Ysong Jang, Weren Sh A Backbone Formaon Algorhm n Wreless Sensor Nework Based on Pursu Algorhm YISONG JIANG, WEIREN SHI College of Auomaon Chongqng Unversy No 74 Shazhengje, Shapngba, Chongqng Chna jys398@6com, wrs@cqueducn Absrac: -In wreless sensor nework, vrual backbone formulaon s a cos effecve mehod o complee he broadcasng Mnmum conneced domnang se s an ousandng canddae of vrual backbone However, s NP-Hard o fnd a mnmum conneced domnang se n an arbrary graph In hs paper, we propose a novel backbone formaon algorhm o consruc a conneced domnang se In he proposed mehod, a domnang se and acon ses are go a frs Then, snk execues he pursu algorhm, n whch nodes are reaed as learnng auomaa and acon probably vecor s changed wh me, and chooses acons from her acon ses o consruc a conneced domnaed se Fnally, he auomaon converges o a common polcy and an approxmae soluon of he mnmum conneced domnang se s go I s also shown ha our mehod s ε- opmaly wh he changng speed of learnng parameer The smulaon resuls show ha our algorhm has a good performance n erms on he sze of backbone, he message overhead and average node degree Key-Words: -Backbone; Wreless sensor nework; conneced domnang se; pursu algorhm Inroducon A wreless sensor nework (WSN) s composed by many sensor nodes and one or mulple snks In WSN, sensors gaher daa from he sensng envronmen and ransfer hose daa o he snk nodes or base saons Generally, WSNs are deployed n some emergen or emporary suaon, for example accdens or envronmen gaherngs Broadcasng s a fundamenal means n WSNs However, he way of packe ransmsson n WSN s dfferen from he way n he wred nework Generally, here are many redundan reransmssons n broadcasng because he omndreconal rado propagaon and a physcal locaon may be covered by he ransmsson ranges of several nodes Vrual backbones are usually used o suppor broadcasng and mulcas n WSNs and a mnmum conneced domnang se (MCDS) s an ousandng canddae o work as a vrual backbone In hs paper, a graph G(V,E) s used o represen a WSN In G, he verexes se V represen he nodes n he WSN, and E represens all he lnks n he nework We also assume ha all nodes have he same ransmsson range Therefore, he graph G s also a un dsk graph A domnang se (DS) of graph G s a subse of V such ha each verex whch s no n se wll be joned o a leas one member of he se by an edge A conneced domnang se (CDS) s a DS ha nduces a conneced subgraph of G A mnmum conneced domnang se (MCDS) s a conneced domnang se wh he smalles cardnaly among all possble conneced domnang ses of G However, s NP-hard o fnd a MCDS of a graph [] We propose a quas-dsrbued algorhm o consruc a CDS of a WSN The proposed mehod consss of wo seps A he frs sep, a DS and he acon ses of he nodes n he DS are go A he second sep, accordng o he cross enropy mehod, he snk ge a CDS based on hose acon ses In deal, he DS s go by a color process Then, each node n he DS (called domnaor) ges s acon se based on he connecon pah beween self and s or 3 hops domnaor neghbors When he snk receves he acon ses from domnaors, reads hose domnaors as auomaa and execues he revsed pursu algorhm o form a CDS In he pursu algorhm, each auomaon chooses an acon from s acon se o consruc a CDS and updaes he esmae vecor and acons probably vecor based on he sze of he feedback Fnally, he auomaon converges o a common polcy ha consrucs an approxmae soluon of he MCDS of he nework E-ISSN: 4-864 53 Volume 3, 04
