Network Coding Does Not Change The Multicast Throughput Order of Wireless Ad Hoc Networks

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1 Network Codig Does Not Chage The Multicast Throughput Order of Wireless Ad Hoc Networks Shirish Karade, Zheg Wag, Hamid R. Sadjadpour, J.J. Garcia-Lua-Aceves Departmet of Electrical Egieerig ad Computer Egieerig Uiversity of Califoria, Sata Cruz, 56 High Street, Sata Cruz, CA 95064, USA Palo Alto Research Ceter PARC), 3333 Coyote Hill Road, Palo Alto, CA 94304, USA {karades,wzgold, hamid, Abstract We demostrate that the gai attaied by etwork codig NC) o the multicast capacity of radom wireless ad hoc etworks is bouded by a costat factor. We cosider a etwork with odes distributed uiformly i a uit square, with each ode actig as a source for idepedet iformatio to be set to a multicast group cosistig of m radomly chose destiatios. We show that, uder the protocol model, the persessio capacity i the ) presece of arbitrary NC has a tight boud of Θ whe m = O ) ad Θ ) whe mlog) log) m =Ω ). Our result follows from the fact that prior work log) has show that the same order bouds are achievable with pure routig based oly o traditioal store-ad-forward methods. I. INTRODUCTION The cocept of etwork codig was first explored by Yeug et. al. [] ad subsequetly geeralized by Ahlswede et. al. [2] for a sigle source multicast i arbitrary directed graphs. Sice the, the iterest i etwork codig has icreased rapidly. A large umber of studies have ivestigated the utility of etwork codig NC) for wireless etworks, ad widely cited experimets [3], [4] have bee reported i which NC has bee used successfully i combiatio with other mechaisms to attai large throughput gais compared to approaches based o covetioal protocol stacks. These results have led may to believe that a combiatio of NC with wireless broadcastig ca lead to sigificat improvemets i the order throughput of wireless etworks. Uderstadably, there is sigificat iterest i idetifyig the true impact of NC o the throughput order of wireless etworks. However, the exact characterizatio of etwork capacity with NC i the presece of multiple access iterferece is a very hard problem, eve for simple etworks, ad limited results have bee reported to date o the subject. Recet work [5] [7] has show that the throughput gai due to the use of NC i a wireless etwork is bouded by a costat whe the traffic i the etwork cosists of multiple uicast sessios. However, the motivatio for the origial work by Ahlswede et. al [2] was improvig etwork performace for multicastig, ot uicastig. Furthermore, may commercial ad defese applicatios, such as coferecig, require multicastig of large amouts of iformatio, ad the study of the multicast capacity of wireless ad hoc etworks is a importat research topic i its ow right. Several works [8] [] have studied the multicast ad broadcast capacity of wireless etworks uder covetioal routig, ad these results show cosistetly that broadcastig ad multicastig sigificatly alter the throughput order of wireless etworks. I light of these fidigs, the importace of multicastig ad broadcastig, ad recet practical results o NC, it is atural to iquire whether the itroductio of NC ca improve the throughput capacity of multi-pair multicastig. I this work, we udertake the characterizatio of the multicast ad broadcast throughput order of wireless ad-hoc etworks i presece of etwork codig, which had remaied a ope problem for the past 0 years. We cosider a etwork cosistig of odes distributed radomly i the etwork space, with each ode actig as source for m radomly chose odes i the etwork. We make two key cotributios. First, we show that i the presece of arbitrary NC, the per-sessio multicast capacity of radom wireless ad hoc etwork uder ) the protocol model has a tight boud of Θ whe m = O mlog) log) ) ad Θ ) whe m =Ω log) ). Secod, we show that, i the presece of arbitrary NC, the per-sessio multicast capacity of radom wireless ad hoc etwork ) uder the physical ) model has a tight boud of Θ m whe m = O log), ad Θ ) whe 2 m =Ω log) ). 