Multicast Capacity for Large Scale Wireless Ad Hoc Networks

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1 Multicst Cpcity fo Lge Scle Wieless Ad Hoc Netwos Xig-Yg Li Dept. of Compute Sciece Illiois Istitute of Techology Chicgo, IL 6066, USA Sho-Jie Tg Dept. of Compute Sciece Illiois Istitute of Techology Chicgo, IL 6066, USA Ophi Fiede Dept. of Compute Sciece Illiois Istitute of Techology Chicgo, IL 6066, USA ABSTRACT I this ppe, we study the cpcity of lge-scle dom wieless etwo fo multicst. Assume tht wieless odes e domly deployed i sque egio with side-legth d ll odes hve the uifom tsmissio ge d uifom itefeece ge R >. We futhe ssume tht ech wieless ode c tsmit/eceive t W bits/secod ove commo wieless chel. Fo ech ode v i, we domly pic odes fom the othe odes s the eceives of the multicst sessio ooted t ode v i. The ggegted multicst cpcity is defied s the totl dt te of ll multicst sessios i the etwo. I this ppe we deive mtchig symptotic uppe bouds d lowe bouds o multicst cpcity of dom wieless etwos. We show tht the totl multicst cpcity is Θ W whe = O ; the totl multicst cpcity is ΘW whe = Ω. Ou bouds uify the pevious cpcity bouds o uicst whe = by Gupt d Kum [7] d the cpcity bouds o bodcst whe = i [, 0]. We lso study the cpcity of goup-multicst fo wieless etwos whee fo ech souce ode, we domly select goups of odes s eceives d the odes i ech goup e withi costt hops fom the goup lede. The sme symptotic uppe bouds d lowe bouds still hold. Fo bity etwos, we povide costuctive lowe boud Ω W fo ggegted multicst cpcity whe we c cefully plce odes d schedule ode tsmissios. Ctegoies d Subject Desciptos C.. [Netwo Achitectue d Desig]: Wieless commuictio, Netwo topology; G.. [Gph Theoy]: Netwo poblems, Gph lgoithms Geel Tems Algoithms, Desig, Theoy The wo of the utho is ptilly suppoted by NSF CCR Pt of the wo ws doe whe the utho visited Micosoft Resech Asi, BeiJig, Chi. Pemissio to me digitl o hd copies of ll o pt of this wo fo pesol o clssoom use is gted without fee povided tht copies e ot mde o distibuted fo pofit o commecil dvtge d tht copies be this otice d the full cittio o the fist pge. To copy othewise, to epublish, to post o seves o to edistibute to lists, equies pio specific pemissio d/o fee. MobiCom'07,Septembe 9-4, 007, Motél, Québec, Cd. Copyight 007 ACM /07/ $5.00. Keywods Wieless d hoc etwos, cpcity, multicst, bodcst, uicst, schedulig, optimiztio, pobbility theoy.. INTRODUCTION I wieless d hoc etwos, wieless odes my coopete i outig ech othes pcets. Lc of cetlized cotol of the fuctiolity d possible ode mobility give ise to my chllegig issues t the etwo lye, the medium ccess lye, d physicl lye of wieless d hoc etwo. At the etwo lye, the mi chllegig poblem is tht of outig, which hs to del with time-vyig etwo topology, possible powe-costits of wieless odes, d the chcteistics of the wieless chel such s ustble, bodcst tue, fdig d so o. The choice of medium ccess cotol is lso esticted by the fct tht the etwo topology is time-vyig, d thee is o cetlized cotol. I the litetue, umbe of esults hve bee poposed to use the TDMA, CDMA, FDMA, d the dymic ssigmet of fequecy bds to impove the etwo thoughput. Notice tht TDMA hs ecetly bee poposed to impove the etwo thoughput fo some etwos o ptil of the etwos [, 3], especilly fo sttic etwos. At the physicl lye impott issue is the powe-cotol, which hs bee studied extesively i the litetue. A ceful selectio of the tsmissio powe of odes c ot oly impove the odl life, but lso impove the sptil euse of fequecy d cosequetly possibly impove the etwo thoughput. I my pplictios, e.g., wieless seso etwos, we ofte eed ough estimtio o the chievble thoughput whe we domly deploy wieless odes i give egio. The mi pupose of this ppe is to study the symptotic cpcity of lge scle dom wieless etwos whe we choose the best potocols fo ll lyes. As i the litetue, we will mily coside oe type of etwos, lge scle dom etwos, whee lge umbe of odes e domly plced i the deploymet egio. We will study the cpcity of give wieless etwo whee the odes positios e give pioi, d how the cpcity of wieless etwos scle with the umbe of odes i the etwos whe give fixed deploymet egio, o scle with the size of the deploymet egio whe give fixed deploymet desity fo multicst. We ssume tht set of wieless odes V = {v, v,, v } e domly distibuted with uifom distibutio i sque egio with side-legth d ll odes hve the sme tsmissio ge. Fo most esults peseted i this ppe, we ssume tht vlues of d e selected such tht the esulted etwo will be coected with high pobbility w.h.p.. The esults deived ude this model lso imply the sme esults fo the dese model, whe odes e distibuted i fixed egio such s uit sque by 66

2 pope sclig d the uifom tsmissio ge of ll odes e selected s the citicl tsmissio ge CTR to get coected etwo with high pobbility. I this ppe, we will cocette o the multicst cpcity of dom wieless etwo, which geelizes both the uicst cpcity [7] d bodcst cpcity [, 0] fo dom etwos. Assume tht subset S V of s = S odes will seve s the souce odes of s multicst sessios. The most esults i this ppe ssume tht S = V. Ech ode v i S hs set of domly chose d = destitio odes to which it wishes to sed dt t bity dt te λ i. The multicst cpcity of dom etwo is defied s Λ = i= λ i whe thee is schedule of tsmissios such tht ll multicst flows will be eceived by thei destitio odes successfully withi fiite dely. To descibe whe tsmissio is eceived successfully by its iteded ecipiet, we will llow oe possible model fo successful oe-hop eceptio: potocol model. We ssume tht ech ode v V hs fixed costt tsmissio ge d fixed costt itefeece ge R >. A ode u c successfully eceive tsmissio fom othe ode v with v u iff thee is o othe ode w such tht w u R d ode w is tsmittig simulteously with ode v. Hee w u is the Euclide distce betwee w d u. We ssume the followig simple wieless chel model s i the litetue: ech wieless ode c tsmit t W bits/secod ove commo wieless chel. Fo pesettio simplicity, we ssume tht thee is oly oe chel i the wieless etwos. We will see tht it is immteil to esults peseted i this ppe if the chel is boe up ito sevel sub-chels of cpcity W, W,, W M bits/secod s log s we hve M i= W i = W. As lwys, we ssume tht the pcets e set fom ode to ode i multihop me util they ech thei fil destitios. The pcets could be buffeed t itemedite odes while witig fo tsmissio. I this ppe, we ssume tht the buffe is lge eough so pcets will ot get dopped by y itemedite ode. We leve it s futue wo to study the sceio whe the buffes of itemedite odes e bouded by some vlues. I some esults, we ssume tht evey itemedite ode hve ifiite buffe size. Fo most of the esults peseted hee, the dely of the outig is ot cosideed, i.e., the dely i the wost cse could be bitily lge fo some esults. Ou Mi Cotibutios: We popose two egimes fo multicst cpcity i tems of. We deive mtchig lyticl uppe bouds d lowe bouds o multicst cpcity of dom wieless etwo. Assume tht the side-legth of the deploymet