Communicating the sum of sources over a network

Communicating the sum of sources o v er a network Aditya Ramamoorthy Department of Electrical an d Computer Engineering Iow a State University Ames, Iow a 5001 1 Email: adityar@iastate. edu Abstract — W e consider a network (that is capabl e of network coding) with a set of sources and terminals, where each terminal is interested in recov ering the sum of t he sources. Considering directed acyc lic graphs with unit capacity edges and in dependent, unit-entropy sources, we sh ow the rate r egion when (a) there ar e two sourc es and n terminals, and (b) n sour ces and two terminals. In these cases as long as there exists at least one p ath fro m each source to each terminal we demonstrate that there exists a valid assignment of coding vecto rs to the edges such that the terminals can reco ver the sum of the sources. I . I N T R O D U C T I O N Network coding is a new paradigm in networkin g whe re nodes in a network have the a bility to proce ss informatio n before forwarding it. T his is un like ro uting wh ere no des in a network primarily operate in a rep licate and forward manner . The p roblem of m ulticast has bee n studied in tensiv ely under the paradigm of network coding. The seminal work of Ahlswede et al. [1] showed that und er network coding the multicast capacity is the minimum of the maximum flows from the sour ce to each individual terminal node. The work o f Li et al. [2] showed that lin ear n etwork c odes w ere sufficient to achieve the multicast cap acity . The algeb raic approach to network codin g p roposed b y Koetter an d M ´ edard [3 ] pr ovided simpler pr oofs of these r esults. In rece nt years ther e has also been a lot of in terest in th e development an d u sage o f d istributed source coding sch emes due to their application s in emerging ar eas such as sen sor networks. Classical distributed sou rce coding results such as the famo us Slepian-W olf theo rem [4] usually assum e a direct link between the sources an d the term inals. Ho wever in applicatio ns su ch as sensor networks, typically the sou rces would communicate with the terminal over a network. Thus, considerin g the d istributed compression jo intly with the net- work informatio n tran sfer is important. Network coding for correlated so urces was first exam ined by Ho et al. [5]. The work of Ramamoor thy et al. [6] showed that in general separating distributed sour ce co ding and network coding is suboptimal except in the case of two sources and tw o termi- nals. A practical appr oach to tr ansmitting correlated sources over a n etwork was considered b y W u et al. [7]. Reference [ 7] also intro duced th e pro blem of Network Arithmetic th at come s up in the desig n of practical systems th at co mbine distributed source co ding and network cod ing. In the network arithm etic problem, there are source node s each of which is obser ving indep endent sources. I n addition This research was support ed in part by NSF grant CNS-0721453. there is a set of term inal n odes that are only interested in the sum of these sour ces i.e. u nlike the multicast scenar io where the termin als are actually interested in recovering all the sources, in this case the te rminals are on ly in terested in the sum of the sources. In th is pap er we study th e ra te region of th e n etwork arith metic pr oblem un der cer tain special cases. In particular we restrict our attention to dire cted acyclic graphs (D A Gs) with unit capacity edges and independen t, unit entropy sources. Mo reover , we co nsider the follo wing two cases. i) Networks with two sources and n terminals, and ii) networks with n sources and tw o terminals. For these two cases we pr esent the rate