2 It has already bee established i the literature that the above bouds are also achievable o the basis of traditioal store-ad-forward routig methods. Cosequetly, our aalysis demostrates that the throughout gai due to NC for mutlicast as well as broadcast is bouded by a costat factor! The remaider of this paper is orgaized as follows. Sectio II surveys relevat prior work. Sectio III describes the etwork models ad other cocepts used proofs. Sectio IV deduces the capacity results uder the protocol model. Sectio V summarizes our coclusios. II. LITERATURE REVIEW Gupta ad Kumar s origial work focused o the uicast capacity of wireless etworks [2], a may subsequet cotributios have bee made o the capacity of wireless etworks subject to uicast traffic. However, the focus of this paper, ad

2 therefore this sectio, is o the capacity wireless etworks uder broadcast ad multicast traffic. Several recet efforts have adressed the multicast ad broadcast capacity of wireless etworks, primarily uder the protocol model. The work by Keshavarz et al. [8] addresses the broadcast capacity of a wireless etwork for ay umber of sources i the etwork. Jacquet ad Rodolakis, [9] proved that the scalig of multicast capacity is decreased by a factor of O m) compared to the uicast capacity result by Gupta ad Kumar [2]. This result implies that multicastig gai, over trasmittig the iformatio from each source as m uicasts, is at least Θ m). Li et al. [0] compute the capacity of wireless ad hoc etworks for uicast, multicast, ad broadcast applicatios. Zheg et. al. [] idepedetly geeralized this work ad itroduced, m, k)-castig as a framework for the characterizatio of all types of iformatio dissemiatio i wireless etworks. This prior work has oly addressed covetioal store-ad forward routig for multicast ad broadcast traffic. Sice Ahlswede et. al. s [2] semial work, most of the theoretical research o NC has focused o directed etworks, where each commuicatio lik is poit to poit ad has a fixed directio. However, a wireless etwork is more appropriately modeled by bi-directioal liks. Li et. al [3], [4] have studied the beefits of NC i udirected etworks. The result shows that, for a sigle uicast or broadcast sessio, there is o throughput improvemet due to NC. I the case of a sigle multicast sessio, such a improvemet is bouded by a factor of two. Nevertheless, the work by Li et. al does ot accout for multiple access iterferece, ad hece caot be directly applied to wireless etworks. Liu et. al. [5], [6] have show that the NC for uicast traffic i a radom etwork i.e. a etwork i which the odes are distributed radomly i a Euclidea space ad the sources ad desitatios are also placed radomly) is bouded by a costat factor. Keshavarz et. al. [7] exteded these coclusios to arbitrary etworks ad a arbitrary uicast traffic patter. Furthermore, they also showed that the NC gai for eve a sigle source multicast is bouded by a costat factor i ay arbitrary etwork. From the above, it is apparet that prior work has ot determied whether NC by itself ca provide ay gais o the multicast order throughput i wireless etworks, which is the subject of this paper. III. PRELIMINARIES For a cotiuous regio A, weuse A to deote its area. We deote the cardiality of a set by S, ad by X i X j the distace betwee odes i ad j. Wheever coveiet, we utilize the idicator fuctio {P }, which is equal to oe if P is true ad zero if P is false. PrE) represets the probability of evet E. We say that a evet E occurs with high probability w.h.p.) if PrE) > /)) as.weemploythe stadard order otatios O,Ω, ad Θ. We assume that the topology of a etwork is described by a uiformly radom distributio of odes i a uit square. Let V =,..., represet the ode-set ad let X i be the locatio of ode i V. To avoid boudary effects, it is typical to assume that the etwork surface is placed upo a toroid or sphere. However, for mathematical coveiece, i this work we igore edge effects ad thus assume that the etwork is placed i a 2-D plae. Further, i