sensor networks topology-control algorithm for heterogeneous wireless An adaptive, energy-aware and distributed fault-tolerant Ad Hoc Networks

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ContentslistsavailableatScienceDirect

Ad Hoc Networks

journalhomepage:www.elsevier.com/locate/adhoc

An adaptive, energy-aware and distributed fault-tolerant topology-control algorithm for heterogeneous wireless sensor networks

Fatih Deniz

^a^,^∗

, Hakki Bagci

^a

, Ibrahim Korpeoglu

^b

, Adnan Yazıcı

^a

a Department of Computer Engineering, Middle East Technical University, Ankara, Turkey

b Department of Computer Engineering, Bilkent University, Ankara, Turkey

a rt i c l e i n f o

Article history:

Received 2 December 2015 Revised 20 February 2016 Accepted 22 February 2016 Available online 3 March 2016 Keywords:

Topology control Fault-tolerance Energy eﬃciency Prolonged network lifetime k -connectivity

Heterogeneous wireless sensor networks

a b s t ra c t

Thispaperintroducesanadaptive,energy-awareand distributedfault-toleranttopology- control algorithm, namelythe AdaptiveDisjoint PathVector (ADPV) algorithm, forhet- erogeneouswirelesssensornetworks.Inthisheterogeneousmodel,wehaveresource-rich supernodes aswell as ordinarysensor nodes thataresupposedto beconnectedto the supernodes.UnlikethestaticalternativeDisjoint PathVector(DPV)algorithm,the focus ofADPVistosecuresupernode connectivityinthepresenceofnodefailures, andADPV achieves thisgoalby dynamicallyadjustingthesensor nodes’transmission powers.The ADPV algorithminvolvestwo phases:a single initializationphase, which occursat the beginning,andrestorationphases,whichareinvokedeachtimethenetwork’ssupernode connectivityisbroken.Restorationphasesutilizealternativeroutesthatarecomputedat theinitializationphasebythehelpofanoveloptimizationbasedonthewell-knownset- packingproblem.Throughextensivesimulations,wedemonstratethatADPVissuperiorin preservingsupernode connectivity.Inparticular,ADPVachieves thisgoaluptoafailure of95%ofthesensor nodes;whiletheperformance ofDPVislimitedto 5%.Inturn,by ouradaptive algorithm,weobtainatwo-foldincreaseinsupernode-connected lifetimes comparedtoDPValgorithm.

1. Introduction

Wireless sensor networks (WSNs) are typically com- posedof largenumbers oftiny sensor nodesthat areca- pable of sensing, processing and transmitting data over wireless channels. Such networkscan be used innumer- ous fields, such as battlefield surveillance [1–3], environ- mental monitoring [4–6] and traffic control [7–9]. Sen- sor nodes collaborate in a distributed, autonomous and

∗ Corresponding author. Tel.: +90 5531866846.

E-mail addresses: fatih.deniz@tcmb.gov.tr , fatihdeniz@gmail.com (F. Deniz), hakkibagci@gmail.com (H. Bagci), korpe@cs.bilkent.edu.tr (I. Korpeoglu),yazici@ceng.metu.edu.tr (A. Yazıcı).

self-organizedmanner to accomplish a certain task,usu- allyinanenvironmentwithnoinfrastructure.

Sensornodes inWSNsshould be low-cost and should have small form-factor. This restricts sensor nodes in many ways as they have limited energy, short transmissionrange, relativelyslow CPU andsmallmemory. These limitations bring out many challenges unique to WSNs, suchasvery low powerconsumption.Since sensornodes are battery powered and these batteries are usually not rechargeable,comingupwithsolutionsthatreduceenergy consumption and prolong network lifetime are very im- portant. Numerous studies address this problem [10–13]

in literature. According to Li andMohapatra [14], 90% of asensornetwork’senergyisstillavailable afterﬁrstnode

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dies.Despitethissubstantialamountofremaining energy, theexistence ofhighly-loadedandbottlenecknodescause earlynetworkdemise.Therearenumerousstudiesthatad- dressbalancing energyconsumption amongnodesto en- sure that all nodes will run out of energy at about the sametime[15].Low-energyAdaptiveClusteringHierarchy (LEACH) [16] is a well-known early study that uses dy- namictransmissionrangesto betterbalance theloadand prolongnetworklifetime.Therearealsosomerecentstud- ieswhichre-establishlostconnectivitybyadjustingtrans- mission ranges. CoRAD [17] and RESP [32] can be listed assome of those studies. With the recent developments inthehardwareofWSNs,dynamictransmissionrangeas- signmenthasbecomeevenmoreeffective[18].

Fault-tolerance is another critical issue in WSNs. Due totheerror-pronenatureofwirelesscommunication,links mayfail,packetscan getcorrupted orcongestionmayoc- cur [19,20]. There are also factors that cause long-term faultsinsensornodes,suchasenergydepletion,hardware failure, link breaks, malicious attacks. Multi-hop communication multiplies the chances of faulty incidents for a packet stream traveling from a source to a destination.

Therefore,fault-tolerancemethods,includingfault-tolerant topologycontrol,areessentialforimprovingWSNreliabil- ityaswellasnetworklifetime.

As stated by Liu et al. [21], most existing works on fault-toleranttopology-controlaimtoobtaink-vertexcon- nectivitybetweenanytwosensornodes,wherethetopol- ogyisguaranteedtoremainconnecteduntilthefailureof thekthsensornode.

In this study, the focus is on two-tiered heterogeneous WSNs, where the network consists of two differ- ent typesof nodes: resource-rich supernodesand simple sensornodeswithlimitedbattery power.In thisnetwork model,sensor nodes are connectedto the set of supern- odesvia multi-hop paths. Toreﬂect thisasymmetry,[22]

proposesk-vertexsupernodeconnectivity,whereeachsen- sor is connected to at least one supernode by k vertexdisjointpaths.Insuchtopologies,thesensornodesremain connectedtothesupernodesaslongasatmostk− 1sen- sornodesfail.

Moststudiesonfault-toleranceproposestaticsolutions, that is, they do not adapt the topology to the changing network conditions. Bagci et al. [23] propose a static al- gorithmcalledtheDisjoint PathVector(DPV) tooptimize total transmissionpower fora givenk-vertex supernode- connectednetwork.Thatstudydoesnot considerresidual batteryenergyanddisregardsthe unbalancedloaddistri- bution on sensor nodes. As a result, k-vertex supernode connectivity is achieved but may not be preserved for a suﬃcientamountoftime.