sque d the tsmissio ge e selected such tht the etwo is coected lmost suely i.e., = Θ. We show tht the ggegted multicst cpcity of dom multicsts is { Θ Λ = W whe = O, ΘW whe = Ω Ou bouds uify the pevious cpcity bouds o uicst whe = by Gupt d Kum [7] d the cpcity bouds o bodcst whe = i [, 0]. Cosequetly, the pe-ode multicst cpcity λ of multicst sessios with eceives pe multicst sessio is { Θ λ = W whe = O, Θ W whe = Ω The bove cpcity bouds e implied by moe geel esult fo the followig etwo settig thee e s multicst sessios, ech with domly selected eceives fom V, d the tsmissio ge d side-legth of the deploymet sque stisfyig tht the esulted dom etwo is coected with high pobbility. Geelly, whe lim s =, we pove tht the ggegted multicst cpcity of s multicst sessios is { Θ Λ = W whe = O 3 ΘW whe = Ω d the pe-souce multicst cpcity of s multicst sessios is { miw, Θ λ = W s whe = O 4 Θ W s whe = Ω We lso study the multicst cpcity fo goup-multicst whee, fo ech souce ode, we domly select goups of odes s eceives d the odes i ech goup e withi costt umbe of hops fom the goup lede. We show tht the symptotic multicst cpcity is still Θ W whe = O ; d is ΘW whe = Ω. Fo multicst i bity etwos, we povide costuctive lowe boud Ω W whe we c cefully plce odes d schedule ode tsmissios. The est of the ppe is ogized s follows. I Sectio we discuss i detil the etwo model d the chel model used i this ppe. I Sectio 3, we fist peset some uppe-bouds o multicst cpcity fo dom etwos. I Sectio 4, we the peset efficiet method fo multicst d pove tht the cpcity chieved by this method symptoticlly mtches the uppe-bouds deived befoe. I Sectio 5, we study the multicst cpcity bouds fo goup-multicst d the multicst cpcity bouds fo bitily etwos. We eview the elted esults o etwo cpcities i Sectio 6 d coclude the ppe i Sectio 7 with the discussio of some possible futue wos.. NETWORK MODEL The cpcity of dom wieless etwos ws fist studied i pioeeig semi wo by Gupt d Kum [7]. Thee e diffeet ppoches to icese the etwo thoughput, such s educig the itefeece, the schedulig o the MAC lye, oute selectio o the outig lye, chel ssigmet if multi-chels e vilble, d powe cotol o the physicl lye. I this sectio, we fist itoduce ou etwo system model, the we discuss i detil the itefeece models we will use d the defie the poblem tht we will study i this ppe. We coside lge scle dom etwos. Typiclly thee e thee wys to icese the umbe of etwo odes to ifiity.. Oe is to fix the deploymet egio d the icese the ode desity to ifiity. This is typiclly clled the dese model. This model is widely studied, e.g., Gupt d Kum studied the citicl tsmissio ge CTR [8] d the cpcity fo uicst [7] usig this model.. Aothe wy is to fix the ode desity to give costt d icese the deploymet egio to ifiity. This is typiclly clled the exteded model. Notice tht to get coected etwo with high pobbility, we lso eed to icese the tsmissio ge of odes. This model is lso used by sevel ppes to study the CTR o cpcity, e.g., [7, 5]. 3. The thid wy is to fix the tsmissio ge of ll odes to some costt, the icese the ode desity symptoticlly sme s the ode degee whe the tsmissio ge is fixed d the deploymet e to icese the umbe of odes i the etwo. We cll this model the costt-ge 67

3 model. Assume tht odes will be deployed. It hs bee poved i [4] tht the miimum ode degee fo coectivity is Θ. This implies tht the e of the deploymet egio is t most Θ. I this ppe, we will dopt the thid model. Notice tht ou esults peseted i this ppe ctully e immteil to the model used. Most esults peseted i this ppe ely o the tio whee is the side-legth of the deploymet sque d is the tsmissio ge, whee eithe o o both could be fuctio of. I this ppe, we ssume tht thee is set V = {v, v,, v } of commuictio temils deployed i egio Ω. We mily focus the sceio whe Ω is sque with side legth. Evey wieless ode hs uifom tsmissio ge such tht ode u c successfully eceive the sigl set by ode v if d oly if u v. The complete commuictio gph is udiected gph G = V, E, whee E is the set of commuictio lis. To schedule two lis t the sme time slot, we must esue tht the schedule will void itefeece. Sevel diffeet itefeece models hve bee used to model the itefeeces i wieless etwos. I this ppe, we will mily focus o the potocol itefeece model. We ssume tht ech ode v i hs costt itefeece ge R. Hee y ode v j will be itefeed by the sigl fom v if v v j R d ode v is sedig sigl to some ode othe th v j. I this ppe, we lwys ssume tht the itefeece ge R is withi smll costt fcto of the tsmissio ge, i.e., R = Θ. Cpcity Defiitio: We ssume tht ech ode v i could seve s the souce ode fo some multicst. Fo ech ode v i, we domly select odes, sy U i V {v i }, fom the emiig odes s the eceives of multicst sessio usig v i s the souce ode. Assume tht ode v i will sed dt to these eceives U i with dt te λ i. Notice tht whe the eceives e f wy fom the souce ode, we eed multiple itemedite odes to ely the dt fo v i. Let λ = λ, λ,, λ, λ be the te vecto of the multicst dt te of ll multicst sessios. Give set of s multicst sessios with the set of souce odes S V, let λ S = {λ i, λ i,, λ is } be the vecto of dt tes of ll souces i S. Whe give fixed etwo G = V, E, whee the ode positios of ll odes V, the set of eceives U i fo ech souce ode v i, d the multicst dt te λ i fo ech souce ode v i e ll fixed, we fist defie wht is fesible te vecto λ fo the etwo G. DEFINITION FEASIBLE RATE VECTOR. A multicst te vecto λ = λ, λ,, λ, λ o geelly λ S bits/sec is fesible if thee is sptil d tempol scheme fo schedulig tsmissios such tht by opetig the etwo i multi-hop fshio d buffeig t itemedite odes whe witig tsmissio, evey ode v i c sed λ i bits/sec vege to its chose destitio odes. Tht is, thee is T < such tht i evey time itevl with uit secods [i T, i T ], evey ode c sed T λ i bits to its coespodig eceives. v i S λ i Give set S of s multicst sessios, the totl thoughput cpcity of such fesible te vecto λ S fo multicst is defied s Λ,S = v i S λi. The pe ode multicst thoughput is defied s λ,s = s, whee is the totl umbe of odes i ech multicst sessio, icludig the souce ode. Whe S is cle fom cotext, we wite Λ,S d λ,s s Λ d λ. DEFINITION THROUGHPUT CAPACITY. A totl multicst thoughput Λ is fesible fo multicst if thee is te vecto λ = λ, λ,, λ, λ tht is fesible d Λ = i= λ i. Give s souces S, pe ode multicst thoughput λ,s bits/sec is fesible if thee is λ S = λ i, λ i,, λ is, λ s tht is fesible d λ,s = i= λ i s. DEFINITION 3 CAPACITY OF RANDOM NETWORKS. The totl multicst cpcity of clss of dom etwos is of ode Θg bits/sec if thee e detemiistic costts c > 0 d c < c < + such tht lim P Λ = cg is fesible = lim if P Λ = c g is fesible < Give s dom souces, we sy tht the multicst cpcity pe ode of clss of dom etwos is of ode Θf bits/sec if thee e detemiistic costts c > 0 d c < c < + such tht lim P λ = cf is fesible = lim if P λ = c f is fesible < Hee the pobbility is te fo ll istces of the dom etwos G d ll set of souces S with cdility s. Useful Kow Results: Thoughput this ppe, we will epetedly use the followig esults fom pobbility theoy litetue. LEMMA CHEBYSHEV S INEQUALITY. Fo vible X, P X µ A VX A, whee µ = EX, VX is the vice of X, d A > 0. LEMMA LAW OF LARGE NUMBERS. Coside ucoelted vibles X i, i with sme expected vlue µ = EX i d vice σ i= X = VX i. Let X = i. ɛ > 0, P X µ < ɛ σ ɛ. LEMMA 3 BINOMIAL DISTRIBUTION. Coside idepedet vibles X i {0, }, p = P X i =, d X = i= X i. P X ξ e p ξ, whe 0 < ξ p. ξ p P X > ξ <, ξ p whe ξ > p. Nottios: Thoughput this ppe, fo cotiuous egio Ω, we use Ω to deote its e; fo discete set S, we use S to deote its cdility; fo tee T, we use T to deote its totl Euclide edge legths; x deotes tht vible x tes vlue to ifiity. 