region for the pro blem. Basically we show that as lon g as ther e exists at least on e pa th from each source to each termina l, there exists an assignm ent of coding vectors to each ed ge in the network such that the terminals can recover the sum of the sou rces. This paper is organized as follows. Section II presents the network coding model that we shall b e assumin g. Section III contains our results for the case w hen ther e are two so urces and n termin als an d section IV contains the results and proofs for th e case when ther e are n sour ces and two te rminals. In section V we outline o ur conclusio ns. I I . N E T W O R K C O D I N G M O D E L In our model, we represen t th e network as a d irected grap h G = ( V , E ) . The network co ntains a set of source nodes S ⊂ V that ar e o bserving indepe ndent, discrete unit-en tropy sources and a set o f terminals T ⊂ V . Our network cod ing model is b asically the one presented in [3] . W e assume th at each edge in the network has un it capacity and can transmit one sym bol fr om a finite field o f size 2 m per unit time (we are free to ch oose m large enoug h). I f a given e dge has a higher capac ity , it can be treated as multiple unit capacity edges (f ractional capa cities can b e treated by choo sing m large enoug h). A directed ed ge e between node s v i and v j is represented as ( v i → v j ) . Thus head ( e ) = v j and tail ( e ) = v i . A pa th between two no des v i and v j is a seq uence of edg es { e 1 , e 2 , . . . , e k } su ch that tail ( e 1 ) = v i , head ( e k ) = v j and head ( e i ) = tail ( e i +1 ) , i = 1 , . . . k − 1 . The sign al on an edge ( v i → v j ) , is a lin ear com bination of the signals on the incom ing ed ges on v i and the source signal at v i (if v i ∈ S ). In this paper we assum e that the source nodes do n ot have any in coming e dges from other nodes. I f this is not the case o ne can always intro duce an artificial sou rce co nnected to the origin al sourc e node th at has no incom ing edges. W e shall only b e conce rned with networks tha t ar e directed acyclic and can theref ore b e treated as delay-fre e networks [ 3]. Let Y e i (such th at tail ( e i ) = v k and head ( e i ) = v l ) den ote the signal on the i th edge in E and let X j denote the j th source. Then , we h av e Y e i = X { e j | head ( e j )= v k } f j,i Y e j if v k ∈ V \ S , and Y e i = X { j | X j observed at v k } a j,i X j if v k ∈ S , where the coefficients a j,i and f j,i are from GF (2 m ) . Note that since the gr aph is directed acyclic, it is po ssible to expr ess Y e i for an edge e i in term s of the sources X j ’ s. Sup pose that there are n sou rces X 1 , . . . , X n . If Y e i = P n k =1 β e i ,k X k then we say that the g lobal coding vector of edg e e i is β e i = [ β e i , 1 · · · β e i ,n ] . W e shall also occasionally use the term c oding vecto r instead of global codin g vector in this paper . W e say that a nod e v i (or edge e i ) is downstream of another nod e v j (or edge e j ) if ther e exists a path from v j (or e j ) to v i (or e i ). I I I . C A S E O F T W O S O U R C E S A N D n T E R M I N A L S In th is section we state and pr ove the rate region f or th e network arith metic pro blem when there are two sources a nd n term inals. The basic id ea of th e proo f is the following. W e sho w that there exist a cer tain set of nodes that can obtain b oth the sources X 1 and X 2 and find a multicast code that multicasts the pair ( X 1 , X 2 ) to these nodes. W e then m odify the set of coding vectors so that all th e terminals can recover X 1 + X 2 while en suring that th e cod ing vectors remain valid. Theor em 1: Consider a dire cted a cylic grap h G = ( V , E ) with unit capac ity edges, two source nodes S 1 and S 2 and n terminal n odes T 1 , . . . , T n such that max-flow ( S i − T j ) ≥ 1 for all i = 1 , 2 an d j = 1 , . . . , n. At each source no de S i , there is a unit- rate so urce X i . The X i ’ s are ind ependen t. There exists an assignment of coding vectors to all edg es such th at each T i , i = 1 , . . . , n can recover X 1 + X 2 . Before emb arking on the proof of this result we define a modified gra ph that sha ll simplify our later arguments. 