Ysong Jang, Weren Sh The organzaon of he res of hs paper s shown as follows Secon shows he relaed works and Secon 3 brefly presens he learnng auomaa and he pursu algorhm The purposed CDS formaon algorhm s presened n Secon 4 and he ε-opmaly of he purposed algorhm s proven n Secon 5 In addon, he performance of he proposed algorhm s shown n Secon 6 The concluson s presened n Secon 7 Relaed works Mos of MCDS approxmaon algorhms consruc a CDS based on a maxmal ndependen se (MIS) A MIS s a se ha every edge n G has a leas one endpon no n he se and every verex no n he se has a leas one neghbor n In [], Alzoub e al proposed wo MIS-based algorhms The frs algorhm requres a spannng ree o complee he process n whch a CDS s generaed The second algorhm does no need he spannng ree and enables he manenance of he weaklyconneced domnang se o be smpler Bo [3] presens a zone-based dsrbued algorhm, n whch every node s assgned Zone and Level marks o ndcae s subree and he dsance o roo of he subree L e al [4] pu forward he S-MIS algorhm, whch s a greedy algorhm and consrucs a CDS wh he help of Sener ree A he same me, a dsrbued verson of hs algorhm s also nroduced Gao e al [5] purpose anoher MIS-based dsrbued algorhm ha has a beer message complexy compared wh Alzoub s algorhm [] and s sable and scalable n large and dense nework In [6], Wu and L purpose a prune-based algorhm, n whch prunes some redundan nodes from he orgnal CDS ha go by Rule and Rule Anoher enhanced prune mehod, called Rule k, s nroduced by Da and Wu n [7] In Rule k, a node wll be removed from CDS f s neghborhoods are covered by a se of k neghbors wh hgher IDs and he node se s srongly conneced Akbar Torkesan and Meybod [8] descrbed an nellgen CDS-based backbone formaon algorhm n whch he learnng auomaa are used o consruc he CDS of he nework The auomaa wll be rewarded, f s he smalles one ha has been consruced so far Oherwse, wll be penalzed However, he learnng auomaa apply a drec algorhm ha only uses he envronmenal feedback of he curren sage o updae he probably vecor Akbar also purposes anoher backbone formaon algorhm o he energy effcen n WSNs In hs algorhm [9], a learnng auomaa-based heursc s purposed for fndng an opmal soluon of he proxy equvalen consraned CDS problem The degree and he backbone delay also consdered n hs algorhm o prolong he backbone duraon and o shoren he delay of ransmssons In [0], a load-balanced vrual backbone consrucon algorhm s proposed by He e al hey consder he sze and he load-balance facors when consrucng he backbone of WSN Afer a backbone s go, hey propose an approxmaon algorhm by usng he lnear relaxng and random roundng echnque o allocae non-backbone nodes o proper backbone nodes wh an objecve o mnmze he maxmum vald degree of all he backbone nodes [0] 3 Learnng auomaa and pursu algorhm A learnng auomaon consss of an adapve learnng agen operang n unknown envronmen [] I akes an acon from s acon se o maxmze he probably of beng rewarded from he envronmen And he learnng auomaon res o fnd opmal acon hrough eraons In each eraon, ges a feedback from he envronmen and updaes s acon probably vecor An auomaon can be represened as a 4-uple (A, P, F, T) and envronmen can be represened as a 3-uple (A, F, D) Where, A= { α, α,, α,, α }, r,s he se r of acons andrs he number of acons T s a scheme ha used o updae he acon probably vecor as follows, P( + ) = TP ( ( ), α( ), β( )) () P s heacon probably vecor of each acon P() = [ p (), p (),, p ()] s he probably vecor r a eraon In each eraon, he auomaon selecs an acon form A wh respecve probables n P() Fs he se of he feedback from envronmen The feedback a he eraon s denoed asβ()(β() F) In hs paper, β()= f he acon ge a reward form he envronmen Oherwse, β()=0 D = { d ( ),, d ( ), d ( )}, r, s he se of r average reward value, where d ( k) = E[ β() α() = α ] sands for he expecaon of reward value when an auomaon chooses he -h acon If d () s ndependen of for all r, E-ISSN: 4-864 54 Volume 3, 04