3. UPPER BOUNDS ON MULTICAST CA- PACITY FOR RANDOM NETWORKS 3. The uppe-boud o We ssume tht wieless odes V with tsmissio ge e domly d uifomly distibuted i sque egio with side legth. We fist study the symptotic boud o / such tht the esulted etwo G = V, E is coected lmost suely, i.e., with pobbility goig to s goes to ifiity. Notice tht fo set of odes, the CTR fo coectivity is lwys the legth of the logest 68

4 edge of the Euclide miimum spig tee EMST of this set of odes [8,6,7]. Cosequetly, studyig the CTR fo coectivity is equivlet to studyig the logest edge of the EMST of set V of odes whe V follows ceti distibutio such s Poisso distibutio o dom uifom distibutio. Assume tht poits e distibuted uifomly t dom i the -dimesiol sque with side legth. Let M, be the dom vible deotig the legth of the logest edge of EMST built o this set of odes. The simple sclig of the esult poved i [6] shows tht, β, lim P π M, β =. Thus, with pobbility, we ow tht the logest edge e e β e legth M e β,, of EMST built o poits distibuted i sque +β with side-legth, is t most. Thus, whe β π d, we ow tht the logest edge of EMST hs π +β legth t most lmost suely. Thus, we hve THEOREM 4. Assume tht odes, ech with tsmissio ge, e domly uifomly deployed i sque egio of side legth. Whe π fo β, the esulted etwo +β G = V, E is coected with pobbility t lest e e β. Fo exmple, we c set = whee β = π. 3. Geel Techiques I pevious studies [7, 8] of cpcity of dom etwos, commo ppoch is to lyze the expected umbe of hops Hb bit b hs to tvel d the totl umbe of simulteous tsmissios S = O possible i the system. If ech souce ode geetes dt t te λ, the umbe of bits geeted by these s souces i time itevl T is simply λt s. Thus, the totl umbe of tsmissios of ll bits to thei destitios is λt s Hb lmost suely. Cosequetly, we hve λt s Hb T S. This implies tht λ = O = O s Hb s. I [7], fo Hb uicst, Gupt d Kum essetilly used Θ ssumed = s estimtio of Hb d deived Θ W s s pe-ode cpcity uppe-boud. I [8], fo multicst, Shotti et l. essetilly used Hb = Θ ssumed = to deive O W = s O s s pe-ode cpcity uppe-boud. Although this tditiol techique is vlid d coveiet fo studyig the symptotic uicst cpcity d the multicst cpcity with some specil cofigutios = ɛ fo some 0 < ɛ < [8], this my poduce uppe-boud smlle th chievble fo symptotic multicst cpcity i geel settig studied i this ppe. The eso fo this possible discepcy is tht fo multicst tee T with totl legth T, vlue Θ T my ot give the lowe boud o the umbe of tsmissios eeded by the tee T due to the multicst tul of wieless tsmissios. To ddess the bove chlleges d discepcies, we use two ew ppoches to lyze the uppe-boud of multicst cpcity:. Ae Agumet: This is bsed o lyzig symptotic lowe boud o the e coveed by the tsmissio diss of ll itel odes i multicst tee;. Dt Copies Agumet: This ppoch is bsed o lyzig symptotic lowe boud o the umbe of odes tht eceive copy of multicst dt duig the tsmissios of ll odes i the tee. The e gumet essetilly wos s follows. Whe we multicst fom oe souce ode v i to ll its eceives U i, ll odes lyig iside the itefeece egio of y tsmittig ode fo this multicst sessio cot eceive dt fom othe odes simulteously. Fo y ode u, let t i u be the time-itevls tht ode u will tsmit dt fo multicst tee T i. Thus, multicst tee will clim umbe of cylides Du, t iu fo itel ode u i T i the spce-time dimesio R T, whee Du, deotes the tsmissio dis of ode u, T is the schedulig peiod. Thus, give multicst tee T i fo multicst oigited fom v i, the pis of Du,, t iu i.e., tsmissio dis Du, will be used fo multicst oigited t v i duig the tsmissio time-itevl t i u climed by this multicst should be disjoit fom the pis climed by othe multicst sessios. Fo s multicst sessios, let A i be the e iside the deploymet egio Ω coveed by ll tsmittig diss of ith multicst. The obviously, λ i A i W Ω. Assume tht P A i. Thus, it is ot difficult to pove the followig lemm: LEMMA 5. Fo y opetio O, such s multicst, let A be the e of the egio defied by uitig the tsmissio egios of ll tsmittig odes. If A is t lest with high pobbility, the, w.h.p., the ggegted cpcity fo this opetio i dom etwo deployed i egio with e Φ is t most Φ W. The dt-copies gumet wos s follows. Whe we multicst fom oe souce ode v i to ll its eceives U i, it is moe liely tht othe odes will lso get copy of the dt. Hee, fo the pupose of lysis, whe ode v seds dt to oe of its eighboig odes, ll its eighboig odes will be chged copy of the dt. Notice tht hee eighboig ode w my ot be the iteded eceive. Howeve, sice whe v is tsmittig, y of its eighboig ode w cot eceive dt simulteously fom y othe tsmittig ode due to itefeece, we will sy tht ode w lso gets copy of the dt. Fo multicst with eceives, clely, t lest odes will get copy of the dt. Geelly, ssume tht C i odes will get copy of the dt whe the eceives e domly selected fo ech possible souce ode v i. Obviously, v i S λi Ci W. Futhe ssume tht Ci C lmost suely, i.e., P C i C s o goes to ifiity. The the totl multicst cpcity stisfies, lmost suely, Λ = v i S λ i W C. 5 Clely, C. Next subsectio is devoted to give bette lowe boud o C. The followig lemm is stightfowd. LEMMA 6. Fo y opetio O, such s multicst, let X be the umbe of odes tht will eceive copy of the dt i.e., fll iside the itefeece egio of y oe of its tsmittig odes. Assume tht X is t lest N with high pobbility. The, with high pobbility, the ggegted cpcity fo this opetio by ll odes i dom etwo of odes is t most W N. I ou poofs, we will utilize these two techicl lemms to give uppe-boud o the cpcity of dom etwo fo opetio tht will be pefomed by ech ode of the etwo, such s multicst. Notice tht the bove lemms equie us to fid lgest such tht P A, o the lgest N such tht P X N. I some cses, such my be much smlle th the me vlue EA of A; such N my be much smlle th the me vlue EX of X. I these cses, we could ely o much stoge techicl lemms bsed o lw of lge umbes whe the umbe s of opetios eeded to pefom goes to ifiity. Fo exmple, LEMMA 7. Fo s = f multicst sessios {O i i s }, whee O i hs ode v i s souce, let X i be the umbe 69