our model, as goes to ifiity, the desity of the etwork also goes to ifiity. Therefore, our aalysis is applicable oly to dese etworks. We do ot cosider mobility of odes ad assume a static statioary distributio of odes. Our capacity aalysis is based o the protocol model itroduced by Gupta ad Kumar [2]. Defiitio 3.: The Protocol Model We assume that all odes use a idetical trasmissio rage r) for all their commuicatio. Node i ca successfully trasmit to ode j if for ay ode k i, that trasmits at the same time as i it is true that X i X j r) X k X j + Δ)r). ) We shall utilize the followig well kow property [5] i our aalysis Lemma 3.2: Coectivity Criteria For a radom distributio of odes i a uit-square, the etwork coectivity uder the protocol model ca be guarateed w.h.p if ad oly if iff) 3log) r) r c ) =. 2) We focus o the traffic sceario i which each ode of the wireless etwork acts as a multicast source for a radomly chose set of m distict destiatios. Defiitio 3.3: Feasible rate ad Throughput Order Our defiitios of feasible rate ad throughput order are similar to those defied i [2]. Fig.. Geeralized Sparsity Cut Defiitio 3.4: Cut Give a ode set V, a cut is the separatio of the vertex set V ito two disjoit ad exhaustive subsets S, S C ). Here, a vertex partitio ca be completely described by partitioig the etwork-area ito two regio A, A c ) as show i Fig., thus

3 we also refer to a closed regio A as a cut. The cut-capacity CA) is defied as the maximum umber of simultaeous trasmissios that ca take place from A c to A. Defiitio 3.5: Multicast Cut-Demad Give a cut A, a source ode i A c is said to have demad across the cut iff at least oe of its destiatio lies i A. The multicast demad DA) across the cut is defied as the total umber of sources i A c such that there is at least oe destiatio i the multicast group across the cut. Defiitio 3.6: Sparsest Cut We defie the sparsity Γ A of cut A as the ratio Γ A = CA) 3) DA) Hece, the sparsest cut is give by A = arg mi Γ A 4) A where A has the least possible sparsity, deoted as Γ A. The covetioal defiitio of Sparsity cut [6] is applicable oly to uicast traffic [6]. Our defiitio geeralizes the covetioal defiitio to multicast traffic. Fially we state the well-kow Cheroff Bouds [7], which shall be repeatedly used i the rest of this paper. Lemma 3.7: Cheroff Bouds: Cosider i.i.d radom variables Y i {0, } with p = PrY i =).LetY = i= Y i. The for ay δ 0 ad δ 2 0 we have Pr Y δ )p) < 2e δ 2 p 3 5) Pr Y + δ 2 )p) < 2e δ 2 2 p 3 6) IV. BOUNDS FOR PROTOCOL MODEL It is well-kow that uder its covetioal defiitio, the sparsity cut ca be used to obtai a upper boud o the uicast traffic flow i a wireless etwork [6], [6]. I a similar way, our geeralized defiitio provides a upper boud for multicast flows. Lemma 4.: Let C m ) be the maximum multicast flow rate i a etwork ad let A be the sparsest cut with sparsity Γ A, the we have C m ) Γ A 7) Proof: Let f be the total maximum feasible average rate at which bits ca be trasmitted from A c to A, where A is ay arbitrary cut. The by Def. 3.4 we have f CA) 8) The total iformatio flow across a cut has to be greater tha or equal to the sum of the data rates associated with idividual multicast sessios that commuicate across the cut. Hece, f C m ) {source i i A c has demad across cut A} i= = C m ) i= {source i i A c has demad across cut A} = C m )DA). 9) Isertig the above equatio i Eq.8, we have C m ) CA) DA) =Γ A Γ A. 0) I the study of etwork capacity, cut argumets are typically employed oly for uicast traffic ad hece the use of such argumets for multicast traffic may seem couter ituitive to some readers. Therefore, some additioal commets are i order. I particular, we ca replace the use of Lemma 4. ad Defiitio 3.6 with the followig alterative result. Lemma 4.2: Cosider a etwork with odes V = {a,...,a } ad multicast sessios. Each sessio cosists of oe of the odes actig as a source with a arbitrary fiite subset of the set V actig as the set of destiatios. Let s i be the source of the i th sessio ad let B i = {b i,...,b imi } be the set of m i destiatios. Now, there exists a joit routigcodig-schedulig scheme that ca realize a throughput of λ i for the i th sessio, i.e., λ =[λ,...