Wireless Inter-Session Network Coding - An Approach Using Virtual Multicasts

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1 Wreless Inter-Sesson Network Codng - An Approach Usng Vrtual Multcasts Mchael Hendlmaer,DesmondS.Lun,DanalTraskov,andMurelMédard Insttute for Communcatons Engneerng CCIB RLE Technsche Unverstät München Rutgers Unversty MIT Munch, Germany Camden, NJ Cambrdge, MA {mchael.hendlmaer, danal.traskov}@tum.de [email protected] [email protected] Abstract Ths paper addresses the problem of nter-sesson network codng to mamze throughput for multple communcaton sessons n wreless networks. We ntroduce vrtual multcast connectons whch can etract packets from orgnal sessons and code them together. Random lnear network codes can be used for these vrtual multcasts. The problem can be stated as a flowbased conve optmzaton problem wth sde constrants. The proposed formulaton provdes a rate regon whch s at least as large as the regon wthout nter-sesson network codng. We show the benefts of our technque for several scenaros by means of smulaton. I. INTRODUCTION Network codng [1] has been an actve feld of research n the past years. Its benefts for uncast and multcast traffc have been well studed [2] [4]. For sngle source network codng problems necessary condtons are also suffcent [1] and dstrbuted lnear codng schemes are avalable [5]. Mult-source network codng s a consderably more dffcult problem: The optmal soluton for nter-sesson network codng mght requre nonlnear encodng functons [6] and even decdng a lnear nter-sesson codng problem can be NP-complete [7]. Nonetheless, suboptmal lnear codng schemes have been proposed both for wrelne and wreless networks. Prevous work on nter-sesson codng for wrelne networks has focused on a graph-theoretc characterzaton of network codes [8], [9]. For eample, n [10], the authors present an approach to translate the soluton for the well-known butterfly network to larger networks. In [11] nter-sesson network codng s combned wth multple descrpton codng. Recent work [12] presented an nterference algnment approach to perform ntersesson codng. We focus on wreless networks, where the most promnent technque s the COPE protocol [13] whch eplots the broadcast nature of the wreless medum. As shown n Fg. 1a, the number of transmssons can be reduced compared to routng f packets from dfferent sessons are combned. Ths technque s called reverse carpoolng n [14] due to the fact that the combned nformaton has to travel n opposte drectons. The nterplay of the COPE protocol wth medum access has been analyzed n [15] and approaches to model ts performance were proposed n [16], [17]. COPE has been successfully mplemented and shows gans especally when the reverse carpoolng stuaton occurs. A A a R R B B A a b a b R B (a) The COPE eample b a a b a b (b) Butterfly eample Fg. 1: In the COPE eample, A and B want to echange the packets a and b. NodeAsendspacketa to the relay node R n the frst transmsson. So does B wth packet b n a second tme step. In the thrd tme step, the relay broadcasts the XOR-ed packet a b to both A and B, so b and a can be decoded, respectvely. Routng would requre four transmsson steps. Smlarly for the Butterfly network, where wants to communcate wth, and s 2 wth. The technque COPE reles on can also be nterpreted as alocalmultcasttoasetofneghbors.wepckupthedea of a vrtual multcast, however, n a dfferent way than COPE does. We are motvated by the Butterfly eample n Fg. 1b, where we can acheve hgher rates for two sessons by creatng avrtualmultcast.furthermore,randomlnearcodes[5]can be used as a codng scheme n ths setup [1]. The dea of our approach s to create vrtual multcast sessons, combne packets from orgnal sessons and delver them to a vrtual termnal set. Our technque s thus a carpoolng approach, as packets are travellng n the same drecton. In the Butterfly eample the vrtual multcast runs from the orgnal sources to both orgnal termnal nodes. However, our approach allows to establsh the vrtual multcast at an arbtrary poston n the network and we wll llustrate n Sec. IV that a multcast to the orgnal termnals s not always optmal. In general, unlke many other technques for nter-sesson codng, the dffculty of our approach s not the codng scheme tself but the proper choce of ths vrtual termnal set. We wll defne the vrtual multcast model precsely n Sec. III-C. The orgnal sessons and the vrtual multcast can be characterzed n terms of flows from sources to termnals, smlar to [3]. Therefore, the whole problem can be stated as a conve flow optmzaton problem. The man contrbuton of ths work s a formulaton that enables nter-sesson network codng by ntroducng vrtual multcast sessons. We evaluate the gans that can be epected wth ths technque compared to an uncoded approach for varous networks. b