5 of odes tht will eceive copy of the dt set by some tsmittig odes i O i. Assume tht vibles X i e idepedet, V X i = σ σ, d EX i = N fo ll i. If lim s = 0 the, w.h.p., the pe-ode multicst cpcity fo this opetio i dom etwo of odes is t most O W. s N PROOF. Let λ be the pe-ode cpcity chievble. The λ s i= Xi W. Let X = s i= X i s d σ = VX i. The the lw of lge umbes Lemm implies tht, fo y ɛ > 0, P X N < ɛ σ σ s ɛ. Thus, s log s lim s = 0, we hve X N ɛ lmost suely. Thus, we hve λ W W s = O N ɛ s lmost suely. N To ou supise, we fid tht the multicst cpcity of dom etwo whee ech multicst sessio hs eceives hs two egimes: whe the umbe of eceives is ove some theshold, multicst cpcity is symptoticlly sme s the bodcst cpcity; othewise, the multicst cpcity deceses liely ove. Next, we will povide uppe-bouds fo ech cse septely. 3.3 Whe = O / We fist study the multicst cpcity whe the umbe of eceives is t most O /. We will peset uppe boud of the totl multicst cpcity. A tivil uppe boud fo totl multicst cpcity is W sice thee e souce odes d ech souce ode c oly sed W bits/sec. A efied uppe boud is W which is deived fom the pespective of ecipiets: ech ode c eceive t most W bits/sec, d mog eceived dt by ll odes, y dt fom y souce ode will hve t lest copies oe copy t ech of the eceives d oe copy t the souce ode. Fom Lemm 6, fo multicst tee T i spig souce ode v i d eceives U i, we would lie to ow the expected, o symptotic lowe boud o umbe of itel odes used i T i. To lyze this vlue, we fist study the symptotic lowe boud of the Euclide legth T i of multicst tee T i. LEMMA 8. [4] Give y odes U, y multicst tee spig these odes my be usig some dditiol ely odes will hve Euclide legth t lest ϱ EMST U, whee ϱ 3 d EMST U is the EMST spig U. Obseve tht the tight boud o ϱ = 3 is the fmous Gilbet d Poll cojectue which ws poved by Du d Hwg i 99 [4]. A boud ϱ c be esily poved s follows. Fo y steie multicst tee T spig these odes, we costuct Eule tou o this tee. Clely the totl legth of the Eule tou EC is times of the legth of the multicst tee T. O the othe hd, the Eule tou hs legth t lest tht of the Euclide miimum spig tee fo these odes. The sttemet follows fom EMST < EC = T. Recll tht i this ppe, T deotes the totl Euclide legth of ll lis i stuctue T. Bsed o Lemm 8, to get lowe boud o T i of y multicst tee T i, we eed study the legth of EMST spig these dom odes. I [9], Steele estblished the followig esult: LEMMA 9. The totl edge legth of the EMST of odes domly d uifomly distibuted i d-dimesiol cube of sidelegth is symptotic to τd d d, whee τd is costt depedig oly o the dimesio d. Thus, bsed o Lemm 8 d Lemm 9, we hve LEMMA 0. The totl edge legth, deoted by T i, of y multicst tee T i spig odes domly plced i sque of side-legth lmost suely is t lest ϱ τ, whe. Fom ow o, fo simplicity, we will deote τ 3τ/. Whe the umbe of eceives does ot go to ifiity whe, we c use the lw of lge umbes to show tht the expected vlue of s multicst tee legths T i, fo v i S is t lest τ lmost suely. Let X = EMST U, whee EMST U is the Euclide miimum spig tee of set of domly selected odes U i sque of side-legth. It ws show i [9] tht VX log. We the show tht X. LEMMA. Fo y odes U plced i sque egio with side-legth, the legth of EMST spig U is t most. PROOF. Give odes i the sque, we will use Pim s lgoithm to costuct EMST: oigilly ech ode is compoet, d the we itetively fid shotest edge to coect two compoets to fom lge compoet util oly oe compoet is left. Coside the + g-th step fo g =,,,, which hs g coected compoets s iput. Fo g, if we ptitio the sque ito g by g gid with side-legth g, the thee is t lest oe cell tht cotis t lest two coected compoets. This implies tht the shotest edge coectig compoets t the + gth step is t most. g Cosequetly, the EMST hs legth t most j= g i= i+ i. i Boud the dt copies: A stightfowd lowe-boud o the expected o symptotic lowe-boud of umbe of odes icludig lef odes eeded i multicst tee spig odes domly selected i sque of side-legth is τ with high pobbility. This boud c be deived s follows: the expected Euclide legth of multicst tee is t lest τ, d the tsmissio ge of ech ode is oly, thus, emovig oe tee edge icidet o lef ode will educe the totl edge legth by t most d we will educe the umbe of odes by. Cosequetly, we hve C τ /. Although this boud o C is much bette th boud C whe = O /, the boud c be futhe impoved bsed o the followig obsevtio. Whe odes o the multicst tee ely dt fom the souce ode to eceives, ot oly its dowstem odes of the multicst tee will eceive the dt, but lso ll its eighboig odes i commuictio gph G will get copy of the dt. We will the lyze the umbe of odes tht will get the copy of the dt. Give multicst tee T, let DT be the egio coveed by ll tsmittig diss of ll tsmittig odes itel odes i the multicst tee T. Obseve tht the lef odes do ot cotibute to DT t ll hee. See Figue fo illusttio. Clely, the e of DT, deoted by DT, is t most DT T + π / whee T is the totl Euclide legth of ll lis i T. We will pove tht the e of DT is lso t lest τ c 0 fo some costt c 0 idepedet of the etwo. LEMMA. The e of the egio DT, deoted by DT is t most τ + π / d w.h.p.is t lest τ c 0 whe < τ 6d+ ρ, fo some costt c 6d++ 0 = /4ρπ, whee 0 < ρ < d costt d 3. d+ PROOF. Fo y multicst tee T spig souce ode v i d the set of eceives U i, fo coveiece, let V T be the set of odes i tee T ; let U i = U i {v i }; let IT be ll the Steie odes used to coect them, i.e., IT = V T \ U i. Clely the commuictio gph defied o V T whee two odes e coected iff thei Euclide distce is o moe th is coected. We use G T to deote such iduced gph. We will the build othe multicst tee T fom G T to coect odes U i. 70

6 IT π 6d+ lt π τ 6d+ π ρπτ. Fo exmple, we c set ρ =. Notice tht t lest /4 d+ of ech tsmittig dis is iside the deploymet sque. Thus, the egio DT tht e iside the deploymet sque is t lest ρπτ /4. This fiishes the poof. Fo coveiece, heefte, we use Regio DT b Dese eceives Figue : Regio DT coveed by tsmittig diss of itel odes i multicst tee T. Hee the solid blc odes e eceives/souce d gy odes e Steie odes. b Ptitio of sque with side-legth ito squelets with side-legth. Hee the solid blc odes e eceives/souce. Shded squelets e squelets with t lest oe eceive. I gph G T, we build coected domitig set CDS usig method descibed i [3, ]. Souce ode v i will be dded to the CDS if it is ot i the CDS. It hs bee poved i [3, ] tht, i the costucted CDS, ech ode o the CDS hs degee bouded by costt, sy d. Fo exmple, it c be show tht the degee of ode i CDS is bouded by 3 if the method peseted i [] is used. The multicst tee T is the simple bedth-fist-sech tee computed fom the CDS, ooted t the souce ode v i. We essetilly will pove tht ech poit fom the egio DT is coveed by t most costt c 0 umbe of diss fom the multicst tee T. Fo ech poit p i the egio, we divide the dis Dp, ceteed t poit p with dius ito 6 equl sized sectos. Thus, y pi of odes fllig ito the sme secto will be withi distce of of ech othe, d thus coected i the oigil commuictio gph. Cosequetly, fo ech poit p, the umbe of diss fom DT tht cove p is t most 6d +. If it is t lest 6d + +, the t lest oe of the sectos will hve t lest d + odes, which implies