes even more noticeable whewe ta e into acco nt the very high complexity of G_CMand the fact that it is a centralized algorithm. Finally, the consistent trend that can be seen from the g res is that as tnode density increases, the difference between PInO t_UDand PInO UD4 decreases and in very dense networ s, they perform almost eq ally well, especially for PULs of 40% alower.

D. Impact of Unidirectional Links

In order to better analyze the impact of PUL on the perf mance of algorithms which we sim lated in o r experimenwe give a di erent presentation of o r res lts in this sectionFig res 7 and 8 show two di erent[T T l ranges and the size of the SCDAS constr cted by each algorithm nd v rying n mber of nodes. As we disc ssed earlier, eac[T T l range corresponds to an almost xed PUL. Morspeci cally, if we ignore the slightly di erent t ends in versparse networ s in the presence of high PULs and ro nd p thaverage PULs for different[T T l transmission ranges,

the transmission ranges [10,50] and [30,50] correspondPULs of 52% and 25%, respectively. As depicted in Fig 9, we also considered the scenario in which PUL is zero ma e it possible to more acc rately predict the trends as PU drops below 12% and tends to zero.

As ill strated in Fig re 7, altho gh W initially o tperforms DAST when 50it does not improve m ch as the node density grows in the presence of high PULs. In other wowhen the ratio of maxim m to minim m t ansmission rangis very high in the networ , W cannot ta e advantage o the increase in node density whereas the other algorithms bene t from increased node density and red ce the relativsize of the SCDAS which they const ct. The reason for W

inability to se increased density in its favor is that its pr ni r les (R les 1 & 2) are not e cient when PUL is high.By comparing Fig res 7 and 8 with Fig re 9, it can b

seen that the existence of nidirectional lin s in the networadversely affects the performance of DAS As can be sein Fig re 9, when there are no nidirectional lin s, DAST performance is m ch closer to that of G CMA compare to Fig res 7 and 8; it constr cts SCDASs which are 128larger than those generated by G_CMA on average. HoweThe performance of DAST degrades as the percentage nidirectional lin s increases. The size of the SCDAS b i by DAST is 153%, 175%, 189%, and 191% l ger than b ilt by G_CMA when the minim m transmission range is 430, 20, and 10 respectively. This shows that DAST is msensitive to the presence of nidirectional lin s comp ed toG_CMA, PInO UD1, and PInO UD4.

The last interesting observation is abo t the relationship b tween the locality of the connectivity testk in o r algorithmand the PUL. As shown in the g res, as the PUL decreaseso does the improvement in the SCDAS size as a res lt increasingk. The reason is that detecting strong connectiviis more dif c lt in smaller neighborhoods (e.g.k 1,2whena large percentage of lin s are nidirectional. In other word



the lower the PUL, the smaller the neighborhood req ired to detect st ong connectivity in the graph.

8 ·..wV< 7QUV 6--, .

"-- . Out

Out_

" 5 0 0 0b

! 3Q 2C

50 1 2 25 3N be of odes

Fig.7 Percentage of unidirectional links % [rm n rmax] [10,50]

87 , wV< . ' <.Q 6uV

O _U)IO _U

' "0 -

03e2C

2 2 3N be of odes

Fig.8. Percentage of unidirectional links % [rm n rmax] [30,50]

5 0

4 5

V <QUV 3 5

3"0 252b5

•. l ',

0

• . .. .

·" ·Wu

PlnO _U-»PlnO _U

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5 2 25 3N be of odes

Fig.9 Percentage of unidirectional links % [rm n rmax] [50-50]

O r sim lations show that when PUL is aro nd 52%(transmission range=[10,50]), there is an average red ctionof 21 % in the size of the SCDAS ask is increased from 1 to4. However, this gain is red ced to 17%, 13%, and 9% when

9

PUL is 40%, 25% and 12% respectively. As seen in Fig 9, when there e no nidirectional lin s in the networ , and the networ is extremely dense, this gain is only aro nd 3In s mmary, sing a larger neighborhood in the connectivi test is helpf l when PUL is relatively high. For lower PU sing a larger neighborhood helps only when the networ not very dense.

V. ONCLUSIONSIn this paper, we st died the problem of virt al bac

bone formation in wireless ad hoc networ s with asymmetlin s by const cting a Strongly Connected Dominating an Absorbent Set (SCDAS) of the nderlying directed grapWe proposed an e cient dis ib ted pr ning-based algorithmfor the constr ction of the SCDAS. Extensive sim lationwere cond cted to st dy the impact of vario s node and lin densities on the performance of the proposed algorithm. T res lts show that o r algorithm consistently o tperforms thother algorithms in the literat re in terms of the size of t constr cted sets. O r algorithm also provides the exibility t

adj st the tradeo between the size of the generated set an the cost of its constr ction; i.e. time complexity and messoverhead.

EFERENCESK. Alzoubi, P.-J. Wan, and O. Frieder. New dist buted algorithm for connected dominating set in wireless ad hoc networks.In Proceedings of the 35th Hawaii Inte ational Co erence onSystem Sciences pages 849 8 Jan.S Basagni, M. Mastrogiovanni, and C. Petrioli. A performancecomparison of protocols for clustering and backbone formationin large scale ad hoc network. In Proceedings of The 1st IEEE Inte ational Conference on Mobile Ad Hoc and Sensor Systems, MASS 2 pages 9 October 4S Butenko, X Cheng, C. A. Oliviera, and P. Pardalos. A new

heuristic for the minimum connected dominating set problemon ad hoc wireless networks. In Recent Developments inCoope tive control and optimization pages 6 4

4 B. N. Clark, C. 1 Colbou , and D S Johnson. Unit disk graphs. Discrete Math. 86( 6 99S Guha and S Khuller. Approximation algorithms for connected dominating sets. Algorithmica (4 4 8 998

6 B. Han. Zone-based virtual backbone formation in wireless adhoc networks. Ad Hoc Netw. ( 8 9H. Kassaei, M. Mehrandish, L. Narayanan, and 1 Opat y.E cient algorithms for connected dominating sets in ad hocnetworks. In Proceedings of the IEEE Wireless Communicationsand Networking Conference (WCNC)

8 M. A. Park, J. Willson, C. Wang, M. Thai, Wu, andA. Farago. A dominating and absorbent set in a wireless adhoc network with different transmission ranges. In Proceedingsof the 8th ACM international symposium on Mobile ad hoc networking and computing (MobiHoc ' 7) pages

9 1 Wu. Extended dominating-set-based routing in ad hoc wireless networks with unidirectional links. IEEE T ns. Pa llelDistrib. Syst. (9 86 88 J. Wu and H. Li. On calculating connected dominating set for e cient routing in ad hoc wireless networks. In DIAL ' :Proceedings of the 3rd inte ational workshop on Discrete algo-rithms and methods for mobile computing and communications pages 4 999

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