Ysong Jang, Weren Sh he envronmen wll be saonary Oherwse, wll be non-saonary In hs paper, he envronmen s saonary, he expresson of D can be smplfed o D = { d,, d, d },m s used o denoe he ndex of r he opmal acon, d = max{d } m I s mporan o choose a scheme of T o ge beer performances, e a beer resul or fas speed of convergence Thahachar and Sasry [] nroduced he esmaor algorhms where he auomaon runs an esmaor o provde gudance for updang he acon probably vecor One of he esmaor algorhms presened by Thahachar and Sasry n [3] s called he pursu algorhm (PA) or connuous pursu reward-penaly (CP RP ) algorhm [4] Table Algorhm PA Inalze =, p () = / r for all r Inalze he esmae vecor d ˆ( ) by akng each acon a small number of mes Repea () A he eraon, he auomaon chooses an acon α n lne wh s acon probably vecor P () Le α() = α () Afer geng he feedback β (), updae he esmae vecor d ˆ( ) accordng o he followng equaons, W ( + ) = W () +β () N ( + ) = N ( ) + () W ( + ) dˆ ( + ) = N ( + ) (3) Updae he acon probably vecor P() on he bass of he followng equaon P( + ) = ( λ) P( ) + λe M() (3) End repea Where d ˆ( ) = [ d ˆ ˆ (),, dr ()] s he esmae vecor and d ˆ () refers o he average reward value of -h acon a he eraon W() = [ W (),, Wr ()] s a vecor and W () s he number of mes ha acon α has been rewarded up o he me N() = [ N(),, Nr ()] s a vecor and N () s he number of mes ha he acon α has been chosen up o he me λs he speed of a learnng parameer and sasfes he condon 0<λ< em() s a r-vecor wh n he M()-h coordnae, and he ohers are 0 Ms () hendex of he maxmal componen of d ˆ( ), and dˆ ( ) = max{ dˆ ( )} M() Noe ha PA sops when he condon s sasfed For nsance, once one of he acon probables s larger han 09 or oher values, he repea wll be sopped I s easy o observe ha he auomaon always pursues he curren acon, e, opmal acon, and hence he name pursu algorhm [5] 4 Backbone formaon algorhm based on he pursu learnng algorhm Our algorhm s cluserng-based In hs algorhm, a MIS s generaed a frs Nex,every node n he MIS fnds s acon se Then, hose acon ses are sen o he snk and he PA algorhm wll be execued o ge a CDS wh small sze Fnally, he opology of he CDS wll send o each node n he MIS 4 Domnaor selecon process Before we ge he domnaor selecon process, each node s marked wh a unque rank, e, (degree, ID), where he degree refers o he number of neghbors and ID s he label of a node We assume ha rank(n )s hgher han rank(n )f he degree of n s larger han he degree of n or he ID of n s larger han he ID of n when he degree of n s equal o he degree of n The way o selec he domnaor s gven n he followng conen () Inal all nodes wh whe color () If a whe node has a hgher rank han all s whe neghbors, wll be marked black and all of s neghbors wll be marked gray When a node s marked black, broadcass a BLACK message o s neghbors As soon as a node receves a BLACK message, changes s color o gray and records he ID of he black node Smlarly, once a node s marked gray, wll broadcas GRAY message o s neghbors o menonhe change of s color The whe node compares s rank wh all of s whe neghbors when receves a GRAY message If he condon () s sasfed, wll be marked blackafer he color process, here are no whe nodes n he nework;all nodes wll bemarked black or gray, and, E-ISSN: 4-864 55 Volume 3, 04
Ysong Jang, Weren Sh he black nodes form a MIS [6] Therefore, each MIS s also a DS of he graph The black node s also called domnaor, whle he gray node s called domnaee 4 The formaon of acon ses Every node n he MIS s reaed as a learnng auomaon To geng a CDS, each domnaor selecs some gray nodes as connecors o buld connecon pahs o oher domnaors, whch are hree hops or wo hops away In hs paper, f he connecon pahs beween hose domnaors are replaced wh edges, o geng a CDS wh a small sze, here s no cycle n he CDS Thus, he opology of he CDS s a ree Each domnaor excep he roo of he ree wll choose anoher domnaor as s faher node An acon of a domnaor s s choce (he choce of he roo s null) The acon se consss of wo pars The frs pars s hose domnaors ha are hree hops or wo hops away (we call hose nodes he domnaors need o be conneced (DNC)) The second par s he connecon pahs, whch ncludes he connecor For example, as shown n Table, d d and d3 are he DNC of domnaor d The connecon pahs form