2 We wll present the system model n Sec. II before we state the problem formulaton n Sec. III. The ntegraton of multple vrtual multcast sessons wll be dscussed n Sec. IV. Smulaton results are presented n Sec. V before we conclude the paper n Sec. VI. II. SYSTEM MODEL A wreless packet network s modeled by a hypergraph H =(N, A), N denotng the set of nodes and A denotng the set of hyperarcs. The demands are gven by a set of connectons, denoted C. Thepar(H, C) defnes the network problem. Ahyperarc(, N()) Amodels a lossy one-tomany broadcast connecton from a node Nto ts neghbors N() N.Throughoutthswork,weassumethatmedum access takes care of schedulng transmssons, so each tme node transmts a packet, all neghbors N() wll be able to hear t. Accordngly, there s one hyperarc per node n the network, so A = N. Aconnectonc C conssts of a par (s c,t c ),thatsasourcenodes c N and a set of destnaton nodes T c N.If T c = 1, c s a uncast connecton, f T c > 1, tsamultcastconnecton.theendto-end throughput of connecton c s denoted by R c.wtheach connecton we assocate a concave utlty functon U c (R c ) dependng on the throughput of that sesson. The rate at whch node njects packets nto ts outgong hyperarc s denoted by z.duetoerasuresthetransmtted packet mght be receved only by a subset K N() of the neghbors of. Wedenotethearrvalrateforpacketseactly receved by K N() by z K.So,z = K N() z K. Let P L N () L K zl b K = z be the probablty that a packet, sent out by node, srecevedbyatleastonenodentheset K N(). Noteverypackettransmttedbynode belongs to the same sesson, so we denote by y (c) the packet njecton rate at for packets of connecton c, soz = c C y(c).the regon Z specfes the set of feasble njecton rate vectors z =(z ) N. Z s assumed to be gven from lower layers. Varables j represent the nformaton flow of connecton c for termnal t T c between nodes N and j N(). III. FORMULATION OF THE DIFFERENT APPROACHES We formulate the problem n three dfferent ways as a utlty mamzaton problem. For smplcty, we consder the problem of mamzng the sum-utlty, c C U(R c). The frst two formulatons wll lead to our man descrpton n Sec. III-C whch bulds up on the latter ones. Frst, we state the network problem f no nter-sesson codng s allowed n Sec. III-A. In Sec. III-B, we present an approach whch combnes all the nformaton of all sources and multcasts t to every termnal n the network, lke n the Butterfly eample. Ths technque wll help us to descrbe the Vrtual Multcast formulaton n Sec. III-C, whch can do both - combne packets from dfferent sessons and delver t to a vrtual termnal set or stck to the non-codng approach. A. No Inter-sesson Codng (NIC) Each sesson c Cperforms ntra-sesson network codng but no nter-sesson network codng. So, flows of dfferent sessons cannot be combned and have to share the network resources. The problem can be wrtten as follows, etendng the formulaton n [3] for multple sessons: mamze c C z y (c) 0, c C N, (1) z =(z ) N Z, (2) y (c) b K j, N,K N(),t T c,c C, j K (3) j 0, N,j N(),t T c,c C, (4) j { R c = s c, j = 0 else, N\{t},t T c,c C. (5) Here, (3) relates the outgong nformaton flow to the packet njecton rates whle (5) assures flow conservaton for each destnaton of the connectons. B. Multsource Multcast (MSM): Completng the Multcast If we allow all connectons to be coded together, as each termnal wants to have all the data, mn-cut condtons are necessary and suffcent [1, Theorem 8]. Agan, t can be formulated as an optmzaton problem wth lnear constrants by ntroducng a super-source and vrtual edges connectng ths super-source wth the orgnal sources. The outgong rate at the vrtual source s c C R c,thecapactyofvrtualedgesfrom the super-source to the orgnal source s equal to R c.in[18] ths fact was used n the contet of correlated sources. In ths setup there s only one vrtual sesson from the super-source to the recever set T,whereT contans all destnatons of all sessons,.e. T = c C T c.tosmplfythenotaton,we can omt the vrtual source and edges and consder sesson c orgnatng at s c agan, knowng that there est codng technques to capture ths setup. The followng formulaton apples: mamze c C z 0, N, (6) z =(z ) N Z, (7) z b K j K (t) (t) j, N,K N(),t T, (8) j 0, N,j N(),t T, (9) (t) j { (t) R c = s c, j = 0 else, N\{t},t T. (10)