λ k ] is a feasible rate vector. The λ is also a feasible vector for ay uicast routig problem i the same etwork, such that the traffic cosists of k uicast sessios with s i beig the source of the i th uicast sessio ad the destiatio z i is ay arbitrary elemet of the set B i. The above lemma basically establishes that, if a multicast capacity from a source to multiple destiatios is feasible, the clearly it is feasible to achieve the same capacity to ay oe arbitrarily chose ode from this set of destiatios. We ca thus deduce the bouds for the case of multi-source multicastig by reducig it to a suitable uicast routig problem. Uder the reductio, a upper boud for the uicast problem also serves for the origial multicast routig problem. Thus, i order to obtai a upper boud o the multicast capacity, we could costruct a uicast problem by choosig destiatios specifically from a suitably chose regio A. To establish the relatioship of the above argumet to Lemma 4., recall the classical defiitio of sparsity used i [6], [6] for the aalysis of uicast traffic. Defiitio 4.3: Uicast Sparsity The uicast) Sparsity of a cut A for a give set of sources S = {s,...,s } ad a chose set of destiatios Z = {z,...,z } is defied as CA) Υ B,A = ) D uicast A) where CA) is the cut capacity ad D uicast A) is the uicast demad across cut A, i.e. the total umber of uicast sources i A c that have a destiatio i A. Now let us cosider a etwork with odes, a set of multicast sources S = {s,...,s } ad set of destiatio sets F = B,...,B, such that B i is the set destiatios for s i. Furthermore, let Z be the set of all possible sets Z = {z,...,z } such that z i B i is a destiatio for source s i.letf Z be a feasible per-sessio flow for a uicast sceario described by S as the set of sources ad Z as the set of destiatios. It is well kow that f Z mi A Υ Z,A for all Z, where A ca be ay arbitrary cut. Hece, Lemma 4.2

4 basically states that f B is a feasible flow rate for the multicast sceario iff f B mi Z Z F Z mi Z Z mi A Υ Z,A 2) Sice CA) does ot deped o our choice of Z, with the exchage of miima s we have CA) f B mi A max Z Z D uicast A) = mi CA) A DA) =Γ A 3) It should be highlighted that the above deductios imply that the maximum multicast flow-rate is less tha the sparsity of ay arbitrary cut. Thus, to obtai a upper boud o the etwork capacity, we are free to choose a regio A of ay arbitrary shape ad size. I this work we shall utilize cuts of square shape as show i Fig.2, with legth L A =4l A, i.e., each side of the square A has legth l A. The parameter l A plays a crucial role i deducig the required upper bouds. I particular, we choose l A so as to guaratee that the demad DA) =Θ). m 4+ɛ )r). Note that if m 2 4+ɛ )r), the 2 2r). +ɛ)m We ivoke the followig importat observatio to obtai a upperboud o the cut-capacity. Remark 4.5: I [2], it was observed that a disk of radius Δr) 2 cetered at each receiver i i ay time slot slot should be disjoit. However, this fact does ot apply to the case i which odes exploit broadcast trasmissios, as is doe whe odes are capable of employig NC. Ideed, as show i Fig.2, the disks ca overlap if the associated odes are receivig idetical iformatio from a commo trasmitter. Nevertheless, as highlighted i [5], eve uder the NC assumptio, the uio of the disks cetered at the receivers of oe trasmissio should be disjoit from the uio of the disks cetered at the receivers of aother trasmissio. Lemma 4.6: If a square-shaped cut A has side-legth l A 2r), the the cut capacity satisfies CA) 6L A πδ 2 7) r) uder the protocol model. Proof: I the protocol model, the distace betwee a trasmitter ad a receiver is bouded by r). Hece, ay ode i A that receives a trasmissio from A c should lie withi a distace r) from the boudary of the cut, i.e., all the receivers must be placed withi a aular regio of area Fig. 2. Cut Capacity