d o d, d and d3 are (d, c, d), (d, c, c3, d) and (d, c4, c5, d3), respecvely Therefore, he connecors of hose pahs are c, (c, c3), (c4, c5), respecvely Table Anllusraon of he acon se of domnaor d beween node and node 8 n Fg, so he connecor of he pahcould be node 4 or node 5 Accordng o he prncple we have menoned before, node 4 wll be chose because has a larger rank han node 5 Ths prncple can decrease he sze of he fnal CDS because he chance ha he connecorwh a larger rank s chose by oher domnaors s larger, for example, node 4 also s chose as a connecor by node 0 A he second sage, every domnaee broadcas a Two-Hop-Domnaor ha ncludes s wo hop domnaor and he connecor ha connec he domnaee and he domnaor Afer hs process, each domnaor wll ge he nformaon of he hree hops away domnaors and fnd connecons pah o s hree hops domnaor neghbors A four-uple (b, b, b3, b4) s used o sgnfy a pah from domnaor b o anoher domnaor b4, whch s3 hops away from b If here are mulple roues beween hem, he roue ha has a greaer amoun of degree (b) and degree (b3)wll be chose For example, here are hree roues from node o node n Fg, e, (,, 6, ), (, 3, 6, ) and (, 3, 7, ) Based on he prncple, he roue (, 3, 7, ) wll be chose DNC d d Connecors c (c,c3) Fg An llusraon for he formaon of acon ses d3 (c4,c5) Generally, he formaon of acon sescan be dvded no wo sagesa he frs, each domnaee broadcas a One-Hop-Domnaor message n whch he IDs of s neghbor domnaor are added Thus, each node wll ge he nformaon of s wo hops domnaor neghbors Then, every domnaor adds he acons of hose domnaors, whch are wohops away,o s acon se We use a rple (a, a, a3) o represen a pah fromdomnaor a o anoher domnaor a3 ha s wo hops away If here are severalpahs beween a and a3, he one wh he maxmum rank wll be chose and added o he acon se For example, here are wo roues 43 Geng he CDS based on he pursu algorhm Afer he formaon of acon ses, every domnaor sends s acon se o he snk (so, our algorhm s quas-dsrbued or quas-global) Based on he acon ses of domnaors, snk fnds nodes as lle as possble o connec he domnaors and ge a small CDS Ths s a Sener ree problem However, hs verson of he Sener ree problem s NP-complee The algorhm PA, n whch he domnaors are reaed as auomaa, s used o ge an approxmae soluon More deals abou he algorhm and s parameers are descrbed as follows E-ISSN: 4-864 56 Volume 3, 04
Ysong Jang, Weren Sh To geng he feedback of he envronmen, he followng prncple s ulzed If he sze of he CDS of curren eraon (we denoe he sze as V_) s no greaer han he bes value (BV) of he szes of CDSs ha have been go so far, he value of he feedback β() wll be se o (as shown n formula (4)) and he values n he array also be updaedβ()= means he acons adoped by he auomaa are rewarded by he envronmen β ( ) = IV { _ BV} (4) V _ BV) Where I{ V _ BV} = 0 oherwse The speed of he learnng parameer λconrols he seps ha can be aken fromp() op(+) In general, for a fxed λ, a large valuewll lead o a hgher rae of convergence; and vce versa We follow [7] o ake he suaon haλ changes wh me (as shown n formula (5)) because hey argue ha a changng λ s conssen wh he noon of convergence / λ() = θ, (5) whereθ (0,) Accordng o value of β() and λ(), each domnaor updaes he values of parameers as shown n formula () Nex, updaes he value of M() and he acon probably vecor P() accordng o formula (6) gven below However, n he process of nalzaon,he acon probably vecor P() keeps conssen P( + ) = ( λ() ) P() + λ() e M () (6) We assume ha he sop condon mees f one of he followng condons s sasfed The frs condon s ha he number of eraons s equal o a hreshold The second one s ha he values of he szes of CDSs, whch are obaned n he las K eraons, are he same (K s a posve neger)when he pursu algorhm fnshes, he CDS wh he smalles sze wll be reurned as he oupu of hs algorhm As menoned n secon 4, he opology of he CDS s a ree f we replace he connecon pahs