3 The Butterfly network n Fg. 1b s an eample where ths technque works well. By multcastng all source nformaton to every termnal we can do better than NIC. C. Vrtual Multcasts (VM) Nether NIC nor MSM s always better than the other strategy for general networks, so t s not clear whch technque to choose. The followng consderatons let us combne both deas: We ntroduce a new, vrtual multcast connecton c whch can etract flow from the other sessons. To dstngush ths new sesson from the prevous ones, we refer to C as the set of orgnal connectons. The vrtual sesson has a set of snk nodes T c,whchreceveallthetraffcabsorbedbythe vrtual sesson. At each node, λ (c) 0 denotes the amount of flow etracted from one of the orgnal connectons c and njected nto sesson c. Let 0 be the amount of traffc re-njected at node nto sesson c for snk t from sesson c. For all c Cand t T c,wehave =0for T c. That s, traffc can only be re-njected at the vrtual termnal nodes. Let C = C { c} denote the set of orgnal and vrtual connectons. mamze c C z y (c) 0 c C N, (11) z =(z ) N Z, (12) y (c) b K j, N,K N(),t T c,c C, j K (13) j 0 N,j N(),t T c,c C, (14) j { R c + λ (c) = s c, j = λ (c) else, N\{t},t T c,c C, (15) =0 T c,t T c,c C. (16) For the vrtual sesson c, we have to add the followng constrants: (t, c) j (t, c) j = λ (c), c C N\{t},t T c, (17) = λ (c), t T c,c C. (18) T c N The modfed flow conservaton constrants n (15) guarantee proper accountng of flow etracton and re-njecton for the vrtual sesson. Flow conservaton for the vrtual sesson s assured n (17). Those nodes, where c C λ(c) > 0 can be consdered as the vrtual sources for the vrtual multsource multcast. Note that each node can act as a vrtual source, however, the set of vrtual termnals T c has to be eplctly specfed. In (18), we make sure that all the collected flow from one sesson s re-njected nto the orgnal sesson at the vrtual termnals. T c (a) Network 1 s 3 t 3 (b) Network 2 Fg. 2: Wrelne eample networks The formulaton above accepts the vrtual termnal set T c as an arbtrary nput. We brefly address consderatons about ths selecton n Sec. IV. For now, we choose the vrtual termnal set to yeld T c = c C T c.thschoceallowsustoperform both the uncoded strategy and the multsource multcast. Let R NIC, R MSM and R VM denote the achevable rate regon of a network problem usng communcaton strategy NIC, MSM and VM, respectvely. Theorem 1: If T c = c C T c,everysolutonthatcanbe obtaned by NIC and by MSM can also be obtaned by VM. Therefore, R VM R NIC and R VM R MSM. Proof: R VM R NIC : If we set all varables related to the vrtual sesson to zero,.e. y ( c) = {y ( c) } N = 0, = { (t, c) j } N,,t T c = 0, λ = {λ (c) } N,c C = 0 and µ = { } N,t Tc,c C = 0, the constrants n VM are eactly the same as for NIC. NIC s thus a specal case of VM. R VM R MSM : If we set λ (c) s c = R c and zero elsewhere and t = N λ(c) = c C R c,zeroelsewhere,thevrtualsesson c carres all the flow of all sessons. The vrtual sesson then corresponds to the sngle sesson n MSM. MSM s thus a specal case of VM. It follows from Theorem 1 that VM acheves at least the throughput of NIC and MSM. As we wll see n Sec. V, VM can often do better than both. IV. MULTIPLE VIRTUAL SESSIONS AND CHOICE OF T c So far only the specal case of multcast of T c = c C T c was consdered. We brefly show that ths can be suboptmal. For smplcty, we use a wrelne network wth unt lnk capacty for the llustraton n ths secton. Consder the network n Fg. 2a, where NIC, MSM and VM wth the choce of T c = c C T c can only provde the rate regon R 1 + R 2 1. Ths s due to the fact that the only paths from to and from s 2 to are not edge-dsjont. However, f we choose the hghlghted nodes to be the vrtual termnal set, we can take advantage of VM and the followng rate regon s achevable: R 1 1, R 2 1. Both sources multcast nformaton wth R 1 = R 2 =1to the nodes n the vrtual termnal set. Then, these nodes forward the demanded nformaton to,,respectvely. The formulaton n Sec. III-C can be easly etended to addtonal vrtual sessons. In order to determne the best termnal set, vrtual sessons for each subset of nodes n the network