In this study, we propose a novel adaptive and dis- tributedtopology-controlalgorithm,AdaptiveDisjointPath Vector (ADPV), which eﬃciently constructs a k-vertex supernode-connected network topology and adapts the topologytonodefailures,whichinturnincreasesnetwork lifetime. The contribution is two-fold. First, the residual batterypowerlevelsofindividualsensornodesareconsid- eredto prolong k-vertex supernode connectivity. Second, anadaptivesolutionisproposedtorestore,ifnecessary,k- vertexsupernodeconnectivityafteranodefailure.

The remainder of the paper is organized as follows:

Section 2 gives some background information and dis- cusses the related studies. In Section 3, we present our proposed adaptive topology-control solution. The results forsimulationexperimentsare presentedinSection4.Fi- nally,Section5concludesthepaper.

2. Relatedwork

Inthissection,wegiveabriefoverviewofsomeofthe prominentrecentworkaddressingfault-tolerance,connec- tivityrestorationandheterogeneityinWSNs.Wealsogive abriefoverviewoftheDPValgorithm[23].

Fault-tolerancetechniquescanbe categorizedintofour [24]:prevention,detection,isolation,andrecovery.Preven- tion attains network connectivity and establishes redun- dant links/nodeswhen necessary.Detection monitorstraf- ﬁcandsendsalertswhenanyindicationoffaulthappens, such as a decrease in packet delivery rate, which would imply a packet loss,interruption, ordelay. Isolation diag- nosesandidentiﬁesthealert.Asforrecovery,afterdetect- ingandidentifyingthefault,thesystemshouldbeableto recoverineitheracentralizedordistributedmanner.Note that dueto thenature ofWSNsit isessential forthe re- coveryschemetobeadistributedmethod.

The replication and redundancy of components prone tofailureisthemostcommonlyusedmethodforfaultpre- ventionandrecovery[21].Forinstance,ifsomenodeshave problemsandfailtosensetheenvironment,theredundant nodesinthevicinitycanstillprovidedata.Keepingredun- dant links ormultiple paths also provides fault-tolerance when some communicationlinksarebroken dueto node failuresorcommunicationerrors.

2.1. ConnectivityrestorationinWSNs

There are three approachesto connectivity restoration in WSNs: mobile node relocation, relay node placement andtopology-controlviatransmissionrangeadjustment.In theﬁrstapproach,asthenodesaremobile,themainidea istorepositiontheexistingalivenodestorestoreconnec- tivity.OneexampleofthismethodisPADRA,developedby Akkayaetal.[25].Inthisapproach,eachnodechoosesone ofitsneighborstobe thefailurehandler,whichwillstart recovery ifthenodedies.Therestorationprocessonlyoc- cursiftheentirenetworkgetsdisconnected,inwhichcase theclosestnodethatcantakethedeadnode’splaceisre- locatedtothatposition.

Intherelaynode placementapproach[26–31],theob- jectiveistoplaceaminimumnumberofrelaynodesina regionwheresensornodesarerandomlydeployedsothat theresultingnetworktopologyisfault-tolerant.

These ﬁrst two approaches, that is, mobile node relocation andrelaynode placement,maynot be practical in real-world scenarios because sensor nodes are often de- ployedinremote andinhospitable regionswithharshen- vironments that render manual node placement or relocation infeasible.Note that duetothe dynamicnature of WSNs,nodeplacementand/orrelocationmustberepeated periodically. In addition, these approachesrequire overall

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Fig. 1. Initial network.

Fig. 2. Optimized network.

network information,something that is also not suitable formostreal-worldapplications.

As a remedy to the problems discussed above, topology-control emerges asathird approachforconnec- tivity restoration. In this approach, the topology is con- trolledbyadjustingthesensornodes’transmissionranges.

One example of this method is RESP [32], which is an energy-aware topology-control algorithm that ensures k- edge connectivity for ﬂat networks. The RESP algorithm assumes sensor nodes are aware of their location information via GPS or other localization techniques, and periodicallyupdatesthenetworktopologytoadapttosen- sor nodes’ residual battery power levels. Because of en- suring k-edge connectivity, but not k-vertex connectivity, RESPcannot keep the networkconnected up tok− 1 node failures.Anotherrecentapproach,Energy-harvesting Heterogeneous WSN (EHWSN) [33], also aims to pre- serve k-vertex supernode-connectivity for heterogeneous WSNs.EHWSNisacentralizedapproachandignoresresid- ual battery power levels, therefore not scalable and not energy-aware.

2.2. DPValgorithm

The aimofthe DPValgorithm [23] istominimize the total transmission power of a WSN while maintaining k- vertexdisjointpaths fromeach sensornode to theset of supernodes.TheDPValgorithmgetsak-vertexsupernode- connected network topology asan input andgenerates a subnetworkconsistingofthesamesetofsensorsbutfewer connections. The output of the DPV algorithm is a to- taltransmissionpoweroptimizedandk-vertexsupernode- connectednetwork topology.Considertheexample topology giveninFig. 1,whichconsists ofone supernode and threesensornodes.Whentheaimistoprovideone-vertex supernodeconnectivity,DPVremovesthreeedgesandop- timizesthegivennetworktopology,asinFig.2.Themain contributionofDPVisitsefficiencyincomputingsuchnet- worktopologies.TheDPValgorithmrequiresO(n²⁾^message transmissions, whereas the best alternative [22] incurs O(⁵⁾ ^messages, ^where ⁿ îs ^the ^number ôf ^sensor nodes and ^refers ^to ^the ^maximum ^degree ôf â ^sensor

node.Note thatwe assume adense network,where îs sufficientlylarge.TheDPV algorithmconsistsoffivemain stages:

1.Collecting path information and calculating disjoint paths,

2.Calculatingthesetofrequiredneighbors,

3.Notifying thenodesinthe disjointpaths andupdating therequiredneighbors,

4.Removingthenon-requiredneighborsand

5.Reducing the powerlevel to apoint suﬃcient only to reachthefarthestrequiredneighbor.

2.3.Powerconsumptionmodel

OurADPV algorithm aims at prolonging network lifetime, andthus it should ﬁrst model the amount oftime untilthebatterypowersofthesensornodesaredepleted.

The ADPValgorithm uses a well-known powerconsumption model, proposed by Heinzelman et al. [34,35]. This approachisbasedontheobservationthatthemainfactor inWSNpowerconsumptionisdatacommunication,which consistsoftwo factors:datatransmissionanddatarecep- tion.In thismodel,the power totransmit a bitto a dis- tanceofdis

Pt

(

^d

)

=

α

¹⁺

α

²^{× d}ⁿ^, ⁽¹⁾

where

α

1and

α

2areparametersthatdependonthetrans- mitter circuitry, and n is the path loss exponent for the environment,whichoftenhasavaluebetween2and4.In ourpowerconsumptionmodel,

α

1,

α

2,andnareassumed tobe50nJ/bit,100pJ/bit/m²and2,respectively.