beween domnaors wh edges On he oher hand, each domnaor sores he connecon pahs n he acon So, he snk jus needs o send each domnaor a message, whch conans he domnaor neghbors of he domnaor n he ree, o make domnaors esablsh necessary connecons ha ncluded n he CDS 5 The convergence of our algorhm The convergence of a learnng algorhm mples ha he auomaon wll always mplemen he opmal acon evenually Such a knd of convergence s ypcally called ε-opmaly Rajaraman and Sasry [5] pu forward a framework o analyse he fne me behavour of he pursu learnng algorhm However, he speed of learnng parameers s a fxed value n her analyss In hs paper, followng he framework, we analyse he convergence of he pursu learnng algorhm when λ changes wh me, as shown n formula (5) Before we sar o analyse ε-opmaly of he algorhm, here are wo lemmas ha can conrbue o he analyss of he convergence Lemma : For any gven posve consan δ>0 and a posve neger 0<n<, here s 0 < T < such ha Pr{mn Z ( ) n} > δ > T (7) =,, r Proof: Provng (7) s equal o: for all {,,, r} Pr{ Z ( ) < n} < δ (8) Tha s, n Pr{ Z ( ) = j} < δ j= 0 If for all j, 0 j n-, he nequaly wll follow δ Pr{ Z ( ) = j} < (9) n A any eraon k, k Pr{ α(k) = α} ( λ() ) p(0) s always enable = because he acon probably decreases by ( λ()) a mos a each eraon Thus, k Pr{ α(k) α} ( λ() ) p(0) =, k / = θ p (0) = where, θ (0,) and p (0) = / r We can ge k /k Pr{ α(k) α} < θ = r = θ / r Based on he foregong nequaly, we can ge he followng nequaly j j k j Pr{ Z( ) = j} < Ck ( θ / r), j k j j k j < k ( θ /r) = k σ whereσ=θ/r n x n Defne a funcon Ψ ( x) = x σ ( >0) If we prove ha here s a value T ha makes Ψ( )<δ/n be always sasfed when > T, he nequaly (9) E-ISSN: 4-864 57 Volume 3, 04
Ysong Jang, Weren Sh wll be proved and Lemma wll be proved as well Where, n x n Ψ ( x) = dψ ( x)/ dx = x σ ( x ln σ + n) I s easy o ge Ψ ( x) < 0 when x> n/ ln( / σ ) = T Consequenly, Ψ( ) s a decreasng funcon when >T If Ψ(T ) δ/n, we can ake T = T o make Formula (9) and Lemma be followed all he me x sands for he smalles neger ha s equal o or larger han When Ψ(T )> δ/n, f we wan o prove Lemma, wll be essenal for us o prove ha here s anoher value T 3 ha s larger han T and Ψ(T 3 ) = δ/n We can ge n n x n x lm Ψ ( x) = lm x σ = lm x + x + x + x n ( / σ ) By vrue of he L' Hôpal's rule and /σ>, we can ge n x lm Ψ ( x) = lm x + x + x n ( / σ ) n! = lm = 0 x + x n n ( / σ) [ ln(/ σ) ] Because Ψ( ) s a decreasng and connuous funcon when >T, here mus be a value T 3 ha s larger han T and Ψ(T 3 ) = δ/n s sasfed Therefore, Formula (9) and Lemma always follow when T = T3 Lemma: For any gven posve consan δ>0, κ>0 and {,,,r}, here s 0< T < so ha ( ) d κ > δ > T { } Proof: Accordng o he defnon ˆ W ( ) { () } () () I α = α β d = = N() N() We can fnd ha d ˆ () [0,] acs as he esmae of d For any eraon, s possble o ge he followng nequaly by applyng he Theorem of Hoeffdng [8] ( ) d > κ < exp( N ( ) κ ) (0) { } In accordance wh he laws of oal probably, we can oban { dˆ d > κ} { dˆ d κ N n} { N n} ˆ { d d κ N n} { N n} ˆ { d d κ N n} { N n} Pr ( ) = Pr () > () Pr () + Pr () > () < Pr () < < Pr () > () + Pr () < () Accordng o Inequaly (0), ( ) d > κ N ( ) n < exp( n κ ) { } 4 Se n = ln κ δ, hen ˆ δ Pr { d() d > κ N() n} < () Accordng o Lemma, δ Pr { N ( ) < n} < (3) The above nequaly always follows when >T Here, we wll oban T = T f Ψ(T ) δ/(n) Oherwse, T = T 4, where he value of T 4 s larger han ha of T and Ψ(T 4 ) = δ/(n) Combne ()-(3), he followng nequaly can be obaned: ( ) d > κ < δ > T (4) { } Tha s, ( ) d κ > δ > T (5) { } Hence, Lemma s proved Theorem : For any gven posve consan δ (0, ) and ε (0, ), here s 0<T < such ha Pr{ pm ( ) > ε} > δ > T Proof: Accordng o Lemma and by akng κ = ( dm d j ), j m, we can ge he probably ha dˆ m () dm < κ and dˆ j () d j < κ s larger han δ when >T Thus, { ˆ ˆ m m j j } Pr d () d + d () d < κ > δ I s easy o ge dˆ () d + dˆ () d > d dˆ () + dˆ () d m m j j m m j j = κ dˆ () ˆ m + d j() Based on he nequales, for all j m and >T, we can ge he followng nequaly () > dˆ () > δ (6) { m j } E-ISSN: 4-864 58 Volume 3, 04