1) W e introdu ce artificial so urce nod es S ′ 1 and S ′ 2 such th at there exists a unit capacity edge S ′ i → S i . Sim ilarly we introdu ce artificial term inal nodes T ′ i and unit capac ity edges T i → T ′ i . Note that we are giv en the existence of at least on e path from S i → T j for all i, j . This in turn implies that m ax-flow ( S ′ 1 − T ′ j ) = max-flow ( S ′ 2 − T ′ j ) = max-flow (( S ′ 1 , S ′ 2 ) − T ′ j ) = 1 . 2) For each v irtual term inal T ′ j , j = 1 , . . . , n there exists a path fr om S ′ i to T ′ j for i = 1 , 2 . Let us den ote this b y path ( S ′ i − T ′ j ) . W e say th at two paths intersect if they have at least o ne node in c ommon. For a giv en terminal T ′ j , in g eneral the path ( S ′ 1 − T ′ j ) and path ( S ′ 2 − T ′ j ) could inter sect in many nod es. Note that th ey hav e to intersect at least once since th e edge T j → T ′ j is of unit capacity . Suppose that th e first intersectio n po int is denoted v j . As demon strated in Fig. 1 it is possible to S 1 ’ S 2 ’ v j T j ’ S 1 ’ S 2 ’ T j ’ Fig. 1. The figure on the left shows p ath ( S ′ 1 − T ′ j ) (in blue) and path ( S ′ 2 − T ′ j ) (in red). The figure on the right sho ws that one can find a ne w set of paths from S ′ 1 and S ′ 2 to T ′ j such that they share edges from v j to T ′ j . The first intersect ion of the new paths is at node v j . find a new set of path s fro m S ′ 1 − T ′ j and S ′ 2 − T ′ j so that they share the set o f edges f rom v j to T ′ j . W e a ssume th at suc h p aths have bee n fou nd f or all terminals. T hus fo r each termin al T ′ j there exists a correspo nding v j which den otes the first vertex wh ere the paths S ′ 1 − T ′ j and S ′ 2 − T ′ j meet. Note that the v j ’ s may not be distinc t. Now , consider the subg raph of G that is defined b y the union o f a ll these paths an d suppose that we call it G ′ . In our discu ssion we shall only be concerned with th e graph G ′ . 3) Note that G ′ is also a d irected acyclic graph. Therefor e a number ing o f the node s exists such that if there exists a path between node v i and v j then i < j . W e now number the n odes in G ′ in this manner . W e shall re fer to the fir st meeting point of path ( S ′ 1 − T ′ j ) and p ath ( S ′ 2 − T ′ j ) under this new numb ering as v α ( T j ) . Lemma 1: In the graph G ′ constructed as above, the fol- lowing pro perties hold f or all j = 1 , . . . , n . max-flow ( S ′ 1 − v α ( T j ) ) = 1 , (1) max-flow ( S ′ 2 − v α ( T j ) ) = 1 , and (2) max-flow (( S ′ 1 , S ′ 2 ) − v α ( T j ) ) = 2 . (3) Pr oof. Obvio us by the co nstruction of the grap h G ′ .  The previous c laim implies that th ere exists a network code so that the pair ( X 1 , X 2 ) can be multicast to each n ode v α ( T j ) , j = 1 , . . . , n using Th eorem 8 in [3] . Supp ose that such a network c ode is fou nd and the glob al codin g vectors for each edge in G ′ are fo und. Le t th ese g lobal codin g vector s be specified by the set β = { β e | e ∈ E ′ } . W e now present an algorithm that mod ifies β so th at each term inal T ′ i , i = 1 , . . . , n can recover X 1 + X 2 . This shall serve as a pro of of Theor em 1. First we sort the set { v α ( T 1 ) , . . . , v α ( T n ) } to o btain { v γ 1 , . . . , v γ n } so that γ 1 ≤ · · · ≤ γ n . Let the terminal n ode cor respondin g to th e node v γ i be denoted T ′ f ( γ i ) . As mentioned before it is possible that there exist terminals T i and T j such tha t α ( T i ) = α ( T j ) . Theref ore the set of γ i ’ s is not distinct. Co nsequently the mappin g f ( γ i ) is on e to many . W e do not make this explicit to av oid the notation becomin g too co mplex. T he steps are presented in Algorithm 1. It is important to no te th at this algorithm may replace the existing coding vector s assigned by the multicast c ode construction on some edges. W e now show that th e new Initialize demand [ i ] = 0 , i = 1 , . . . n ; 1 for k ← 1 to n do 2 if d emand[ f ( γ k ) ] == 0 then 3 for e ∈ path ( v γ k − T ′ f ( γ k ) ) do 4 β e = [1 1] ; 5 end 6 demand [ f ( γ k ) ] = 1; 7 for m ← k + 1 to n do 8 if deman d[ f ( γ m ) ] == 0 then 9 if the r e exists a pa t h ( v γ k − T ′ f ( γ m ) ) 10 then for e ∈ path ( v γ k − T ′ f ( γ m ) ) do 11 β e = [1 1] ; 12 end 13 demand [ f ( γ m ) ] = 1; 14 end 15 end 16 end 17 end 18 end 19 Algorithm 1 : Algorithm for assigning coding vecto rs so that each term inal can recover the sum of the two sources. global coding vector assignment is valid a nd is such that each terminal re ceiv es X 1 + X 2 . Pr oof of Th eor em 1. W e claim that the assignment of codin g vector s is valid at each stage of the algorith m an d by stage 1 ≤ k ≤ n , demand [ f ( γ k )] = 1 . • Base case (k =1). Note th at by the construction of G ′ there exists a path fr om v γ 1 to T f ( γ 1 ) . Th e algorithm shall assign coding vector [1 1] to those ed ges and set demand [ f ( γ 1 )] = 1 . W e only nee d to ensure that the assignment is valid. T o see the validity o f the assignment note that the g raph is acyclic, therefor e the coding vectors on path ( S ′ 1 − v γ 1 ) and path ( S ′ 2 − v γ 1 ) do not change. The assignm ents are only done on edges do wnstream of v γ 1 and are therefore valid. • Inductio n Step. Assume that th e c laim is true for all j = 1 , . . . , k and consider stage k + 1 . If for a g i ven j , the algorithm enters the fo r loop on lines 4-6, we call the node v γ j an active no de. 1) Case 1. If there e xists a path between some acti ve node v γ j in the set { v γ 1 , . . . , v γ k } and T ′ f ( γ k +1 ) then de mand [ f ( γ k +1 )] will be set to 1 at o ne o f the earlier stages. By the in ductive hyp othesis, the assignment is v alid. 2) Case 2. If demand [ f ( γ k +1 )] is still zero after k iter- ations of the algo rithm, th is implies that there doe s not exist a path between an activ e node and T ′ f ( γ k +1 ) i.e. there does not exist a path from a n active no de to any no de on path ( S ′ 1 − T ′ f ( γ k +1 ) ) and path ( S ′ 2 − T ′ f ( γ k +1 ) ) . Therefore the coding vectors on the edges in path ( S ′ 1 − T ′ f ( γ k +1 ) ) ∪ path ( S ′ 2 − T ′ f ( γ k +1 ) ) are unchan ged at the end of iteration k and are such that v γ k +1 receives ( X 1 , X 2 ) . This implies that setting β e = [1 1] for e ∈ path ( v γ k +1 − T f ( γ k +1 ) ) will ensure tha t demand [ f ( γ k +1 )] = 1 . This assign ment is valid since the co ding vector [1 1] lies in the span of the cod ing vector space of v γ k +1 . Furth ermore, there do es not e xist a path fro m v γ k +1 to any node on S k j =1 path ( S ′ 1 − v γ j ) ∪ path ( S ′ 2 − v γ j ) since the graph is acyclic. Therefore the assignment of codin g vectors to the p revious ed ges remains valid.  