Inour model,theenergyconsumption fordata recep- tionisaconstantvalueperbit.Werepresentthisconstant with

β

ândâssumeîtêquals⁵⁰^nJ/bit.

Forour experiments, we assume all sensor nodes are sensingthe environment andgeneratingtraﬃc at aﬁxed rate.Wealsoassumethat dataaggregationisappliedand thatallnodesonapathcarrythesameload.Therefore,to- talpowerconsumptionforreceivingabitandtransferring ittothenexthopequals:

Pf

(

^d

)

=

β

+

α

1+

α

2× dⁿ. ⁽²⁾

Iftheresidualbatteryenergylevelofsensornodei isde- notedase_i,thenthelifetimeofnodeiequals:

li=ei/

((

^rri×

β )

+

(

^rri+ rgi

)

×

( α

1+

α

2× d_iⁿ

))

, ⁽³⁾ wherer_riistheincomingdataratetonodei,r_giisthedata rategeneratedinnodeiandd_iisthetransmissionrange.

3. Adaptivedisjointpathvectoralgorithm

Inthissection, wepresentournoveladaptiveanddis- tributed algorithm, ADPV, which aims to construct and maintainak-vertexsupernode-connectedtopologytopro- longthek-vertexsupernode-connectedlifetimeofthenet- work.TheADPValgorithmcontrolsthetopologybyadjust- ingthetransmissionrangesofsensornodes,andtocomply withreal-lifesituations itconsiders node failures.Theal- gorithm requires only one-hop neighborhood information andconstructs the network topology by a series ofmes- sageexchanges.

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Fig. 3. Sample scenario.

TheADPV algorithmconsists oftwo phases:initializa- tionandrestoration.Itcollects necessaryinformationand builds an initial topology during the initialization phase.

Whenevera node failure breaks k-vertex supernode connectivity, ADPV restores connectivity within the restorationphase.SimilartoDPV,ADPVutilizesdisjointpathsand within each restoration phase each sensor node decides whetherornotto changeitsdisjointpaths.Attheendof eachrestoration phase, sensornodes’transmission ranges areadjustedaccordingtotheintendedtopology.Themain differencesbetweenADPVandDPVareasfollows:

• ADPVisanadaptiveapproach,adaptingtonodefailures and remaining energy levels, whereas DPV is a static one.

• ADPVconsidersresidualbatterypowerlevelsofsensor nodes,thereforeitisanenergy-awaresolution.DPVon theotherhandignoressensornodes’remainingenergy levels.

• ADPV balances energyconsumptionandoptimizes the lifetimeofdisjointpaths,asopposedtoDPV,whichop- timizesthetotaltransmissionpowerofsensornodes.

• ADPV signiﬁcantly prolongs both one-vertex and k- vertex supernode-connected lifetimes of the network with its solutions for restoration path selection, k- vertexsupernode-connectivityveriﬁcationandconnec- tivityrestoration.

3.1. Networkmodel

Consider a mission critical border surveillance system thatisintegratedwithatwo-tieredheterogeneouswireless sensornetwork.Inthisnetwork,thereare supernodeslo- catedoneachtowerandregularsensornodesthatareuni- formlydistributedintothe targetarea asshowninFig.3. In thisnetwork, sensor nodes are responsible for detect- ingpotentialintrusionactivities andinformthetowersby forwardingdatatothesupernodeslocatedatthosetowers.

Sinceitiscommontolosesome sensornodesbecauseof energydepletion, harshenvironmental conditionsorhos- tile activities of intruders, it is desired for every sensor nodetohavemorethanacertainnumberofindependent

paths to the supernodes. Inthe ﬁgure, we cansee a sol- dier crossing the border, and a sensor node close-by in- formssometowersviathreedisjointpaths.

Thisnetwork modelisﬁrstdescribedin[22],andalso used by the DPV algorithm. In this model, the network consistsof Msupernodesthat are deployedatknownlo- cations andNsensornodesthat arerandomly distributed in the 2D plane so that M < <N. We assume the supernodes have transmission ranges long enough to communicate with the base station or any other supernode in the network. Therefore, we do not model andare not concerned with supernode-to-supernode communication.

We areonly interestedinsensor-to-sensorandsensor-to- supernodecommunication.

We represent the initial network topology with an undirected weightedgraphG=(^V,E),where Vis the set of nodesand E=

{ v

_i_,

v

_j | dist(

v

_i_,

v

_j₎_<R_max

}

^is ^the^set ^of

edges; dist(

v

i,

v

j) ^deﬁnes ^the ^distance ^between ^nodes

v

i

and

v

j.

3.2. Problemdeﬁnition

We ﬁrstgive the formal deﬁnitionofk-vertex supernodeconnectivity.

Deﬁnition 1 (k-vertex supernode connectivity [22]). An heterogeneous WSN is said to be k-vertex supernode- connected ifremoval ofany k− 1 sensornodes doesnot disconnect any sensor node from all the supernode(s), that is, each sensor node is still connected to some supernode(s).

Initially we are given a k-vertex supernode-connected network with M supernodes and N sensor nodes, where thesensornodetransmissionrangecanbeadjustedupto apredeﬁnedconstantRmax.Aswemodelnodefailures,the number ofactive sensor nodes decreases duringthe net- worklifetime.WeuseN_ttodenotethesetofactivesensor nodesattimet,wheretimeisrepresentedbydiscretetime intervals. Our problem is to determine the transmission rangesofallactivesensornodesatanytime,suchthatthe resultingtopologyisstillk-vertexsupernode-connected,so that networklifetimecan beimproved.Now,we formally statetheproblemofmaximizingfault-tolerantlifetime.

Definition 2 (Fault-tolerantlifetime maximization). Given a k-vertex supernode-connected WSN G=(^V,E) ^with â set M⊂ Vof supernodevertices andaset Nt⊂ Vofactive sensor node vertices, such that M∩Nt=∅, find a set of edges F⊂ E such that G(^V,E− F) îs ^k^-vertex ^supernode- connected and_|N_t|

i=1l_i is maximized, where l_i is the lifetime of the minimum lifetime path among the disjoint pathsof

v

∈Nt.

3.3. Residualbatterypowerlevel-awaredisjointpath selection

The ADPV algorithm adapts the network topology dy- namicallyduringnetworkoperationby adjustingthe sensor nodes’ transmission ranges according to residual energy levels. For instance, if a node has low remaining energy, it should choose closer neighbors; otherwise, it

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maychoosefartherones. Inthisway,weattainafairdis- tributionoftotalresidualenergyamongsensornodes.