tht y ode i tht secto will hve degee t lest d + i the iduced CDS gph. This is cotdictio to the fct tht the degee of iduced CDS is bouded by d. Thus, the e of the egio DT is thus t lest IT π, whee d is the 6d+ degee boud o the iduced CDS gph costucted. Hee IT is the umbe of itel odes i multicst tee T. Notice tht some lef odes i T my become itel odes i T ; some itel odes my ot be used by tee T t ll. Let AT be the egio coveed by ll diss ceteed t ll odes of tee T, icludig the lef odes. Let lt be the umbe of lef odes i tee T. Obviously, DT + lt π AT AT DT. Thus, DT DT lt π IT π lt π. 6d + Obviously, lt. Fo multicst tee T, thee e t most lef odes. If we emove ll edges i T icidet o lef odes, the totl edge legth of ll edges left is t lest T. Thus, the umbe of itel odes IT i T is t lest T. Notice tht T is tee spig the souce ode v i d ll eceives U i. Thus, with high pobbility, T τ sice U i hs odes. Thus, with high pobbility, we hve IT τ. Assume tht < τ 6d+ ρ, which implies DT is t lest 6d++ θ = τ 6d + ρ 6d to deote the theshold vlue such tht Lemm is tue if < θ. Bsed o Lemm, we ow tht the expected umbe, deoted by C, of odes fom V tht is i the egio DT is t lest DT τ = τ c 0 c 0 Recll tht we ssume tht thee is oly oe sigle chel i the etwo. It is the ot difficult to show the followig lemm: LEMMA 3. With high pobbility, the umbe C of odes tht get copy of the multicst dt stisfies C > τ c 0. PROOF. Coside multicst tee T. Notice tht wieless odes will be domly distibuted i sque egio of side-legth. Let X i = {0, } be idicto vible whethe the ith ode v i will fll iside the egio DT fo multicst tee T. Clely P X i = =. Recll tht, we ledy poved tht, with DT high pobbility, DT τ. Thus, we hve P X i = c 0 τ. Obviously, X = c 0 i= X i is the expected umbe of odes fllig iside the egio DT, which is lso the umbe C of odes tht will get copy of the dt by multicst. The the expected vlue EX τ. Bsed o Lemm 3, we hve DT P C c 0 DT DT DT 4. e = e Notice tht to gutee coected etwo with high pobbility, we hve < π with high pobbility. Thus, P C DT e τ c 0 e τ πc 0 = DT Cosequetly, whe, P C P C > τ c 0 This fiishes the poof. P C > Cosequetly, we hve the followig theoem: τ πc 0 0. Thus, DT. THEOREM 4. The multicst cpcity with eceives fo odes tht e domly d uifomly deployed i sque with side-legth is t most c W fo some costt c whe < θ /. PROOF. Notice tht the multicst cpcity is t most W d, C with high pobbility, C τ c 0 whe < θ /. Thus, the multicst cpcity Λ is t most W c 0 = c W τ fo costt c = c 0. This fiishes the poof. τ 7

7 Recll tht we hve poved tht, to gutee tht we hve coected etwo with high pobbility, we eed π +β fo β. Thus, lettig c = c π, we hve the followig theoy: THEOREM 5. The multicst cpcity fo dom etwo of odes, whe < θ / = O, is t most Λ c W = O W. With s multicst sessios, the pe ode multicst cpcity is λ = miw, Λ = OmiW, W. s s Notice tht Theoem 4 ws poved ude the ssumptio tht. Whe this is ot the cse, we c pove tht the pe-ode multicst cpcity λ whe ech souce ode geetes multicst dt t te λ lso stisfies tht λ = O s W whe s. Sice is costt i this cse, we ow tht the peode multicst cpcity is uppe-bouded by the pe-ode uicst cpcity with s uicst sessios. Thus, the pe-ode multicst cpcity is lmost suely t most O s W, which is sme s O s W sice is costt. 3.4 Whe = Ω I the pevious subsectio, we showed uppe boud of the multicst cpcity whe < θ /. I this subsectio we will peset uppe boud o multicst cpcity whe θ /. We will essetilly show tht i this cse, multicst is symptoticlly equivlet to bodcst. Bodcst cpcity of sigle-souce of bity etwo hs bee studied i [, 0]. I this ppe, we will pove tht the chievble itegted multicst cpcity is oly ΘW if bity subset of the odes will seve s eceives fo ech possible souce ode v i. We ptitio the sque of side-legth ito squelets, ech with side legth. The sque will be ptitioed ito M = / squelets, sy B, B,, B M. Recll tht we will domly select θ / eceives i the sque egio. See Figue b fo illusttio. LEMMA 6. With high pobbility, t lest ρ M squelets will hve t lest oe eceive whe θ / fo costt θ. PROOF. Let X be the umbe of squelets tht do ot hve y eceives iside, d A be fixed fctio of squelets, sy A = ρ M fo costt 0 < ρ <. Let X i be idicto vible whethe squelet B i is empty of eceives X i = is empty. The X = M i= Xi. Notice VX = V M i= Xi = M M i= j= CovXi, Xj, whee CovXi, Xj = EXi Xj EX iex j is the covice of vible X i d X j. We the compute such CovX i, X j fo ll possible pis of i d j: CovX i, X i miimum = coected domitig set MCDS of dom etwo. EX i EX i d EX i = M ; d EX i X j = Fo ech domly geeted etwo istce, we will fist costuct ppoximtio of MCDS. Fo exmple, we c use the M if i j. Cosequetly, we hve VX = MM [ M M ] + M[ M method itoduced i [3] Algoithm i [3]. This method fist M ]. Sice [ M M + M ] 0, we hve VX M[ M M ]. Fom Lemm, we hve P X EX ρ M M[ M M ] ρ M Fom θ M, EX = M M M e θ. Thus, P X e θ + ρ M e θ e θ ρ M Whe M, the pobbility goes to zeo. We c lso show tht, with high pobbility, thee is t most costt fctio of squelets tht will be empty of eceives. This fiishes the poof. We the pove tht the uio of the tsmissio diss of these odes i multicst will cove t lest costt fctio, sy 0 < ρ, of the deploymet egio. LEMMA 7. The uio of the tsmissio diss of these odes eceives d souce ode i multicst will cove t lest costt fctio, sy 0 < ρ, of the deploymet egio. PROOF. Bsed o lemm 6, we ow tht mog M squelets ptitioed fom the deploymet egio, thee e t lest ρ M squelets, ech of which cotis t lest oe eceive o souce ode iside. I ech such squelet B j, thee is t lest oe eceive d thus t lest oe tsmittig ode i the multicst tee tht coves this eceive. The tsmittig ode must lie iside this squelet o 8 djcet squelets. O the othe hd, ech tsmittig dis c cove eceives fom t most 9 squelets. Cosequetly, we must hve t lest ρ M/9 tsmittig diss to cove eceives fom ρ M squelets. Recll tht the squelet side-legth is, which implies tht ech poit i the deploymet egio is coveed by t most 9 such epesettive tsmissio diss. Cosequetly, the totl e coveed by these epesettive tsmissio diss is t lest ρ M π /8. Recll tht the deploymet egio hs e d M = /. Thus, the e of ll tsmissio diss of ll these odes is t lest ρ = ρ π fctio of the totl 8 e of the deploymet egio. This fiishes the poof. Bsed o Lemm 7 d Lemm 5, we hve THEOREM 8. Whe θ / fo costt θ, with high pobbility, Λ W ρ = W ρ = OW,x whee ρ is costt depedig oly o θ. Notice tht fo bodcst, it hs bee poved i [, 0] tht the bodcst cpcity is oly ΘW. Hee we essetilly pove tht fo multicst, whe the umbe of eceives is lge eough t lest Ω, the symptotic multicst cpcity is lso oly OW. 4. LOWER BOUNDS ON MULTICAST CA- PACITY WITH RANDOM NETWORKS I this sectio, we will povide multicst scheme d pove tht the multicst cpcity chieved by ou scheme mtches the symptotic uppe bouds. 4. Good Appoximtio of MCDS Ou multicst scheme is bsed o good ppoximtio of fids mximl idepedet set usig geedy ppoch, d the coects them usig ely odes. A mximl idepedet set MIS c be costucted s follows: oigilly ll odes e med s white odes d ech ode v is ssiged uique v; ode v is selected to MIS d med s blc ode cosequetly if d oly if it hs the smllest mog ll its white eighbos; Cle such MIS is domitig set. The, we coect evey pi of domitig odes tht e t most 3 hops wy usig the lest-hop pth. The odes o such lest-hop pths will be med s coectos. The set of ll MIS odes d coectos will fom 7