Ysong Jang, Weren Sh Accordng o he law of oal probably, Pr { pm ( ) > ε} (7) > Pr () > () > () Pr () > () { p ˆ ˆ } { ˆ ˆ m ε dm d j dm d j } Theorem wll be proved f Pr p () > ε dˆ () > dˆ () =, { m m j } for all >T T Assume T = + T, where >0 pm() = pj() j m Therefore, we can prove pj () < ε when j m dˆ () ˆ m > d j() and >T p (T j ) = pj(t ) [ λ(t + q)] j m j m q= /(T + q) < [ λ(t + q)] = θ < θ q= q= /(T + ) /(T + ) If θ < ε, he followng nequaly should be sasfed (lnθ ln ε) < T lnε Take θ<ε, T lnε ln >, ln θ ln ε T ε = ln θ ln ε and T = + T Accordng o (7), we could ge Pr p ( ) > ε > δ, { } m for all >T Based on Theorem, can be concluded ha he choce probably of he expeced acon converges a an approxmae soluon o he MCDS for a larger eraon and he value of K s se o 0 The value of SN, he rado ransmsson range of nodes (presened by Tr), and he speed of learnng parameers (presened by λ) are he parameers ha nfluence he backbone of he nework We compare he performance of BFA-PA wh Zone-based vrual backbone formaon (ZVBF- MD) [3] and Torkesan s DAL-BF algorhm [8] n erms of he sze of CDS, message overhead and he average node degree n he CDS On he oher hand, Buenko s algorhm [9] s used o gve a reference o he bes soluon of CDS In our smulaon, he domnaor n ZVBF-MD uses he quas-global verson o ge a smaller CDS [3] In he smulaon, he learnng rae [8] of DAL-BF s 0, he PCDS [8] of DAL-BF s 09 and he maxmum eraons of DAL-BF s 00 Accordance o he defnon of λ, we change he value of θ o ge he nfluences of λ In he followng expermen, he value of θ s se as 0, 0, 03, 04, 05 and 06, respecvely A he same me, SN s se as 300 and he ransmsson range s vared from 5 o 30 uns The sze of CDS and he number of eraons are he mercs The smulaon resuls are shown n Fg and Fg 3 Based on Fg, we ge ha he sze of CDS decreases slghly when θ ncreases Fg 3 shows ha, n general, he number of eraons ncrease wh he value of θ Ths can be aached o he fac ha he smaller value of θ make he algorhm has a larger value of λ, and led converges quckly On he oher hand, rapd convergence leads o greaer resuls abou he sze of CDS and he smaller resul abou he number of eraons In he followng smulaons, we se he value of θ o 04 o make a rade-off beween he convergence rae and he sze of he CDS 6 Expermen resuls To nvesgae he performance of our algorhm, several expermens are shown n hs secon (he smulaor s wren n M language of Malab 00b) We assume ha an deal MAC layer s used n hose smulaons The packe can be ransmed whou conenon, packe losses or collson In hose smulaons, nodes are unformly dsrbued n a square area of 00 uns by 00 uns The number of nodes n he nework ranges from 00 o 500 (he number of nodes s presened by SN), and more han 50 conneced graphs are random generaed for each gven number of node (he smulaon resul are he average values of hose graphs) In he smulaon of BFA-PA, he hreshold of eraons s se o 3000 FgThe sze of CDS when he value of θ s changed E-ISSN: 4-864 59 Volume 3, 04
Ysong Jang, Weren Sh Fg3The number of eraonswhen he value of θ s changed In he nex smulaon, we ge he resuls when SN and Tr are changed The ransmsson range Trchanges from 5 o 50 uns andhe sze of he nework SN vares from 00 o 500 In Fg4, he x- axs denoes he value of SN and dfferen curves have dfferen values of Tr Accordng o he resuls shown n Fg 4, we can ge ha he sze of CDS ncreases wh SN when Tr s small and he ascendan rend s no sgnfcanwhen Tr s large, e, Tr = 35 When SN s fxed, he sze of CDS decreases wh he ncrease of Tr becausea domnaor node can cover more nodes when Tr ncrease However, he declne rend of he sze of CDS s slgh when Trs greaer han 45 when SN s more han 300) We also make a comparson among hose