TheDPValgorithmisnotanenergy-awaresolution,and ignores sensor nodes’ residual energy levels. This design maycauseearly batterydepletion,since anode withlow residual energy may be assigned to a high transmission powerrange.TheADPValgorithm,ontheotherhand,takes residual energy levels into consideration when selecting disjointpaths.Estimatingthelifetimeofeachsensornode onapathliesatthecoreofourapproach.Themotivation behindthismethodisthatachainisonlyasstrongasits weakestlink, andthus apath survivesonlyaslongasall nodessurviveinthepath.Therefore,theshortestnodelife- time onthepathdeterminesthelifetimeofthepath.The ADPV algorithm chooses aset ofdisjoint paths such that theminimumlifetimeofthosepathsismaximized.

Weformallydeﬁnethelifetimeofapathasfollows:Let apathP consistsofnodesn₀,n₁,..,n_l,inwhichn₀ isthe startingsensor node andn_l is asupernode. Letb_idenote the residualenergylevelofsensor node n_i andd_idenote thedistancebetweenn_iandn_i₊₁foreach0≤ i<l.Then, thelifetimeofPisdeﬁnedas:

Lifetime

(

^P

)

=min

0≤i<l

{

^bi/

( β

+

α

1+

α

2× d_iⁿ

) }

, ⁽⁴⁾ where

β

^,

α

1,

α

2 and n are the constant parameters of powerconsumption,deﬁnedinSection2.3.

3.4. Initializationphase

Thissectiondescribesourproposedapproachforselect- ing alternativeroutesintheinitializationphaseofthe algorithm,wherethoseroutesaretobeusedtorestorecon- nectivity duringrestoration phases. InADPV, each sensor node keepsalternativeroutes, herereferred toasrestora- tionpaths,thatstartwiththatnode.

Theprimary goalistoconsumetheminimumpossible resources while attaining high-quality restoration paths.

Theresourcesincludememory,CPU,andnetwork.Regard- ingmemory,forinstance,ifallpossiblepathsfromsensor nodestosupernodeswere held,thememory requirement wouldbeintractable.In[36],Valiantdiscussestheaverage numberof paths froma node toa givenset ofnodes.In terms of CPU,Bagci et al.[23] show that the complexity ofselectingkdisjointpathsfromapoolofpalternativesis O(p^k).Therefore,withahighernumberofrestorationpaths ofsizer,ittakeslongerto computeadisjointpath setof size k duringeach restoration phase. As forthe network, which is last butnot least, we aim to communicate using minimumnumber ofmessages. Eachrestoration path incurs communicationbetween its nodes inorder to up- date its lifetime. As a result, we should maintain a very restrictedsetofrestoration pathsforthe sakeofnetwork performance, but atthe same time, the amount ofthose pathsshouldbehighenoughtorestoreconnectivitywhen- everneeded.

Toovercometheserestrictions andeﬃcientlyconstruct restoration paths, ADPV employs a well-known method, called maximum set packing (MSP) [37]. This method is the optimization version of theset packing (SP) problem and asks for the maximum number of pairwise disjoint

setsamongafamilyofsets.Moreformally,foragivenuni- verseU andafamilyS ofsubsetsofU,MSPisasubfamily C⊆ S ofsets such that all setsinC arepairwise disjoint, andC uses asmany sets aspossible, so that the size of thepacking

C

^is^maximum.^Maximum^set^packing^is^NP-

hard[38]andcannotbeapproximatedwithinanyconstant factor[39].

Algorithm3.1 MaximumSetPacking(MSP).

Input: S Output: M

1: M←∅;

2: whileS=∅ do

3: m←MinIntersectingPath(S);

4: M←M

m;

5: for allPathp ∈S do

6: if p

m=∅ then

7: S←S− p;

8: endif

9: end for

10: endwhile

There is a well-known greedy heuristic, shown in Algorithm 3.1, to solve the MSP problem and it runs in polynomial time. We employ this heuristic to construct restorationpaths.Atthebeginning,wehaveapoolofcan- didate paths of a relatively large size. The heuristic per- forms with many iterations, where each iteration selects themostdiversepathfromthepool.Weusethetermdi- verseasbeingdisjointwithothers,thatis,theone thatis disjoint to the largest number of paths among others in the pool. We add the selected path into the restoration pathset andremove all thepaths fromthe poolthat in- tersectwiththeselectedpath.Theiterationscontinueun- tilthepool becomesempty orthe numberof restoration pathsreachesapredeﬁnedthreshold.Sincetheinitialsen- sornodeandthedestinationsupernodedonotviolatedis- joincy,ADPV representseach pathby the setof itsinter- mediatesensornodes.

Algorithm3.2 PathInformationCollectioninADPV.

Input: I,L,k Output: D,R

1: T←∅;

2: R←∅;

3: forallreceivedPathInfomessageI do

4: D←MinDisSet(T,k);

5: c←Cost(D);

6: U←I.T∪T;

7: R←MaxSetPacking(R∪U); (Algorithm3.1)

8: U←MaxSetPacking(U);

9: Sort(U);

10: T←

{

^pⁱ^∈^U

|

ⁱ^<⁼^L

}

^;

11: D←MinDisSet(T,k);

12: c←Cost(D);

13: if c<cthen

14: T←T;

15: TransmitPathInfo(T);

16: end if

17: endfor

Algorithm 3.2 shows path-information-collection and restoration-path-selection procedures. The variables

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Table 1 ADPV notations.

I Received pathInfo message

L Maximum number of paths to be stored k Disjoint connectivity degree

R Set of restoration paths T and T Set of local paths D and D Set of disjoint paths

c and c Cost of disjoint paths, which equals the minimum lifetime of the disjoint paths U Union of two path sets

S Set of paths

M Set of paths in MSP m , p , r Variables referencing paths sr Supernode ratio

n Total number of sensors

Maximum degree of a node r Amount of a node’s restoration paths l Average path length in the restoration set n 0 , n ¹ , .., n i Number of remaining sensor nodes after each

restoration phase

used in the pseudo codes are deﬁned in Table 1. In Algorithm3.2,eachsensornodemaintainsalocalpathset alongwithdisjointandrestoration pathsets.Asan input, the algorithm takes a ‘PathInfo’ message that contains the localpath set of the sender node and generates two outputs, whichare the disjointpath andrestoration path sets of size k and a relatively large size, respectively.