8 coected domitig set [3]. We the show tht we c schedule the tsmissios of ll odes i CDS i costt time-slots without itefeece; the CDS is legth spe. TDMA Schedulig with Costt Slots: Fo ech ode o the CDS, thee e t most costt umbe of eighboig odes o CDS. I othe wods, the degee of y ode o CDS is bouded by costt, sy D. Usig this popety, it is esy to show tht the umbe of odes i the CDS costucted bove tht c itefee with y ode i the CDS is t most costt. Fo ech ode v, coside two cicles both ceteed t the ode d with dii R d R +. Coside ode u whose tsmissio will itefee with the tsmissio of ode v. Clely ode u will be completely iside the dis ceteed t v with dius R +. O the othe hd we ow ech cicle ceteed t u with dius cotis t most D + odes. Let be the mximum umbe of odes i CDS whose tsmissio will itefee with the tsmissio of ode v i CDS. Usig the e gumet, we c show tht π R + D + = + R π D +. This popety esues tht we c schedule the tsmissios of ll odes i CDS by TDMA me such tht ll odes will be ble to tsmit t lest oce i evey + time slots. Cosequetly, the pe-ode dt tsmissio dt te chieved by odes o the CDS is t lest W +. Notice tht hee is costt. Legth Spe Popety: Fo y two odes u d v i the etwo, if u v >, the the shotest pth coectig u d v vi the CDS costucted bove hs legth t most 3 times the legth of the shotest pth coectig them i the oigil dom commuictio gph G = V, E see Lemm 5 of [3]. Notice tht hee whe u o v o both is ot i the CDS, we will fist coect u o v o both to oe of its domitos sy u d v i the CDS. The we fid the shotest pth coectig these coespodig domitos u d v i the CDS. 4. Whe = O / u w w Squelets ptitio v b Mhtt Routig Tee Figue : Ptitio deploymet sque ito squelets with side-legth / 5. Fo edge uv EMST U, fid ode w eithe ode w o ode w which hs sme ow s u d sme colum s v to coect them. b multicst tee costucted usig Mhtt ppoch, whee dotted lies deote oigil EMST of odes i multicst sessio. Whe the umbe of eceives, plus the souce ode, is oly O, we will costuct multicst tee fom CDS. Coside istce of dom etwo G = V, E d lso istce of multicst with v s the souce ode d U = {v, v 3, v } s the eceive odes. Let U = {v, v, v 3, v }. We will costuct multicst stuctue s follows: Algoithm Multicst Cpcity Achievig Mhtt Routig Bsed o Squelet fo Nodes U : We ptitio the deploymet sque ito squelets, ech with side legth / 5 s i [8], see Figue b fo illusttio. Thus, we hve squelets. Ech squelet is deoted / 5 by i, j whe it is the ith colum d jth ow. : We build the Euclide miimum spig tee, deoted s EMST U, coectig odes i U, usig followig method lso descibed i Lemm : Oigilly, odes U fom compoets; fo the gth step, whee g =,,,, ptitio the deploymet sque ito t most g sque-shped-cells, ech with side legth g ; 3 fid cell tht cotis two odes of U tht e fom diffeet coected compoets d the coect them usig Mhtt outig; mege these two coected compoets. 3: Fo ech li uv i the tee EMST U, ssume tht u d v e iside squelet i u, j u d squelet i v, j v espectively. Fid ode w i squelet i v, j u o squelet i u, j v, i.e., uwv is Mhtt pth coectig u d v. We fid the shotest pth with miimum Euclide legth coectig uw d wv vi the CDS costucted peviously. 4: The esulted stuctue by uitig ll such shotest pths fo ll lis i EMST U will seve s multicst. Notice tht hee such stuctue my ot be tee. If this is the cse, we could emove the cycles tht do ot coti odes fom U. Deote the esulted tee s MT U. Usig Mhtt pth to coect odes is to void the hot-spot e i the cete of egio poduced by diected shotest pth outig, vi id of lod blcig. Due to such pseudo-lod-blcig ppoch, we c lte show tht the lod totl dt tes of ll outig equests o ech ode o CDS equivletly ech squelet is t most fctio of W lmost suely. To show tht the bove outig chieves the symptoticlly optimum multicst cpcity, we eed show tht the totl umbe of dt copies of multicst bit is t most O, which will be deived bsed o the uppe boud o the e coveed by ll tsmis- sio diss i the multicst tee MT U. We will pove tht, with high pobbility, the tee MT U hs Euclide legth t most costt fcto of the Euclide legth of tee EMST U. THEOREM 9. With high pobbility, the totl Euclide legth of the multicst tee MT U is withi costt c 5 fcto of EMST U, i.e., MT U c 5 EMST U. PROOF. Fo odes U i multicst sessio with souce ode v, we will fist costuct the Euclide miimum spig tee EMST U. The fo ech edge uv i EMST U, we will eplce it with Mhtt pth uwv Hee ode w is i the sme colum with v d sme ow with u i the squelet ptitio. The esultig stuctue is cll Mhtt tee MHU. The fo ech edge uv i MHU, we fid the shotest pth, deoted s P Gu, v, coectig them i the oigil commuictio gph G. Hee we deote the fil outig stuctue s ST U. We will pove tht the totl legth of ST U is costt fcto of the totl legth of EMST U, with high pobbility. Fo evey edge uv i the Mhtt outig stuctue, we descibe method to fid outig pth coectig them. Whe u v, o dditiol ode is eeded. Othewise, we will ty to fid ode w stisfyig the followig coditios. w u σ, fo some smll costt σ > 0. 73

9 . Node w is close to the diectio of uv, i.e., wuv ϕ fo some pe-defied gle ϕ < π/3. We c show tht ode w mes o-egligible pogess, i.e., v u v w ϱ fo some costt ϱ > 0. Afte ode w is foud, we the ecusively fid odes to coect w d v. Obviously, the e of the egio A to select w is ϕ σ d the e of the deploymet egio is. Let l be the umbe of ely odes we eed to fid to ech ode v fom u usig the bove ppoch. The the ovell pobbility tht t lest oe such itemedite ode w used to coect u d v is empty is t most ϕ σ / l. The the pobbility tht evey time we c fid ely ode is t lest l. We deote the ϕ σ / fist ely ode foud by ode u s w. The u v v w is t lest + σ σ cos ϕ. By iductio, the umbe of u v ely odes coectig u d v is t most l +σ. σ cos ϕ Thus, the pobbility tht we c coect u d v usig the bove ppoch is t lest ϕ σ / u v +σ. σ cos ϕ Let χ = +σ. σ cos ϕ The pobbility we c fid sequece of odes w fo ll edges i MHU bsed o pevious ppoch is t lest [ ϕ σ EMST U ] / + σ σ cos ϕ. Recll tht, with high pobbility, we hve EMST U τ. The, whe c, the bove pobbility is t lest l 3/ ϕ σ c l l c χ τ ϕ σ c 3 c 3/ χ θ τ. The iequlity comes fom the fct tht θ /. To esue tht the bove pobbility goes to s, it is sufficiet d ecessy to equie tht ϕ σ c We c esily fid such ϕ, σ d c to me it stisfible. Notice tht hee we eed 0 < c < 4π/9. Thus, with high pobbility, the totl legth of ST U hs Euclide legth t most EMST U +σ σ cos ϕ. Becuse fo y two odes u d v i the etwo, if u v >, the the shotest pth coectig u d v vi the CDS hs legth t most c 