algorhms when he sze of nework s fxed and Tr s changed Relaed resuls are shown n Fg7 and Fg 8, n whch SN s se as 00 and Tr s changed from 5 o 50 wh a sep of 5 uns The sze of CDS decreases whentr ncreases because he ncrease of Tr resuls n hgher densy of he nework Consequenly, a domnaor can cover more nodes so ha fewer domnaors are needed When Tr s a large number, he performance of ZVBF-MD, BFA-PA and DAL-BF are close Fg5 The sze of CDS when SN s changed and Tr equals 5 uns Fg6 The sze of CDS when SN s changed and Tr equals 30 uns Fg4 The number of CDSs when SN and Trs changed In he followng expermens, we compare our algorhm BFA-PA, DAL-BF algorhm [8], Buenko s algorhm and ZVBF-MD [3] n erms of he sze of CDS, message overhead and average node degree n he CDS Fg5 and Fg6show he resulswhen ransmsson range s 5 and 30 uns, respecvely, and SN s changed from 00 o 500 Those resuls show ha he sze of CDS ncreases when SN ncreases However,when SN s a large number, he ncrease rae slows down because makes he nework denser and a domnaor covers more nodes Furhermore, can be found ha BFA-PA consrucs he smaller CDS compared wh ZVBF- MD and DAL-BF when he nework s dense (e, Fg7The sze of CDS when Tr s changed and SN equals 00 E-ISSN: 4-864 530 Volume 3, 04
Ysong Jang, Weren Sh Fg8 The sze of CDS when Tr s changed and SN equals 00 Message overhead s anoher merc of he CDS formaon algorhm In he wreless sensor nework, he hoss suffer from src resource lmaons so ha he communcaon overhead should be kep as low as possble Fg9 and Fg 0 show he message overhead of dfferen algorhms when he sze of he nework s changedand Tr s se o5 and 30, respecvelythe message overhead of DAL-BF s no shown n he fgures because s message overhead s large For example, when SN s 00 and Tr s 5 uns, he message overhead DAL-BF s more han 400 housands byes Based on hose resuls, we can fnd ha BFA-PA has alarge message overhead han ZVBF-MD Ths s cause by he fac ha, n BFA-PA, each domnaor wll ge s acon se and send he se o he snk On he oher hand, n ZVBF-MD, only he domnaor a he zone border execues a smlar process We also could fnd ha he number of message overhead ncreases when he number of nodes or Tr ncreases Fg0 The Average messageoverhead when SN s changed and Tr equals 30 uns The average node degree n he CDS s anoher parameer proposed by Bo [3], whch s also he average node degree n he nduced sub graph of he consruced CDS Bo beleves ha a low degree may cause less nerference for communcaon Boh Fg and Fg show he expermen resuls when he value of SN changes from 00 o 500, and Trs se as 5 and 30 uns, respecvely Based on hose resuls, we can ge ha he node degree n he CDS ncreases when he sze of nework ncreases and Trncreases When he nework s dense (Tr s se as 30 uns and SN s more han 300), he rsng rends of hose algorhms are no conspcuous any more, and BVF-PA has smlar performance o ZVBF-MD FgAverage node degree n he CDS when SN s changed and Tr equals 5 uns Fg9 The Average messageoverhead when SN s changed and Tr equals5 uns Fg Average node degree n he CDS when SN s changed and Tr equals 30 uns E-ISSN: 4-864 53 Volume 3, 04
Ysong Jang, Weren Sh 7 Concluson In hs paper, based on he pursu algorhm, we propose a novel backbone formaon algorhm, called BFA-PA In hs algorhm, a DS s go a frs and each node n he DS ges s acon se based on he connecon pah beween he domnaor and her or 3 hops domnaor neghbors Snk execues he pursu algorhm and reads domnaors as auomaa wh o ge an approxmae soluon of MCDS of nework Moreover, a changng speed of learnng parameer s used o avod choosng a specal learnng rae, whch should be carefully seleced o make a rade-off beween he convergence speed and he sze of backbone I s also shown ha our mehod s ε-opmaly o connec he DS wh he changng speed of learnng parameer Accordng o he resuls of he smulaon, our algorhm generaes a smaller CDS han ZVBF-MD and