Local paths are logical paths that are used for informing neighbor nodes about the paths they can use over the sender node. Therefore, local paths have a very critical roleindetermining disjointandrestorationpath setsand need to be selected very carefully. When a sensor node receives a ‘PathInfo’ message containing a local path set, itﬁrst calculates the unionof the sender’s andreceiver’s localpathsets.ItthenexecutestheMSPprocedureonthis uniontoeliminatepathsthathavetoomanysensornodes in common. The procedure then determines a candidate localpath set T asthe ﬁrst Lminimum-cost pathof the remainingset.Whiledoingso,theprocedurealsoupdates therestoration paths by executing theMSP procedure on the union of local sets and the current restoration path set.

Using the candidate local path set, the set of disjoint paths with minimum cost is calculated using Algorithm3.3. In thisalgorithm, all disjoint subsetswith kelementsare traversed andthe onewiththe minimum costisselected.Iftheminimum-costdisjointpathsethas asmallercostthanthecurrentdisjointpathset,boththe disjointand the local paths are updated and a ‘PathInfo’

messagecontainingthe newlocalpath setis transmitted tothesetofneighbors.Thisprocess continuesuntilthere arenomoreupdatesinthedisjointpathsets.

Afterdetermining thedisjointpaths,each sensornode determines its required neighbors, which include the neighborsthatdisjointpathsusetheedgesbetween.After determiningtherequiredneighbors,eachnode adjustsits transmissionpowerto reach itsfarthest neighboraccord- ingtotheresultingtopology.

Algorithm 3.3 Finding Disjoint Paths to Supernodes (MinDisSet).

Input: T andk Output: D

1: D←∅;

2: if

|

^T

|

>kthen

3: Q←

{

^q^{⊂ T}

| |

^q

|

=k

}

^;

4: c←∞;

5: qmin←∅;

6: forallq∈Q do

7: if qconsistsof disjointpathsthen

8: if Cost(q)<cthen

9: c←Cost(q);

10: qmin←q;

11: end if

12: end if

13: endfor

14: D←qmin;

15: end if

3.5. Connectivityrestorationphase

We start the connectivity restoration procedure only when k-vertex supernode connectivity is broken due to node failure. Thus, the ﬁrst step after a node failure is tocheck whetherthenetworkisstillk-vertexsupernode- connectedornot.As thisisacostly operation[40],ADPV employsasimpledistributedgreedyheuristicwithnofalse positives. That is, if ADPV postulates the network is k- vertexsupernode-connected,thenthenetworkisdeﬁnitely connected. However, the network can still be connected even if ADPV claims it is not. Therefore, ADPV ensures strongk-vertexsupernodeconnectivity.

When a node failure occurs, ADPV ensures all the node’sneighborsinitiateafailuremessagetoinformothers aboutthefailure. Uponreceivingafailuremessage,asen- sornoderemovesallpathsincludingthefailednodefrom its restoration set. Sincefrequent transmission powerad- justmentisdiﬃculttorealizeinpractice,weemployperi- odicaltransmissionpowercontrol,andduringeach period wecheckwhetheranyfailednodesexistonanyofthedis- jointpaths.Ifafailednodedisconnectsadisjointpath,the restoration process takes place.Notethat this eventdoes not necessarily imply k-vertex supernode disconnectivity, yet because ADPV takes early action it never allows the connectivity to break. After deciding k-vertex supernode connectivity must be restored, ADPV applies a two-step process:updatingthelifetimesoftherestorationpathsand computingminimum-costdisjointpathsfromtherestora- tionset.

In the ﬁrst step, path lifetimes in the restoration set areupdatedviamessagestransmittedalongthepathfrom the source node tothe destination supernode. Eachnode redirects a received messageto the next hopin thepath and returnsa message that containsupdated lifetime in- formationofthesensornodesbackalongthatpath.Inthe second step, minimum-cost disjoint paths are computed using the previously discussed disjoint-path-selection algorithm, Algorithm3.3. An overview of the connectivity- checkingandconnectivity-restorationproceduresaregiven inAlgorithm3.4.

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x y

A z Supernode Sensornode

Residual Energy: 100

Consumpon: 1.2 Residual Energy: 100 Consumpon: 2

Residual Energy: 100 Consumpon: 1

x x->A x->y->A

Disjoint Paths y y->A y->x->A

z z->y->A z->x->A

a

x

A z

Residual Energy: 40 Consumpon: 1.2

x x->A x->z->A

Disjoint Paths

z z->A z->x->A

b

x

A

x x->A Disjoint Paths

c

Fig. 4. Sample connectivity restoration for k = 2 .

Algorithm3.4 ConnectivityRestorationinADPV.

Input: k, R,D Output: D

1: FailedNodes←∅;

2: forall receivednodefailuremessage

δ

^do

3: forallPathr ∈R do

4: if r contains

δ

.FailedNodethen

5: R←R− r;

6: end if

7: endfor

8: FailedNodes←FailedNodes∪

δ

.FailedNode

9: ifcertaintimeelapsed sincelastperiod then

10: for allPathp ∈D do

11: if

(

^p∩FailedNodes

)

=∅then

12: UpdateCosts(R);

13: D←MinDisSet(R,k);

14: break;

15: endif

16: end for

17: FailedNodes←∅;

18: endif

19: end for

For instance, continuing from the example given in Section2.2,fork=2,ADPVoptimizesthetopologyshown inFig.1,asinFig.4(a).In thistopology,allinitial energy levelsareequal.Assumingthedatagenerationrateisuni- formfor all nodes,thepower consumption ofnodes x, y andzare 1.2,2and1,respectively. Withthispower consumption, node ydies ﬁrst(100/2=50 slater), both node xandnodezloseoneoftheirdisjointpathsandthenet- workbecomesone-vertex,butnot two-vertex,supernode- connected.TheADPValgorithmrestoresconnectivity,asin Fig. 4(b),by adjustingthe transmissionrange ofz,which introduces alinkfromnodez tosupernodeA.Because of its increasing power consumption, node z happens to be thesecond dyingnode (50/5=10 slater)andthusnode x losestwo-vertexsupernodeconnectivityoncemore.How- ever, because it has no alternative routes, it adjusts its transmissionpoweragainandworksasaconnectednode, as in Fig. 4(c), for the rest of its life (28/1=28 more seconds). For this network, the two-vertex supernode- connectedlifetimeisbroken whennodezdies.Therefore,

n_i

Fig. 5. One-hop neighbors of node i .

the two-vertex supernode-connected lifetimeof this net- workequals60sandtheone-vertexsupernode-connected lifetimeequals88s.

Lemma1. Theconnectivity restoration processof ADPVen- suresk-vertexsupernodeconnectivity.

Proof. Bydeﬁnition,thenetworkgetsk-vertexsupernode- connected ifeach sensor in the network is connected to atleast one supernode withk-vertex disjoint paths. This translatesintothedisjointpathsetofeachsensornodeof beingsize k, andifthere exist more thank paths in the restorationset,ADPVchoosesadisjointsetandensuresk- vertexsupernodeconnectivity.