4 would have to be ntroduced, resultng n an eponental number of vrtual sessons. However, the computatonal complety already becomes too hgh for small networks, so we have to restrct us to use only a small number of vrtual connectons. As mentoned, VM wth T c = c C T c can combne NIC and multsource-multcast to all termnals. However, for many connectons n the network t mght restrct us too much requrng every termnal of all sessons to receve all data. Consder the network n Fg. 2b wth three uncast sessons (, ), (s 2, ), (s 3,t 3 ).Whletsusefultomultcastthe nformaton from and s 2 to both and,weshouldnot nclude t 3 to the vrtual termnal set as t would reduce R 3. To capture ths dffculty, we can ntroduce one vrtual sesson for each subset of termnal nodes. The orgnal sessons can nject flow to vrtual sessons f a multcast to the respectve termnal set s helpful. So, the number of vrtual sessons s eponental n the number of termnal nodes n the network whch s stll tractable for small networks. V. PERFORMANCE EVALUATION We smulate the algorthm for wreless networks to see what gans can be epected. In all the smulatons we used proportonal farness as utlty functon for each sesson,.e. U c (R c ) = log(r c ) c C.As c U c(r c ) s a concave functon and all constrants are lnear, all the presented optmzaton problems are conve. Note that whle the eamples n Sec. IV were wrelne networks, wreless networks wth broadcast transmssons are smulated here. We restrct the representaton of wreless networks to a connectvty graph. That s, nodes whch are n mutual rado range are connected. The hyperarcs presented n Sec. II have to be consdered as aone-to-manyconnectonfromeachnodetoalltsneghbors and are omtted for the sake of clarty n the followng fgures. The relaton of hyperarcs and connectvty model s brefly shown n a small network n Fg. 3b. We evaluate the performance of VM compared to NIC wth respect to two ndcators. Frst, f VM can mprove the optmal sum-utlty, the correspondng network problem s called utlty-mproved (UI). IfVMprovdesahgherratefor all sessons, that s R c,v M k R c,nic, c C,wecall the network problem rate-mproved (RI), wthk>1 denotng the gan factor. Of course, every rate-mproved problem s also utlty-mproved. A. Mesh Networks We consder the type of wreless mesh networks shown n Fg. 3a. Sources and termnals of two uncast connectons are placed on a crcle wth radus r 2 whereas the other nodes of the network are randomly placed wthn a crcle of radus r 1. Nodes are n mutual rado range f ther dstance s below a certan threshold value. The whole nformaton has to travel through the center of the network, so we assume that all center nodes are able to overhear all packets from all sessons anyway. So, we can multcast to the whole center network and let the nodes at the border of the center forward the nformaton demanded by connected snks, smlar to the r 2 r 1 s 2 (a) Connectvty graph of a mesh network. (b) Relaton between connectvty graph and hyperarcs. There s one outgong hyperarc per node. Fg. 3: Connectvty graph and hyperarcs. #nodes #smulaton #UInetworkproblems #RInetworkproblems average gan factor k for rate TABLE I: Smulaton results for mesh networks. eample n Fg. 2a. However, t turned out that ths choce of T c dd not lead to observable gans of VM over NIC. However, f the vrtual termnal set t chosen to be T c = c C T c,we obtan convncng results: For the partcular case of 6 nodes n the network,.e. two nodes n the center, VM returns a hgher sum-utlty n nearly 75% of the cases. For 282 out of 1000 network problems, VM provdes a hgher rate for both sessons. The average gan for the rate-mproved 6-node networks 4%. TableIsummarzesresultsforthssetupand networks of larger sze. The number of UI network problems decreases wth more nodes n the center. In that case, the center of the network has enough capacty to forward both