Note that conversely if any of the cond itions in the stateme nt of Theorem 1 is violated then there e xists some terminal that cannot ob tain the value of X 1 + X 2 . T o see this note that since the graph has unit-cap acity ed ges the max-flow between any pair of no des has to be an in teger . Furth er , if for examp le max-flow ( S 1 − T j ) = 0 , then the rece i ved signal at T j cannot depend on X 1 . Since , X 1 and X 2 are indep endent, X 1 + X 2 cannot be computed at T ′ j . I V . C A S E O F n S O U R C E S A N D T W O T E R M I NA L S W e now p resent the rate region for the situation when there are n sources and two terminals suc h that each terminal wants to recover the sum of the sources. T o show the main result we first demo nstrate that the original network can be transfor med in to a nother network where there exists e xactly one pa th from each source to each terminal. This en sures that when network co ding is perfo rmed on this tra nsformed g raph the g ain on the path from a sou rce to a terminal can be specified by a monomial. By a simple argument it then follows that coding vectors can be assigned so tha t the ter minals recover th e sum of the sou rces. Theor em 2: Consider a dir ected acylic grap h G = ( V , E ) with unit capacity edges. T here are n so urce no des S 1 , S 2 , . . . , S n and two term inal node s T 1 and T 2 such that max-flow ( S i − T j ) ≥ 1 for all i = 1 , . . . , n an d j = 1 , 2 . At the so urce n odes there are indepe ndent un it-rate sou rces X i , i = 1 , . . . , n . Ther e exists an assignmen t of cod ing vectors such that each termin al can recover the modulo -two sum of the source s P n i =1 X i . As before we mod ify the graph G b y introducing virtu al source nodes S ′ i , i = 1 , . . . n , virtu al terminals T ′ j , j = 1 , 2 and v irtual unit- capacity edges S ′ i → S i , i = 1 , . . . , n an d T j → T ′ j , j = 1 , 2 . Let th e set of sources be d enoted S = { S ′ 1 , . . . , S ′ n } . W e deno te the modified gr aph by G ′ . W e also need the follo wing defin itions. Definition 1: Exactly one path condition. Con sider two nodes v 1 and v 2 such ther e is a p ath P between v 1 and v 2 . W e say that th ere exists exactly o ne path between v 1 and v 2 if there does not exist another path P ′ between v 1 and v 2 such that P ′ 6 = P . Definition 2: Minimality . Consider the dire cted acyclic graph G ′ defined above, with sour ces S ′ 1 , . . . , S ′ n and terminals T ′ 1 and T ′ 2 such that max-flow ( S ′ i − T ′ j ) = 1 ∀ i = 1 , . . . , n and j = 1 , 2 . (4) The graph G ′ is said to b e minimal if the re moval of any edge fr om E ′ violates o ne of the e qualities in (4). T o sho w that Theorem 2 holds we first n eed an aux iliary lemma that we state and prove. Lemma 2: Consider th e graph G ′ as con structed above with sources S ′ 1 , . . . , S ′ n and termin als T ′ 1 and T ′ 2 . The re exists a subgrap h G ∗ of G ′ such that G ∗ is minimal and there e xists exactly one path from S ′ i to T ′ j for i = 1 , . . . , n and j = 1 , 2 in G ∗ . Pr oof. W e proceed by induction o n the number of so urces. • Base case n = 1 . In th is case ther e is only one sou rce S ′ 1 and b oth the terminals need to recover X 1 . Note that we are given the existence of path ( S ′ 1 − T ′ 1 ) an d path ( S ′ 1 − T ′ 2 ) in G ′ . In gene ral these paths c an inter sect a t mu ltiple nodes which m ay imply that there exist multiple paths (for example) fro m S ′ 1 to T ′ 1 . Now , f rom path ( S ′ 1 − T ′ 1 ) and path ( S ′ 1 − T ′ 2 ) we can find th e last n ode where these two paths meet. Let this last node be denote d u 1 . Then as sho wn in Fig. 2 we can find a new set of pa ths from S ′ 1 to T ′ 1 and S ′ 1 to T ′ 2 that overlap fr om S ′ 1 to u 1 and have no overlap thereafter . Cho ose G ∗ to be the unio n of these n ew set o f paths. It is e asy to see that in G ∗ there is exactly one path from S ′ 1 to T ′ 1 and exactly one path from S ′ 1 to T ′ 2 . Moreover removin g any e dge fro m G ∗ would cause at least one p ath to not