Lemma2. Inthe restoration pathset,there areatmost paths,whereîs^the^maximum^degreeôfâ^sensor^node.

Proof. Wearegoingtoprovethisbycontradiction.Asdis- cussedinSection3.5,eachsensornodekeepsamaximum setpackofsomesizeintheirrestorationsets,sothateach pathinthesetispairwisedisjointwiththeothers.Let denotethemaximumdegreeofa nodeandassume there exists a node, say node i, that has more than ^paths inits restoration set.Since there are more than ^paths that are usingat most ^neighbors,^according ^to^the ^pi- geonholeprinciple, there existtwo restoration paths that usethesameneighbor.LetFig.5representnode iandits one-hopneighbors.Iftheneighborthattwopaths havein commonisasupernode,thennodeiwillhaveexactlythe same two paths in its restoration set, which is not pos- sible,because the MSP procedure calculatesthe union of theselected pathsto guaranteediversity. Ifthatneighbor

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is a sensornode, those paths will not be disjoint, which violatestheMSPdeﬁnition.Therefore,noneighbor,neither sensornode nor supernode,can havetwo paths incom- mon, andthe number ofelements in the restoration set cannotexceed^.

3.6.Runningtimeanalysis

Wecomputethelifetimeofarestorationpathviames- sagestransmittedalongthepath fromthesourcenode to thedestinationsupernode. Eachnoderedirects areceived messageto the next hop inthe pathand returnsa message that containsthe sensors’updated lifetimeinforma- tionbackalongthesamepath.Therefore,foreachrestora- tion path,the number ofmessages equals two times the lengthofthepath.Weassumethatpathlengthisbounded by a constant, say l,following previous studies [23]. No- tice that the number of restoration paths is less than or equalto ^, ^where îs ^the ^maximum^degree ôf â ^node.

Then, there are at most l × ^messages ⁱⁿ ^total, ând thus,themessagecomplexityisO(⁾âtêachconnectivity- restorationphase.Intheworstcase,foreachsensornode, connectivity restoration is carried out for O(⁾ ^times,âs the restoration path set embodies at most l × ^nodes.

Therefore, at each sensor node, total message complexity becomes O(²⁾ ^for connectivity restoration. For path informationcollection,ADPV hasthe messagecomplexity of O(n^), ^which âlso êquals ^that ôf^DPV ^[23]^. ^Therefore, thetotal message complexitybecomes O(²⁾ ⁺Ô(ⁿ⁾ = O(n^).

TheADPValgorithmconsumescomputationalpowerin the initialization phase for disjoint and restoration path constructionandduringtheconnectivityrestorationphase for determining new disjoint paths from the restoration set.Duringtheinitializationphase,whensensornodesre- ceivea‘PathInfo’message,theycalculatetheunionofthe local path information and the received paths in the in- comingmessage.Therunningtimecomplexityofthisstep dependson thenumberofpaths (p)in thelocalpathin- formationtable. InADPV, since themaximum numberof pathsthatcan bestoredina sensornode’spathinforma- tion table isset to a constant value, both calculating the unionofthe two pathinformationtablesand sortingthe pathsaccordingtotheircoststakeconstanttime.

In the initialization phase, there are two more procedures that consumes processing power: maximum set packinganddisjoint-path-selectionalgorithms.Thegreedy heuristic forMSP,shown inAlgorithm 3.1,is usedtwice:

onceforconstructingtherestorationpathandagainforse- lectingthelocalpathinformationtable.Asdiscussedinthe secondlemma,thenumberofrestoration pathsislimited bythe maximumdegreeofnode (^), ^and^the^number^of pathsinthelocalpath informationtable isaconstant (l).

Therefore,the MSP algorithmconsists ofnumerous iterations, each consistingoftwo steps:i) selecting themini- mumintersectingpathandii)removingthepathsthatin- tersect withit. In thelatter step, thealgorithm traverses all path pairs and determines the minimum intersecting one. The activity of removing the intersecting paths also traversesthe set once more. Considering set size is rep- resented by s, the running time complexity of the MSP

algorithm equals O(s²+s). Therefore, the MSP running time complexityineach stepisO(²++l²+l), which canbereducedtoO(²^).

To calculate the minimum disjoint set, Algorithm 3.3 enumeratesallsubsetsofsizekandﬁndsthesetwiththe minimumcost. EnumeratingallthesesubsetstakesO(p^k), whereprepresentsthe numberofpathsinthe givenset.

Since the input of the minimum disjoint set procedure is the localpath information table, which hasa constant number ofelements, therunning time complexity ofthe minimumdisjointsetcalculationisalsoaconstant.

Considering that the messagetransmission complexity of ADPVis O(n⁾ ând^the^dominating^step ^(MSP) îsêxe- cutedonceforeveryincomingmessage,therunningtime complexityofthetotalinitializationphaseisO(n³^).

Fortherestorationphases,asdiscussedabove,themaximum number of restoration phases a node can execute isO(l^),ândⁱⁿêach ^phase^thereâre^two operations:up- datingpathcosts,whichonlyusesmessagetransmissions, andcalculatingtheminimumdisjointsetfromtherestora- tion set. Since the maximum number ofelements inthe restorationsetis^,^the^running^time^complexity^for^deter- miningtheminimum-costdisjointpathsfromtherestora- tion set will be O(^k^), ând ^the ^total ^running ^time ^com- plexityoftherestorationphaseswillbeO(^k⁺¹^).

Sincen>>^,ând^the^commonlyâccepted^valuesôf^k are2and3[22],theADPVrunningtimecomplexityequals O(n³^).

3.7. ExpectednumberofrestorationsinADPV

In this section we discusstheoretical expectationsre- sulting from theADPV algorithm andanalyze how many times ADPV can restore k-vertex supernode connectivity fora givennode. SinceADPV can restore suchconnectiv- itywhen thereare atleastk pathsin therestoration set, we willdetermine the expected number ofnode failures beforeanodecannotrestore itsconnectivity.Letndenote the number of sensor nodesin the network and assume the sensor node batteries deplete uniformlyin any order withthesame probability

ρ

^.^The ^parameters^used ⁱⁿ^this

sectionaregiveninTable1.

Considering that the number of sensor nodes in the restoration setequals r× l,theexpectednumberof sen- sornodesthat diebeforeoneoftheser× lsensornodes diesequals:

n

r× l. ⁽⁵⁾

Forexample, ifthereare 100 nodesin theentire network and20takepartintherestorationset,thentheex- pected number of node failures before one ofthe nodes in therestoration setfailsequals ﬁve.Whena node ona pathdies,thenthatpathwillnolongerbevalidandthere- fore will be removed from the restoration sets available.