4 5 times the legth of the shotest pth coectig them i the oigil dom commuictio gph G = V, E, see [3] fo poofs. Thus, with high pobbility, the totl legth of ST U hs Euclide legth t most c 4 EMST U +σ. σ cos ϕ Recll tht fo CDS costucted peviously, we ow tht the shotest Euclide pth coectig u d v vi CDS is oly t most 3 times of the legth of the shotest pth coectig u d v i the etwo G = V, E. Thus, the multicst tee MT U bsed o CDS hs Euclide legth t most 3 MT U. The theoem follows by lettig c 5 = 3 c 4 +σ. σ cos ϕ Cosequetly, we hve Euclide legth MT U < c 5η with high pobbility sice EMST U η fo η = see Lemm. Hee we deote the egio coveed by ll tsmissio diss of ll itel odes i MT U s DT d the umbe of odes lyig i DT s C. We the show tht with high pobbility, the multicst cpcity chieved usig bove outig ppoch is withi costt fcto of the symptotic optimum. We essetilly show tht, with high pobbility, the umbe C of odes tht will eceive copy of the multicst dt is withi EC. THEOREM 0. The totl multicst cpcity Λ chievble W by ll multicst flows is t lest c 6, whe θ d / fo some costt c 0, 4π/9. Hee c6 is costt. c l PROOF. Coside set of eceives U fo souce ode v. Let tee T be the multicst tee MT U costucted bove. Let X i {0, } be idicto vible whethe the ith ode v i will fll iside the egio DT fo multicst tee T. Clely p = P X i = = DT. Notice tht the e of DT is t most T +π /, d edge legth T c 5 η with high pobbility. Obviously, X = i= X i is the umbe of odes fllig iside the egio DT, d X is biomil distibutio. Usig Lemm 3, = P C > DT DT DT DT [ ] DT DT c 0 τ = c 0 τ c 0 c l τ DT DT The lst iequlity comes fom the ssumptio tht / c. l The secod to lst iequlity comes fom Lemm tht DT τ c 0. Cosequetly, P C DT l c 0 c τ. Thus the umbe of odes tht c get copy of the dt fo multicst withi odes U, with high pobbility, is t most DT c 5η 4 +π 4c 5η+π θ, The lst iequlity comes fom θ. Recll tht, by pefomig multicst bsed o CDS stuctue, we c gutee tht ech ode will be ble to tsmit oce evey + time-slots. This implies tht the totl bits/sec chieved by ll odes is t lest W/ +. Cosequetly, the multicst cpcity is t lest W/ + 4c 5 η+π θ This fiishes the poof by settig c 6 = 4c 5 η+π W θ +. 4c 5 η+π. θ + Obseve tht the coectess of Theoem 0 elies o the fct tht c d l θ /. Hee costt 0 < c < 4π/9. Cosequetly, by lettig = c fo 0 < c < 4π/9, d l c = c 6 c, bsed o Theoem 0, we hve = COROLLARY. The multicst cpcity fo dom etwo of odes, whe < θ /, is t lest Λ c W = Ω W. The multicst cpcity pe ode with souces is λ = Λ = Ω W. Obseve tht the coectess of Theoem 0 equies tht the lod of evey outig ode is o moe th costt fcto of W bits/sec, due to the equiemet of TDMA ode schedulig. Ufotutely, it is uow ow whethe we c pove whethe such coditio is stisfied with high pobbility. We will pove wee cpcity lowe boud: with high pobbility, the tffic lod o y outig ode d ll itefeig odes is o moe th OW. 74

10 Give squelet, we defie its lod s the totl umbe of multicst sessios tht will be outed though odes iside this squelet. We show tht ude ou outig lgoithm, fo y squelet, with high pobbility, its lod is o moe th Θ. To pove ou clim, we fist study simple uicst cse. Coside gid of L L squelets. Coside specific squelet s tht is of ith ow d jth colum i the squelet-gid. Rdomly pic two odes u d v fom the gid d coect them vi Mhtt outig. Let p sl deote the pobbility tht the Mhtt outig will use odes fom the squelet s. The p sl = i L i + + j L j +. 8 L L L L j L j+ Hee i L i+ esp. is the pobbility tht L L L L squelet s is used whe u esp. vis o the sme ow esp. colum s s. It is esy to show tht p L sl. L Let us ow study the umbe of times tht specific squelet s is used by ou outig stuctue fo multicst. Recll tht we will costuct the Euclide miimum spig tee s the method descibed i Lemm d the fid multicst outig stuctue s Algoithm. Fo give multicst sessio, this squelet s my be used i y oe of the steps to build the spig tee. Fo step g with g, ecll tht we will ptitio the sque with side-legth ito g g cells, ech with side-legth. Fom pigeohole piciple, thee exists cell g tht cotis two odes, sy u d v, fom two diffeet coected compoets. We will coect them d mege these two coected compoets. Hee we will coect u d v usig Mhtt outig s illustted i Figue. Let X s,g be the idicto whethe the specific squelet s is used i this gth step. Clely, whee P X s,g = = g p s g s is used, d p s g /, 9 5 is the pobbility tht the cell cotiig squelet g / 5 is the pobbility tht s is used whe tht cell cotiig s is used. Hee g / is the umbe of 5 squelets pe ow i cell, i.e., the vlue of L i fomul 8. Thus, p = P X s = P X s,g = = g= g= g p g s / Notice tht, to chieve lge multicst cpcity, we will set = c fo some costt 0 < c < 4π/9 see poof of Theoem l 9. Thus, P X s = 4 0 l. The we hve 5 c LEMMA. Give s multicst sessios, the expected umbe of multicst outig flows tht use specific squelet s is t most l c s. Whe s =, it is t most l. c Recll tht ll multicst sessios will domly select its eceives. Let µ = l. Usig Lemm 3, we c c show tht with pobbility t lest p = O µ the umbe of multicst flows outig though squelet is t most µ. Thus, we hve the followig theoem THEOREM 3. With pobbility t lest O, the umbe of multicst flows tht pss though squelet is t most µ = l = c c. Thus, by lettig λ i = c W, i, fo some pope costt 0 < c <, the with high pobbility, the totl tffic lod t y specific squelet is t most c c W bits/sec. Sice the size of the squelet is / 5 d the itefeece ge R = Θ, we c show tht the totl flow equiemets of ll squelets tht could cuse itefeece to give squelet is lso t most c c W with high pobbility. Hee = Θ is the umbe of squelets withi distce R fom this squelet. Thus, by choosig c c, we c schedule flows t this squelet with high pobbility t lest O. Recll tht w.h.p., flow will pss though O / = O / squelets. The the pobbility tht give flow c be scheduled vi TDMA is t O / lest O = O. 4.3 Whe = Ω I this cse, we hve poved tht the uppe boud o the totl multicst cpcity is oly ΘW. Obviously, the totl multicst cpcity is t lest the lowe boud of the cpcity fo bodcst. I [], they peset bodcst scheme to chieve cpcity ΘW. Thus, we hve the followig theoem THEOREM 4. The totl multicst cpcity Λ chievble by ll multicst flows is t lest c 7W whe = Ω /, whee c 7 = d costt is the mximum umbe of CDS odes + tht e withi itefeece ge R of ode. 5. OTHER MULTICASTS 5. Cpcity Boud fo Goup Multicst I pevious sectios we hve studied the symptotic multicst cpcity by ssumig tht we domly select eceives fo ech multicst sessio. I this sectio, we study the multicst cpcity of so-clled -goup multicst: fo ech souce ode v i, thee e goups of eceives g i,, g i,,, g i,. The eceives i ech goup g i,j e coveed by dis with dius δ fo costt δ d ceteed t oe of the eceives i the goup. We ssume tht the cete ode i ech goup is domly selected. The umbe of odes i ech goup could be bity. Fo simplicity, let ode z i,j be the cete ode of goup g i,j. We the study the multicst cpcity fo goup-multicst whe ech ode v i will hve domly selected goups d it wts to sed dt with te λ i to ll eceives i these goups. As the cse whe ech goup hs oly oe ode, whe θ /, it is esy to pove tht the cpcity fo goup-multicst is t most W/ϱ s Theoem 8. Clely, simple bodcst bsed o the coectig domitig set costucted peviously will lso W chieve cpcity fo goup-multicst t lest. Cosequetly, + we hve THEOREM 5. Fo goup-multicst, whe θ / fo y costt θ > 0, the cpcity of goup-multicst is t most W/ϱ d t lest W +. Fist, fo goup-multicst, multicst tee hs to ech the cete ode z i,j of ech goup g i,j. The fom Theoem 4, the cpcity of goup-multicst is t most c W with high pobbility whe < θ /. We the show how to desig multi- cst outig fo the goup-multicst poblem: we fist pply ou multicst scheme fo tditiol multicst whe odes z i,j, j, e eceives fo souce ode v i. We the let ode z i,j multicst loclly to ll eceives i the goup g i,j. We ledy poved i Theoem 9, the totl legth of the multicst tee to sp 75