exist. S 1 ’ T 1 ’ T 2 ’ S 1 ’ T 1 ’ T 2 ’ u 1 Fig. 2. The figure on the left shows path ( S ′ 1 − T ′ 1 ) (in blue) and p ath ( S ′ 1 − T ′ 2 ) (in red). The figure on the right shows that one can find a new set of paths from S ′ 1 to T ′ 1 and T ′ 2 such that they share edges from S ′ 1 to u 1 and hav e no intersec tion thereafte r . • Inductio n Step . W e now assume the indu ction h ypothesis for n − 1 sources. i.e. ther e exists a minimal subgraph G ∗ n − 1 of G ′ such th at there is exactly on e path f rom S ′ i to T ′ j for i = 1 , . . . , n − 1 a nd j = 1 , 2 . Using this hypoth esis we shall show the result in the case when there are n sources. As a first step colo r the edg es in the subgraph G ∗ n − 1 , blu e (the remaining edges in G ′ have no color). The conditions on G ′ guaran tee the existence o f path ( S ′ n − T ′ 1 ) and path ( S ′ n − T ′ 2 ) . Note th at these path s may intersect at m any nodes. W e prepr ocess them in the f ollowing manner . Find the last n ode not in G ∗ n − 1 belongin g to both path ( S ′ n − T ′ 1 ) and path ( S ′ n − T ′ 2 ) . Suppose that this n ode is deno ted v r . Find a new set of paths su ch that they share edges f rom S ′ n to v r and ca ll these new paths path ( S ′ n − T ′ 1 ) and path ( S ′ n − T ′ 2 ) . Color all edges on path ( S ′ n − T ′ 1 ) an d path ( S ′ n − T ′ 2 ) r ed. Th is would im ply that some ed ges have a pair of colors. Now , conside r the subgra ph induced by the u nion of the blue and red subgrap hs th at we d enote G br . Find the fir st nod e at which path ( S ′ n − T ′ 1 ) intersects the blue subgraph and call that node u 1 . Similarly find the first n ode at wh ich path ( S ′ n − T ′ 2 ) intersects th e blue subgrap h and call that n ode u 2 . Observe th at in G ∗ n − 1 there has to exist a path ( S ′ i − T ′ j ) fo r some i = 1 , . . . , n − 1 and j = 1 , 2 that passes throug h u 1 . T o see th is a ssume o therwise. Th is implies that u 1 does not lie on any path connecting one of th e sources to one o f the terminals. Therefore the incoming and the ou tgoing ed ges of u 1 can be r emoved without violating th e max-flow cond itions in (4). This contrad icts the minimality of G ∗ n − 1 . Therefore we are guaranteed t hat there exists at least one source such that there exists an exclusi vely blue path from it to u 1 in G ∗ n − 1 . A similar statement hold s for th e nod e u 2 . W e no w establish the statement of the lemma wh en there are n sources. – Case 1. I n G br there exists a p ath fro m u 1 to T ′ 2 such that all edg es on this path have a blue compon ent. First, we remove the color red f rom all edges on path ( S ′ n − T ′ 2 ) \ path ( S ′ n − T ′ 1 ) . Next, for m a subset of the sources denoted S ( u 1 ) in the following ma nner . For each source S ′ i , i = 1 , . . . , n d o the following. i) If there exists a path ( with edges o f color red or blue) from S ′ i to u 1 , ad d it to set S ( u 1 ) 1 . Let G ( u 1 ) denote the subgra ph induced by S S ′ i ∈ S ( u 1 ) path ( S ′ i − u 1 ) . Consider the graph obtained by removing the sub - graph G ( u 1 ) from G br . W e denote this graph G − br . W e claim that the max-flow co nditions in (4) co ntinue to h old over G − br for the set o f sou rces S \ S ( u 1 ) . Furthermo re there still exist p ath ( u 1 − T ′ 1 ) an d path ( u 1 − T ′ 2 ) in G − br . T o see this note th at the max-flow cond itions f or a source S ′ i ∈ S \ S ( u 1 ) can b e v iolated only if an edge e belong ing to a path fr om S ′ i to T ′ j , j = 1 , 2 is removed. This happ ens only if there exists a path from e to u 1 which contrad icts the fact that S ′ i ∈ S \ S ( u 1 ) . Next, ther e still exist path s from u 1 to the terminals since the edges on these p aths are downstream