sessons, so t s not the bottleneck anymore. In many of these cases the maflow bounds can acheved for both connectons wth NIC, whch essentally means that both sessons do not nterfere. B. Random Geometrc Wreless Networks In ths eperment we ask whether the gans observed for the mesh networks consdered before also translate to random networks wthout specal structure. Accordngly, nodes are randomly placed on a square wth constant node densty. Nodes can hear each other f ther dstance s smaller than a certan threshold. Sources and termnals are randomly chosen. For a network consstng of 10 nodes wth two multcast sessons and two termnals for each connecton, UI and RI network problems occur less frequently. However, for the average RI problem the rate regon s ncreased by 33%. The network n Fg. 4 wth two uncast sessons s an eample for such an RI network problem. Here, R 1,NIC =0.8 <R 1,MSM =0.91 <R 1,V M =0.92 and R 2,NIC =0.8 <R 2,MSM =0.91 <R 2,V M =0.97, whch corresponds to a gan of 15% and 21%, respectvely. We add further sessons to evaluate the behavor of VM f there s more nterference n the network. For a 12-node network wth three uncast sessons, VM can mprove the utlty for 259 out of 1000 network problems. As proposed n Sect. IV, one vrtual sesson s created for each subset of orgnal termnals. We count 37 RI network problems, wth an average gan of 38%. AssummarzednTableII,welose 9% on average f only one vrtual sesson s consdered. The

5 s 2 Fg. 4: Connectvty graph of an RI network problem wth 10 nodes. Arrows represent sources and termnals of sessons. #node #smulaton #sessons #UInetworkproblem #RInetworkproblems average gan factor k for rate average gan wthout recever subsets TABLE II: Smulaton results for random networks. more sessons are added, the more frequently UI network problems can be observed. For a 12-node network wth 4 uncast sessons, VM ncreases the utlty n more than 50% of the cases. Gans are shown n Table II. C. Wreless Grd Networks We are further nterested about how the postons of source and destnaton have an mpact on the performance of VM. Therefore, we analyze ts behavor on more structured wreless networks shown n Fg. 5. In ths setup, we are not partcularly nterested n the specfc gans of VM but rather at whch poston t turns out to be useful. Consder the network n Fg. 5a: One sesson (, )andthepostonofthetermnal for the second sesson,,arefed.wevarythepostonof the source for the second sesson and nvestgate f we can acheve hgher rates by usng VM wth T c = c C T c.we dstngush four cases: Postons of the source whch let all the sessons acheve the maflow bounds are marked gray. Postons leadng to UI network problems are marked wth acrosswhlepostonsleadngtorinetworkproblemsare marked by a square. Other postons are unchanged. In ths case, only two postons close to the termnal lead to auinetworkproblem.formostpostonswecanachevethe maflow bounds for each sesson ndvdually whch means that the sessons do not nterfere. In the network n Fg. 5b we consder four connectons. Three sessons (, ), (s 2, ), (s 3, t 3 )arefed,wevarythepostonofthesourceforthe fourth sesson, s 4.Now,astheresmorenterferencenthe network, we can do better wth VM for every possble poston n the network. We conclude that VM s partcularly useful f nterference reduces the rates achevable by NIC. VI. CONCLUSION AND FURTHER WORK We presented an approach for ntersesson network codng by creatng vrtual multcast sessons and leveragng the suffcency of random lnear network codng for ths setup. Vrtual sessons can be defned for an arbtrary vrtual termnal set, however, our smulatons showed that the unon of all orgnal termnals s a good choce. The technque turns out to be benefcal for many networks, wth a rate ncrease of more (a) Two sessons. t 4 t 3 s 3 (b) Four sessons. maflow bounds achevable utlty-mproved rate-mproved Fg. 5: Connectvty graphs of grd networks. than 30% for each connecton, at the cost of computatonal complety. There are elegant ways to solve the optmzaton problem by dual decomposton, whch wll be presented n future work. REFERENCES [1] R. Kötter and M. Médard. An algebrac approach to network codng. Networkng, IEEE/ACM Transactons on, 11(5): ,oct [2] D.S. Lun, M. Médard, and R. 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