As a result, with a node failure, the number of restoration pathswilldiminishby one.Therefore,whentheﬁrst node on a restoration set dies, r− 1 paths, which con- sist of (^r− 1)× l sensornodes, will remain. At the same time, thenumberofremainingsensornodesintheentire

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networkwillequal:

n− n

r× l. ⁽⁶⁾

Continuing from the previous example, 100− 5=95 sensor nodeswill remain in theentire network afterthe ﬁrstnodeintherestorationsetdies.Theremainingsensor nodesaftertheithrestorationpathremovalcanbegener- alizedasfollows:

n_i₊₁=n_i− n_i

(

^r− i

)

× l, (7)

whichalsoequals:

ni+1=ni×

1− 1

(

^r− i

)

× l

(8) andwhichcanalsobewrittenasaproductof:

ni+1=n×

i j=0

1− 1

(

^r− j

)

× l

. ⁽⁹⁾

Since ADPV can restore k-vertex supernode connectivity whenthereare atleastk pathsintherestorationset,the numberofsensornodeswhenk-vertexsupernodeconnectivity ofthe givennode cannot be restoredequals n_r_−k₊₁ andcanbecalculatedas:

n_r−k₊₁=n×

r−k

j=0

1− 1

(

^r− j

)

× l

. ⁽¹⁰⁾

Thenbychangingtheparametertot=r− j, n_r_−k₊₁=n×

r

t=k

1− 1 t× l

. ⁽¹¹⁾

Accordingtotheaboveformula,thenumberofsuccess- ful restorations will be proportional to the sensor node count. Also, withthe increasing average path length, the numberofremainingsensorsincreases,whichinturnde- creases the possibility of successful restorations. There- fore, choosing paths with smaller path lengths may be preferable.

4. Experimentsandresults

In thissection we report ourmeasurements regarding lifetime and other metrics for the DPV and ADPV algorithms andtry to evaluate ADPV’s success. For thiseval- uation, we implemented ADPV using an extended ver- sionof acustom simulator,which hasalsobeenused for evaluating the DPV algorithm. We added a time dimen- sionandabatterymodelintotheexistingframework and thusprovidedanenvironmentthatcouldevaluatenetwork lifetime.

4.1. Experimentalsetup

In our experiments, we assumed that sensor nodes and supernodes are uniformly and randomly deployed in an area of 600 m x 600 m and that the initial maximum transmission range Rmax of the sensor nodes is set at 100 m. We repeated our experiments for

{

¹⁰⁰,150,...,500

}

^sensor^node, ^for ^k=2,3(as theseare

Table 2

Simulation parameters.

Deployment area 600 m x 600 m

Initial transmission range of sensor nodes: R max 100 m Number of sensor nodes: N [10 0 . . . 50 0]

Number of supernodes: M 5% and 10% of N

Degree of disjoint connectivity: k 2 and 3

Packet loss rate 10%

commonly accepted k values), and for a supernode ratio (sr) of5%and10% overtheregion.Finally,we assumeda packetlossrateof10% foreach messagetransmission.As aresult,wehad9× 2× 2experimentalinstances,andon each weexecuted both algorithms 20timesandreported the averages. Oursimulation parameters are summarized inTable2.

4.2.Results

InFig.6,wecomparethenodefailuretoleranceofDPV and ADPV. For each algorithm, we measure performance in terms of the fraction of dead sensor nodes when the networkgets (i) supernodedisconnected and(ii)k-vertex supernode disconnected. If there exists a path (single or multi-hop)betweenasensornodeandanyoneofthesu- pernodes, then the sensor node is said to be connected.

Ifevery (alive)sensornode inthe networkhask disjoint paths to theset ofsupernodes, then the network iscon- sideredask-vertexsupernode-connected.Withthesemea- surements we determine the maximum number of node failures that can occur before supernode connectivity is broken. Here,we observethe moststrikingresult, andat thesametime,evidenceofthisstudy’smotivationregard- ingtheinstabilityofstaticalgorithms andeffectivenessof ADPV for keeping the network supernode-connected. As seenintheﬁgure,evenbeforethefailureof5%ofthesensornodes,thenetwork’ssupernodeconnectivityisbroken whenweemploytheDPValgorithm.Thisresultlimitsus- ingDPVasafault-tolerantalternative.Ontheotherhand, ADPVsuccessfullykeepsthenetworksupernode-connected uptofailureofabout95%ofthesensornodes.

In the figure, we observethat when the network becomes denser, ADPV keeps it supernode-connectivity for longer. This result can be attributed to ADPV becoming moreeffectiveatfindingalternative routesdueto thein- creasingnumberofsensornodes.Forinstance,inoneex- treme,when we examine the results ofa 500-node network for k=2 and sr=10%, as shown in Fig. 6(a), we see that the network is still supernode-connected up to a failure of 95% of the sensor nodes. In the other ex- treme, where the number of initially deployed sensor nodes equals 100, ADPV sustains supernode connectivity uptothefailure of20%ofthesensornodes.Looking into eachofthesub-figures,whentheinitialnumberofsensor nodesisbetween250and300,wenoticethat ADPVsuc- ceedsinkeepingsupernodeconnectivityevenaftertheac- tivesensornodesarehalved.Consideringallexperimental instances,onaverage,ADPVmaintainssupernode connec- tivityuptoafailureof52%ofsensornodesfork=2,and 55% of sensor nodes for k=3. Since the optimized network topologiesfor k=3 contain moreconnections, it is

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0,00%

10,00%

20,00%

30,00%

40,00%

50,00%

60,00%

70,00%

80,00%

90,00%

100,00%

100 150 200 250 300 350 400 450 500

Node Failure Tolerance (%)

Number of Sensor Nodes ADPV Connected

ADPV 2-Connected DPV Connected DPV 2-Connected

(a) k=2, sr=10%

0,00%

10,00%

20,00%

30,00%

40,00%

50,00%

60,00%

70,00%

100 150 200 250 300 350 400 450 500

(b) k=2, sr=5%

0,00%

10,00%

20,00%

30,00%

40,00%

50,00%

60,00%

70,00%

80,00%

90,00%

100,00%

100 150 200 250 300 350 400 450 500

(c) k=3, sr=10%

0,00%

10,00%

20,00%

30,00%

40,00%

50,00%

60,00%

70,00%

100 150 200 250 300 350 400 450 500

(d) k=3, sr=5%

Fig. 6. Percentage of failed sensor nodes when the network becomes (i) supernode disconnected and (ii) k -vertex supernode disconnected.

expectedthatthosenetworkswillhaveahighertolerance fornodefailures.Ontheotherhand,moreconnectionswill consume more battery power, which will affect network lifetime.Wethereforealsoexaminelifetimemeasurements ofthenetworksforthesamesetofscenarios.