of u 1 . If any of th ese w as removed b y the pr ocedure , this would contr adict th e acyclicity o f the g raph. Note tha t the subgraph G ( u 1 ) contains a set of sources S ( u 1 ) and a single node u 1 such that there exists exactly one p ath fro m each so urce in S ( u 1 ) to u 1 . Th is has to b e true fo r the sour ces in S ( u 1 ) \{ S n } otherwise the min imality of G ∗ n − 1 would be contr a- dicted and is true for S n by con struction. Next, introd uce an artificial source S a and an ed ge S a → u 1 in G − br . Note that | S \ S ( u 1 ) | ≤ n − 2 , which means that the total n umber of sour ces in G − br (includin g S a ) is at m ost n − 1 . T herefor e the induction hy pothesis can b e app lied on G − br i.e. th ere 1 A path from S ′ i to u 1 cannot have a (red,blue ) edge since u 1 is the first node where a red path inte rsects the blue subgraph exists a subgraph of G − br such that there e xists e xactly one path from ( S \ S ( u 1 ) ) ∪ { S a } to each term inal. Suppose th at we find this subg raph. Now r emove S a and the edge S a → u 1 from this subgr aph and augmen t it with the subg raph G ( u 1 ) found ear lier . W e claim that th e resulting gr aph has the p roperty that th ere exists exactly one path fro m each source to eac h terminal. T o see this n ote that there exists only one path from a source S ′ i ∈ S \ S ( u 1 ) to T ′ j , j = 1 , 2 . This is beca use ev en after the introd uction of G ( u 1 ) there does not exist a pa th from S ′ i to u 1 in this graph. There fore the in troduction o f G ( u 1 ) cannot in troduce add itional paths between S ′ i ∈ S \ S ( u 1 ) and the terminals. Next we argue for a source S ′ i ∈ S ( u 1 ) . Note that there exists exactly one p ath f rom u 1 to b oth the terminals so th e co ndition can be violated o nly if there exist m ultiple p aths from S ′ i ∈ S ( u 1 ) to u 1 , but the co nstruction of G ( u 1 ) rules this out. – Case 2. I n G br there exists a path fro m u 2 to T ′ 1 such that all edg es on th is path h av e a blue compon ent. This case can be han dled in exactly the same manner as in case 1 b y removing th e color red from all edges on path ( S ′ n − T ′ 1 ) \ path ( S ′ n − T ′ 2 ) and applying similar argu ments for u 2 . – Case 3. In G br there (a) does not exist a path with blue ed ges from u 1 to T ′ 2 , and (b) does no t exist a path with blue edges f rom u 2 to T ′ 1 . As shown pre viously u 1 lies on some path from S ′ i to T ′ j for some i and j in G ∗ n − 1 . In the current case there does not exist a blue path from u 1 to T ′ 2 . Th erefore there has to exist a b lue path f rom u 1 to T ′ 1 in G ∗ n − 1 . A similar argument shows that th ere has to exist a blue path from u 2 to T ′ 2 in G ∗ n − 1 . Note that the exclusi vely red paths from S ′ n to u 1 and u 2 are such that they overlap u ntil their last inte rsec- tion po int. No w , ch oose the d esired subgrap h to be the union of G ∗ n − 1 and the red paths, path ( S ′ n − u 1 ) and path ( S ′ n − u 2 ) i.e. G ∗ n = G ∗ n − 1 ∪ pa th ( S ′ n − u 1 ) ∪ path ( S ′ n − u 2 ) . By the indu ction hypo thesis th ere exists exactly one path between S ′ i , i = 1 , . . . , n − 1 and T ′ j , j = 1 , 2 . Th is continu es to be true in G ∗ n , since the red edges cann ot be reached from th e blue edges. T o see that there is exactly on e path from S ′ n to T ′ 1 , assume other wise and ob serve that there is exactly one pa th from S ′ n to u 1 by the construction of the red paths. Thus th e on ly way there can be multiple p aths from S ′ n to T ′ 1 is if there are m ultiple paths f rom u 1 to T ′ 1 , but this would contradict the induction hypo thesis since this would imply that ther e exists so me S ′ i , i = 1 , . . . , n − 1 that has m ultiple paths to T ′ 1 . A similar argument shows that there exists exactly one path from S ′ n to T ′ 2 .  