has a smlar resul wh DLA-BF abou he sze of CDS However, s effcen han DLA-BF because has a small overhead However, n hs algorhm, we assume ha he nework s saonary When some new sensor nodes are added o he nework or removed, he algorhm should be run agan o ge a backbone of nework Because he opology of nework s changed, BFA- PA s dffcul o work on a dynamc nework Is here a smple mehod o fx he backbone when here are some addons or falures of nodes? The answer o hs queson s sgnfcan o he furher applcaon of hs mehod, and wll be a gude for our fuure work 8 Acknowledgemens Ths work s suppored parally byhe scence and echnology projec of CQ CSTC (No csc0jja40037) References: [] Yu J, Wang N, Wang G, e al Conneced domnang ses n wreless ad hoc and sensor neworks A comprehensve survey [J] Compuer Communcaons, 03, 36(): - 34 [] Alzoub K M, Wan P J, Freder O Maxmal ndependen se, weakly-conneced domnang se, and nduced spanners n wreless ad hoc neworks [J] Inernaonal Journal of Foundaons of Compuer Scence, 003, 4(0): 87-303 [3] Han, Bo Zone-based vrual backbone formaon n wreless ad hoc neworks [J] Ad Hoc Neworks 7 (009): 83-00 [4] L Y, Tha MT, Wang F, Y C-W, Wang P-J, Du D-Z On greedy consrucon of conneced domnang ses n wreless neworks [J] Specal ssue of Wreless Communcaons and Moble Compung (WCMC), 005 [5] Gao B, Yang Y, Ma H A new dsrbued approxmaon algorhm for consrucng mnmum conneced domnang se n wreless ad hoc neworks [J] Inernaonal Journal of Communcaon Sysems 005, 8(8), 734 76 [6] Je Wu, Halan L On calculang conneced domnang se for effcen roung n ad hoc wreless neworks [C] The Thrd ACM Inernaonal Workshop on Dscree Algorhms and Mehods for Moble Compung and Communcaons (ACM DIALM 999), Augus 999, 7 4 [7] Fe Da and Je Wu An exended localzed algorhm for conneced domnang se formaon n ad hoc wreless neworks [J] IEEE Transacons on Parallel and Dsrbued Sysems 5 (0) (004), 908 90 [8] J Akbar Torkesan, MR Meybod An nellgen backbone formaon algorhm n wreless ad hoc neworks based on dsrbued learnng auomaa [J] Compuer Neworks 54 (00) 86 843 [9] Akbar Torkesan J Energy-effcen backbone formaon n wreless sensor neworks [J] Compuers & Elecrcal Engneerng, 03, 39(6): 800-8 [0] He Jng, JShoujng, Pan Y, e al Approxmaon algorhms for load-balanced vrual backbone consrucon n wreless sensor neworks [J] Theorecal Compuer Scence, 03, 507: -6 [] Narendra, K S and Thahachar, M A L Learnng auomaa: An nroducon Englewood Clffs, NJ: Prence Hall (989) [] M A L Thahachar and PS Sasry A Class of Rapdly Convergng Algorhms for Learnng Auomaa [C] IEEE In Conf on Cybernecs and Socey, Bombay, Inda, Jan 984 [3] M A L Thahachar and PS Sasry Esmaor Algorhms for Learnng Auomaa [J] Proc Planum Jublee Conf on Sys Sgnal Processng, Dep Elec Eng, Indan Insue of Scence, Bangalore, Inda, Dec 986 [4] Oommen B J, Agache M Connuous and dscrezed pursu learnng schemes: Varous E-ISSN: 4-864 53 Volume 3, 04
Ysong Jang, Weren Sh algorhms and her comparson [J] Sysems, Man, and Cybernecs, Par B: Cybernecs, IEEE Transacons on, 00, 3(3): 77-87 [5] Rajaraman, K and Sasry, P S Fne me analyss of he pursu algorhm for learnng auomaa [J] IEEE Trans Sysems Man CyberneB(996), 6, 590 598 [6] Kaled M Alzoub, Peng-Jun Wan, Ophr Freder Maxmal ndependen se, weakly conneced domnang se, and nduced spanners for moble ad hoc neworks [C] Inernaonal Journal of Foundaons of Compuer Scence 003 4(), 87 303 [7] Tlak, O, Marn, R, and Mukhopadhyay, S,A decenralzed, ndrec mehod for learnng auomaa games [J] IEEE Trans Sysems Man Cyberne B (0), 4, 3 3 [8] W Hoeffdng Probably nequales for sums of bounded random varables [J] Journal of he Amercan Sascal Assocaon, 963, vol 58, pp 3-30 [9] Buenko S, Cheng X, Olvera C A, e al A new heursc for he mnmum conneced domnang se problem on ad hoc wreless neworks [M] Recen Developmens n Cooperave Conrol and Opmzaon Sprnger US, 004: 6-73 E-ISSN: 4-864 533 Volume 3, 04