Further, aswe can observe in Fig. 6, network topolo- giesgeneratedby theDPValgorithmbecome k-vertexsu- pernodedisconnectedafterthefailureofatmost1%ofthe sensornodes.Eventhoughtheinitialoptimizedtopologies generatedby DPVarek-vertexsupernode-connected,after the failure of a few sensor nodes, the remaining topolo- giesbecome atmost(^k− 1)^-vertex supernode-connected.

The ADPV algorithm, on the other hand, maintains two- vertex supernode connectivity up to a failure of 32% of sensornodes,andthree-vertex supernodeconnectivityup toa failureof21% ofsensornodes,onaverageamong all experimentalinstances.

InFig.7, we comparethe lifetimeresultsofthe same setofexperimentalinstanceswiththoseofFig.6.Apromi- nent aspectof ADPV isconsidering the sensor nodes’re- mainingenergylevels;asaresult,energydepletionoccurs lessfrequently. Thisfactor, when coupledwiththe adap-

tivenatureofthealgorithm,resultsinlongernetworklife- times.

As we observe in Fig. 7, in terms of one-vertex supernode-connected lifetimes, on average, ADPV results in atwo-foldincrease withrespect toDPV. Forthe same experimental instances, ADPV provides respectively 65%

and 46% longer two-vertex and three-vertex supernode- connectedlifetimesthanDPV.Asweobserveintheﬁgures, network density has almost no effect on the lifetimes of thetopologiesgeneratedbyDPV.Ontheotherhand,ADPV successfullyprolongs thenetwork lifetimealmost propor- tionallytothenetworkdensityforallk=1,2,3.

We observe inFig. 7(c)and(d) that if thenumber of sensornodesdropsbelowacertainthreshold(inourcase, 150sensornodesina600mx600mareawhensr=10%, and200sensornodeswhensr=5%),itis hardtorestore three-vertex supernode connectivity.Thisﬁnding suggests that aminimum numberofsensornodesforeveryk and sr value is necessary to restore k-vertex supernode connectivity. Figs. 6 and 7, respectively compare node failure tolerance andnetwork lifetimefordifferentvalues of sr=5%,10%andk=2,3.Accordingtotheresults,withthe

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15 20 25 30 35 40 45 50 55 60 65

100 150 200 250 300 350 400 450 500

Lifeme (s)

(a) k=2, sr=10%

15 20 25 30 35 40 45

100 150 200 250 300 350 400 450 500

Lifeme (s)

(b) k=2, sr=5%

15 20 25 30 35 40 45 50

100 150 200 250 300 350 400 450 500

Lifeme (s)

(c) k=3, sr=10%

15 17 19 21 23 25 27 29 31 33 35

100 150 200 250 300 350 400 450 500

Lifeme (s)

(d) k=3, sr=5%

Fig. 7. Lifetime comparison of the DPV and ADPV algorithms.

15 20 25 30 35 40 45 50 55 60

150 200 250 300 350 400 450 500

emitefiL(s)

(a) Lifetime comparison

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

150 200 250 300 350 400 450 500

NodeeruliaFecnareloT(%)

(b) Node failure tolerance comparison

Fig. 8. Lifetime and node failure tolerance of DPV and ADPV algorithms for k = 4 .

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0 5000 10000 15000 20000 25000 30000

100 150 200 250 300 350 400 450 500

Number of Message Transmissions

Number of Sensor Nodes

ADPV DPV

(a) k=2, sr=10%

0 5000 10000 15000 20000 25000 30000 35000 40000

100 150 200 250 300 350 400 450 500

ADPV DPV

(b) k=2, sr=5%

0 5000 10000 15000 20000 25000 30000 35000 40000 45000

100 150 200 250 300 350 400 450 500

ADPV

DPV

(c) k=3, sr=10%

0 10000 20000 30000 40000 50000 60000

100 150 200 250 300 350 400 450 500

ADPV

DPV

(d) k=3, sr=5%

Fig. 9. Number of message transmissions in DPV and ADPV algorithms.

increasing number of supernodes, lifetime also increases, however,therelationbetweentheincreaseinthenumber of supernodesand the increase in the lifetime is sublin- ear,andtherefore,weexpectthat theincreaseinthelife- time becomesinsubstantial asthenumber ofsupernodes exceedsacertain threshold.Wealsonoticethat,withthe increasing k value, more disjoint paths are required and thismakesprovidingalternativeroutesharder.InFig.8(a) and(b),we compareDPV andADPV algorithms fork=4 intermsofnetworklifetimeandnodefailuretolerance,respectively.AsseeninFig.8(a),ADPVsuccessfullyprolongs bothone-vertexandfour-vertexsupernode-connectedlife- timesofthenetwork.Also,inFig.8(b),we seethat ADPV canpreservefour-vertexsupernode-connectivityup tothe failure of 50% of the sensor nodes on dense networks, whichinturn,achievesalmost a two-foldincrease inthe four-vertexsupernode-connectedlifetime.

Anotherimportantmetricwemeasureduringouranal- ysis is the number of message transmissions. Message transmissionisanimportantmetricbecausewemustnot onlyconsiderpowerconsumptionintheresultingtopolo- giesbutalsoconsiderthepowerrequiredtogeneratethose topologies, which can be viewed as a fixed cost of ob- taining the final topologies. If this cost is high, then the power efficiencyof the resulting topology mightbecome meaningless. In Fig. 9, we compare the number of DPV andADPV messagetransmissions. Tosimulate theworst-

0 50 100 150 200 250 300 350 400

100 150 200 250 300 350 400 450 500

Number of Connecvity Restoraons

for k=3

for k=2

Fig. 10. Number of connectivity restorations.

case scenario of ADPV, we set the waiting periodin the restoration phase to zero, which means that every node failure that affectsdisjointpathswilltrigger arestoration phase for that node. According to the results, for k=2, ADPV makes atleast 2.25timesand atmostthree times the messagetransmissions than DPV does, andfork=3, ADPVmakesatleastthreetimesandatmost3.5timesthe messagetransmissionsofDPV.As seeninthesub-ﬁgures,

sensor networks topology-control algorithm for heterogeneous wireless An adaptive, energy-aware and distributed fault-tolerant Ad Hoc Networks

Ad Hoc Networks

An adaptive, energy-aware and distributed fault-tolerant topology-control algorithm for heterogeneous wireless sensor networks

Fatih Deniz

, Hakki Bagci

, Ibrahim Korpeoglu

, Adnan Yazıcı

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