11 these domly selected odes z i,j is t most c 5 EMST U i with high pobbility. Recll tht EMST U i 3τ with high pobbility. Notice tht the totl e coveed by tsmittig diss of ely odes used fo elyig dt fom ech z i,j to eceives i its goup is t most πδ +. The, the e coveed by ll tsmittig diss fo multicst sessio is t most MT U + πδ +, which is w.h.p.t most 3τ c 5 + πδ + θ πδ + + 3c 5τ. The lst iequlity comes fom the fct tht θ Fo coveiece, let c 8 = θ πδ + + 3c 5τ. The simil to Theoem 0, we c pove tht the umbe of odes tht will get copy of the dt fom oe multicst sessio is t most c 8 = c 8. Thus, we hve THEOREM 6. Whe θ /, the ggegted multicst cpcity fo goup-multicst with goups is t most c W, W d is t lest c 9, with high pobbility. Hee costt c 9 = c Bouds fo Abity Netwos I pevious studies we cocetted o the multicst cpcity fo dom etwos whe odes will be domly plced i the deploymet egio. I this sectio we will study wht is the symptotic mximum multicst cpcity tht c be chieved by specific coected etwo whe odes positio c be cefully selected. We fist peset costuctive lowe boud o the multicst cpcity. Assume tht odes e deployed i by gid, ech cell hs side-legth, i.e. the side-legth of the sque is =. The eceives e domly selected fom the gid poits. We the pefom multicst s befoe: the multicst tee is costucted bsed o the Euclide miimum spig tee coectig souce ode d eceives. Let L be the totl legth of the Euclide MST costucted bove. Simil to pevious studies i Subsectio 4., we ow tht the multicst cpcity Λ stisfies Λ c 6 W L fo some costt c 6 depedig oly o R/. Lemm gives uppe boud o L fo the EMST. Thus, COROLLARY 7. The multicst cpcity Λ fo bity etwo we c choose ode positios, is t lest W. 6. LITERATURE REVIEWS c 6 Netwo cpcity hs bee extesively studied ecetly. Fo give sttisticl desciptio of the etwo, set of costits such s powe pe ode, li cpcity, etc., d list of desied commuictio pis, the cpcity egio is the closue of ll te tuples tht c be chieved simulteously. Hee te tuple specifies the te fo ech of the desied commuictios. Kysu d Vidy [3] studied the cpcity egio o give multi-hop multi-dio multi-chel wieless etwos whe thee e totl c chels vilble d ech ode hs m wieless itefces with m c. O the othe spect, sevel ppes [, ] ecetly studied how to stisfy ceti tffic demd vecto fom ll wieless odes by joit outig, li schedulig, d chel ssigmet ude ceti wieless itefeece models. Gupt d Kum [7] studied the symptotic uicst cpcity of dom multi-hop wieless etwos fo two diffeet models. Whe ech wieless ode is cpble of tsmittig t W bits pe secod usig costt tsmissio ge, the thoughput obtible by ech ode fo domly chose destitio is W Θ bits pe secod ude the potocol-itefeece model, whee i umbe of odes. If odes e optimlly ssiged d. tsmissio ge is optimlly chose, eve ude optiml cicumstces, the thoughput is oly Θ W bits pe secod fo ech ode. Simil esults lso hold fo physicl itefeece model. Gossgluse d Tse [6] ecetly showed tht mobility ctully c help to impove the uicst cpcity if we llow bity lge dely. Thei mi esult shows tht the vege log-tem thoughput pe souce-destitio pi c be ept costt eve s the umbe of odes pe uit e iceses with the id of mobility d dymic powe djustmet. Notice tht this is i shp cotst to the fixed etwo sceio whe odes e sttic fte dom deploymet. The mi ide used i [6] is to use some itemedite ode to seve s fey ode: this ode will cy the dt fom the souce ode d move oud d it will dump the dt to the tget ode whe it is withi its commuictio ge. I othe wods, essetilly, the esult peseted i [6] still obey the cpcity boud poposed i [7]: the cpcity is impoved becuse the vege distce L pcet hs to be tsmitted is educed fom Θ i [7] to Θ i [6]. I summy, fo dom etwos, ude the potocol model, the chievble pe-ode thoughput cpcity λ W d the vege tvel distce L stisfies λ L Θ. Simil pheomeo hs lso bee obseved i [4]. Bodcst cpcity of bity etwo hs bee studied i [, 0]. They essetilly showed tht the bodcst cpcity of give etwo is ΘW fo sigle souce bodcst d the chievble bodcst cpcity pe ode is oly ΘW/ if ech of the odes will seve s souce ode. The uppe boud ΘW o bodcst cpcity tivilly holds sice ech ode c eceive t most W bits/sec. The cpcity ΘW is chieved by costuctig coected domitig set i which we c schedule evey ode i CDS to tsmit t lest oce i costt time slots. This cpcity bouds lso pply to dom etwos. Keshvz-Hddd et l. [0] studied the bodcst cpcity with dymic powe djustmet fo physicl itefeece model. Multicst cpcity ws ot fully studied i the litetue. Jcquet d Rodolis [9] studied the sclig popeties of multicst fo dom wieless etwos. They essetilly studied the omlized multicst cost, which is defied s the tio of the umbe of lis i the multicst tee ove the vege oute legth fom dom souce i the multicst goup to dom destitio i the multicst goup. They biefly showed tht the mximum te t which W ode c tsmit multicst dt is O. Tee e simil effots peseted i this ye s MobiHoc [8]; howeve, ou esults subsume the othes s they e specil cse of ous. Shotti et l. [8] studied the multicst cpcity of dom etwos whe the umbe of multicst souces is ɛ fo some ɛ > 0, d the umbe of eceives pe multicst flow is ɛ. They ssume the sme potocol itefeece model d use the dese dom etwo model. They show tht the sum of the souce tes Λ tht the etwo c suppot is O ɛ w.h.p., with pe flow thoughput cpcity of O ɛ w.h.p.. This esult c be implied by ou esults usig s = ɛ d = ɛ. To chieve the uppe boud, they popose ovel outig chitectue, clled the multicst comb, to tsfe multicst dt i the etwo. They lso show tht with high pobbility, the lod of evey squelet is withi smll fctio of the dt te. I [7], the cpcity of wieless etwos e solved ude umbe of ssumptios, mog them poit-to-poit codig which excludes fo exmple the multi-ccess d bodcst codes. Gstp d Vetteli [5] d Liu et l. [5] studied the cpcity of wieless etwos whe etwo codig c be used to impove the cpcity. Gstp d Vetteli demostted the powe of etwo codig: ude the poit-to-poit codig ssumptio cosideed i [7], 76