Pr oof of Theo r em 2. Fro m Lemma 2 we know that it is possible to find a subg raph G ∗ of G such that there exists exactly one path from S ′ i to T ′ j for all i = 1 , . . . , n and j = 1 , 2 . Suppo se that we find G ∗ . W e will show that each terminal can recover P i =1 X i by assigning a ppropr iate local encod ing responsibilities for every node . Consider a node v ∈ G ∗ and let Γ o ( v ) and Γ i ( v ) represent the set of outg oing edges from v and inc oming edges into v r espectiv ely . Let Y e represent the symbol transmitted on edge e . Each node operates in the following m anner . Y e = X e ′ ∈ Γ i ( v ) α × Y e ′ for e ∈ Γ o ( v ) (5) i.e. e ach nod e scales the symbo l on each in put edg e by α (note that α is the sam e f or every nod e) and the fo rwards the sum of the scaled inputs on all output edges. W e shall see that the setting α = 1 will ensure that each te rminal recovers P n i =1 X i . T o see this we exam ine the transfer matrix from the inp uts [ X 1 . . . X n ] 1 × n to the o utput Z T j → T ′ j denoted M j which is of dimensio n n × 1 i.e. Z T j → T ′ j = [ X 1 . . . X n ] M j . Note that the i th entry of M j correspo nds to the sum of the gains fr om all p ossible paths from S ′ i to T ′ j . The construction of G ∗ ensures that the re is exactly one such path. The refore the i th entry of M j will be a non-zero m onomial in α for all i = 1 , . . . , n . Now setting α = 1 will ensur e that all the monom ials evaluate to 1 i.e. M j = [1 · · · 1] , which implies that Z T j → T ′ j = P n i =1 X i .  As in the p revious sectio n it is cle ar that if any o f th e condition s in the statemen t of Theorem 2 are violated then either term inal T 1 or T 2 will be unable to find P n i =1 X i . For example if max -flow ( X j − T 1 ) = 0 then the received signal at T 1 cannot dep end o n X j . Thus, T 1 cannot compu te any function tha t depend s on X j . V . C O N C L U S I O N W e considered the pr oblem of finding the rate r egion for the p roblem of commu nicating the m odulo- 2 sum of a set of indepen dent unit rate sources to a set o f termin als in the case when the un derlying network c an be mo deled as a directed acyclic grap h with un it cap acity ed ges. The rate region h as been presented for the cases wh en there are (a) tw o sources and n terminals, and (b) n sou rces an d two termin als. Rate r egions for arbitrar y number of sources and termin als over gen eral network top ologies po ssibly co ntaining cycles are cur rently under investi gation . R E F E R E N C E S [1] R. Ahlswede, N. Cai, S.-Y . Li, and R. W . Y eung, “Network Informatio n Flo w, ” IEEE T rans. on Info. Th. , vo l. 46, no. 4, pp. 1204–1216, 2000. [2] S. -Y . Li, R . W . Y eung, and N. Ca i, “Linear Network Coding, ” IEEE T rans. on Info. Th. , vol. 49, no. 2, pp. 371–381 , 2003. [3] R. Koett er and M. M ´ edard, “Bey ond Rout ing: An Algebraic Approach to Network Coding, ” in IEEE Infocom , 2002. [4] D. Slepia n and J. W olf, “Noiseless coding of correlat ed information sources, ” IEEE T rans. on Info. Th. , vol. 19, pp. 471–480, Jul. 1973. [5] T . Ho, M. M ´ edard, M. Effros, and R. K oetter , “Netw ork Coding for Correla ted Sources, ” in CISS , 2004. [6] A. Ramamoorthy , K. Jain, P . A. Chou, and M. Effros, “Separa ting Distrib uted Source Coding from Network Coding, ” IEEE T rans. on Info. Th. , vol. 52, pp. 2785–2795, June 2006. [7] Y . Wu, V . Stanko vi ´ c, Z. Xiong, and S. Y . Kun g, “On practical design for joint distributed source coding and netwo rk coding, ” in P r oceedin gs of the Fi rst W orkshop on Network Coding, Theory and Applications, Riva del Garda, Italy , 2005.