uder Protocol Model l 2 A l A 2r)) 2 =4l A r) 4r) 2 4l A r) =L A r) 8) where legth L A of the cut is the perimeter of the regio A. We observe that each trasmissio across the cut will ot allow ay more trasmissio withi a area of at least πδ 2 r) 2 4. Additioally, at least 4 of this area has to fall withi the aular regio ear the cut boudary. Therefore, Lemma 4.4: I a radom etwork with odes, each actig as source for m radomly chose odes, for every ɛ 0 if l A = for m + ɛ)m 4 + ɛ)r) 2 4) l A = 2r) for m 4 + ɛ)r) 2 5) the for ay δ 0 ad such that log2) 3, w.h.p δ 2 c we have DA) δ )c 6) ) ) where c = +ɛ. e +ɛ Proof: The proof is omitted due to space limitatios. A choice of l A = ca be used i the above lemma +ɛ)m for all m, ad such a coditio would be sufficiet to prove the required result that demad DA) δ )c w.h.p. However, i the followig aalysis we require that l A 2r). Therefore, we itroduce the coditio that l A = 2r) for CA) = max. o. of trasmissios from A c to A Area of aular regio = 6L A πδ 2 r) πδ 2 r) Theorem 4.7: I a radom geometric etwork with NC the multicast capacity uder the protocol model has the followig uppr boud w.h.p. C m ) = C m ) = 2c 2 whe log2) 3 c ɛ )mlog) if m + ɛ ) 2log) if m + ɛ ) 2log) 9) 64+ɛ )e +ɛ πδ 2 ɛ δ )e +ɛ ) ad δ, where c 2c 2 = δ,ɛ 0 Proof: O accout of Lemma 4., we ca obtai a upper boud o the etwork capacity by just providig a boud for

5 the sparsity Γ A. Furthermore, ote that L A =4l A. Hece, due to Lemma 4.6 we ca say that, for all l a 2r), wehave 64l A C m ) πδ 2 r)da). 20) Cosider m 4+ɛ )r). If we choose l 2 A =2r), the from Lemma 4.4 we have that DA) δ )c w.h.p. for all such that. Therefore, c log2) 3 δ 2 28 C m ) πδ 2 δ )c 2) Similarly, if we choose l A = +ɛ)m for all m 4+ɛ )r) 2,wehave 64 +ɛ)m C m ) πδ 2. 22) δ )r)c Note that, for all m 4+ɛ )r), C 2 m ) is maximized by choosig the smallest possible value of r). Nevertheless the coectivity criteria Lemma 3.2) requires that r) 3log). The fial result is obtaied by substitutig the value 3log) of c ad r) = i Eqs The multicast capacity uder pure routig has bee characterized i [0], []. Theorem 4.8: [0], [] I a radom geometric etwork with pure routig, the multicast capacity uder the protocol model has a tight boud ) ) C m ) = Θ if m = O mlog) log) 23) C m ) = Θ ) ) if m =Ω log) 24) NC is a geeralizatio of routig ad thus ay capacity achieved by routig is ecessarily achieved by NC. Hece, Theorem 4.9: I a radom geometric etwork with NC, the multicast capacity uder the protocol model has a tight boud C m ) = Θ if m = O ) ) if m =Ω ) mlog) C m ) = Θ log) log) ) 25) 26) Fially, we ca arrive at the followig coclusio. Corollary 4.0: The multicast throughput order gai provided by etwork codig over pure routig i a radom geometric etwork uder the protocol model is O) V. CONCLUSION Network codig NC) has received cosiderable attetio, ad recet results for specific istatiatios of NC have led may to ifer that NC could lead to order throughput gais for multicastig i wireless etworks. I this work, we used the protocol model to show that the order throughput gai derived from NC for multicastig ad broadcastig i wireless etworks is bouded by a costat. That is, as the etwork size icreases, NC reders the same order throughput as traditioal store-ad-forward routig. A similar argumet ca be proved uder the more realistic physical model assumptio, ad this is the subject of aother paper. VI. ACKNOWLEDGMENTS This work was partially sposored by the U.S. Army Research Office uder grats W9NF ad W9NF , by the U.S. Navy Aviatio/Ship Itegratio uder grat , by the Natioal Sciece Foudatio uder grat CCF , by the Defese Advaced Research Projects Agecy through Air Force Research Laboratory Cotract FA C-069, ad by the Baski Chair of Computer Egieerig. The views ad coclusios cotaied i this documet are those of the authors ad should ot be iterpreted as represetig the official policies, either expressed or implied, of the U.S. Govermet. REFERENCES [] R. W. Yeug ad Z. 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