Linear Network Coding

Transcription

1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 2, FEBRUARY Linear Network Codin Shuo-Yen Robert Li, Senior Member, IEEE, Raymond W. Yeun, Fellow, IEEE, Nin Cai Abstract Consider a communication network in which certain source nodes multicast inormation to other nodes on the network in the multihop ashion where every node can pass on any o its received data to others. We are interested in how ast each node can receive the complete inormation, or equivalently, what the inormation rate arrivin at each node is. Allowin a node to encode its received data beore passin it on, the question involves optimization o the multicast mechanisms at the nodes. Amon the simplest codin schemes is linear codin, which reards a block o data as a vector over a certain base ield allows a node to apply a linear transormation to a vector beore passin it on. We ormulate this multicast problem prove that linear codin suices to achieve the optimum, which is the max-low rom the source to each receivin node. Index Terms Codin, network routin, switchin. I. INTRODUCTION DEFINE a communication network as a pair,, where is a inite directed multiraph is the unique node in without any incomin edes. A directed ede in is called a channel in the communication network,. The special node is called the source, while every other node may serve as a sink as we shall explain. A channel in raph represents a noiseless communication link on which one unit o inormation (e.., a bit) can be transmitted per unit time. The multiplicity o the channels rom a node to another node represents the capacity o direct transmission rom to. In other words, every sinle channel has unit capacity. At the source, a inite amount o inormation is enerated multicast to other nodes on the network in the multihop ashion where every node can pass on any o its received data to other nodes. At each nonsource node which serves as a sink, the complete inormation enerated at is recovered. We are naturally interested in how ast each sink node can receive the complete inormation. As an example, consider the multicast o two data bits,, rom the source in the communication network depicted by Manuscript received November 4, 1999; revised September 12, The work o S.-Y. R. Li was supported in part by an MTK Computers Grant. The work o R. W. Yeun N. Cai was supported in part by the RGC Earmarked Grant CUHK4343/98E. The material in this paper was presented in part at the 1999 IEEE Inormation Theory Workshop, Metsovo, Greece, June/July S.-Y. R. Li R. W. Yeun are with the Department o Inormation Enineerin, The Chinese University o Hon Kon, N.T., Hon Kon ( bobli@ie.cuhk.edu.hk; whyeun@ie.cuhk.edu.hk). N. Cai was with Department o Inormation Enineerin, The Chinese University o Hon Kon, N.T., Hon Kon. He is now with the Fakultät ür Mathematik, Universität Bieleeld, Bieleeld, Germany ( cai@mathematik.uni-bieleeld.de). Communicated by V. Anantharam, Associate Editor or Communication Networks. Diital Object Identiier /TIT Fi. 1. Two communication networks. Fi. 1(a) to both nodes. One solution is to let the channels,,, carry the bit channels,,, carry the bit. Note that in this scheme, an intermediate node sends out a data bit only i it receives the same bit rom another node. For example, the node receives the bit sends a copy on each o the two channels. Similarly, the node receives the bit sends a copy into each o the two channels. In our model, we assume that there is no processin delay at the intermediate nodes. Unlike a conserved physical commodity, inormation can be replicated or coded. Introduced in [1] (see also [5, Ch. 11]), the notion o network codin reers to codin at the intermediate nodes when inormation is multicast in a network. Let us now illustrate network codin by considerin the communication network depicted by Fi. 1(b). In this network, we want to multicast two bits rom the source to both the nodes. A solution is to let the channels,, carry the bit, channels,, carry the bit, channels,, carry the exclusive-or. Then, the node receives, rom which the bit can be decoded. Similarly, the node can decode the bit rom. The codin/decodin scheme is assumed to have been areed upon beoreh. It is not diicult to see that the above scheme is the only solution to the problem. In other words, without network codin, it is impossible to multicast two bits per unit time rom the source to both the nodes. This shows the advantae o network codin. In act, replication o data can be rearded as a special case o network codin. As we have pointed out, the natures o physical commodities inormation are very dierent. Nevertheless, both o them are overned by certain laws o low. For the network low o a physical commodity, we have the ollowin. The law o commodity low: The total volume o the outlow rom a nonsource node cannot exceed the total volume o the inlow /03$ IEEE

2 372 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 2, FEBRUARY 2003 The counterpart or a communication network is as ollows. The law o inormation low: The content o any inormation lowin out o a set o nonsource nodes can be derived rom the accumulated inormation that has lown into the set o nodes. Ater all, inormation replication codin do not increase the inormation content. The inormation rate rom the source to a sink can potentially become hiher hiher when the permitted class o codin schemes is wider wider. However, the law o inormation low limits this inormation rate to the max-low (i.e., the maximum commodity low) rom the source to that particular sink or a very wide class o codin schemes. The details are iven in [1]. It has been proved in [1] that the inormation rate rom the source to a set o nodes can reach the minimum o the individual max-low bounds throuh codin. In the present paper, we shall prove constructively that by linear codin alone, the rate at which a messae reaches each node can achieve the individual max-low bound. (This result is somewhat stroner than the one in [1]. Please reer to the example in Section III.) More explicitly, we treat a block o data as a vector over a certain base ield allow a node to apply a linear transormation to a vector beore passin it on. A preliminary version o this paper has appeared in the conerence proceedins [3]. The remainder o the paper is oranized as ollows. In Section II, we introduce the basic notions, in particular, the notion o a linear-code multicast (LCM). In Section III, we show that with a eneric LCM, every node can simultaneously receive inormation rom the source at rate equal to its max-low bound. In Section IV, we describe the physical implementation o an LCM irst when the network is acyclic then when the network is cyclic. In Section V, we present a reedy alorithm or constructin a eneric LCM or an acyclic network. The same alorithm can be applied to a cyclic network by expin the network into an acyclic network. This results in a timevaryin LCM, which, however, requires hih complexity in implementation. In Section VI, we introduce the time-invariant LCM (TILCM) present a heuristic or constructin a eneric TILCM. Section VII presents concludin remarks. II. BASIC NOTIONS In this section, we irst introduce some raph-theoretic terminoloy notations which will be used throuhout the paper. Then we will introduce the notion o an LCM, an abstract alebraic description o a linear code on a communication network. Convention: The eneric notation or a nonsource node will be,,,,, or. The notation will st or any channel rom to. Deinition: Over a communication network a low rom the source to a nonsource node is a collection o channels, to be called the busy channels in the low, such that 1) the subnetwork deined by the busy channels is acyclic, i.e., the busy channels do not orm directed cycles; 2) or any node other than, the number o incomin busy channels equals the number o outoin busy channels; 3) the number o outoin busy channels rom equals the number o incomin busy channels to. In other words, a low is an acyclic collection o channels that abides by the law o commodity low. The number o outoin busy channels rom will be called the volume o the low. The node is called the sink o the low. All the channels on the communication network that are not busy channels o the low are called the idle channels with respect to the low. Proposition 2.1: The busy channels in a low with volume can be partitioned into simple paths rom the source to the sink. The proo o this proposition is omitted. Notation: For every nonsource node on a network,, the maximum volume o a low rom the source to is denoted as, or simply when there is no ambiuity. Deinition: A cut on a communication network, between the source a nonsource node means a collection o nodes which includes but not. A channel is said to be in the cut i. The number o channels in a cut is called the value o the cut. The well-known Max-Flow Min-Cut Theorem (see, or example, [2, Ch. 4, Theorem 2.3]) still applies despite the acyclic restriction in the deinition o a low. Max-Flow Min-Cut Theorem: For every nonsource node, the minimum value o a cut between the source a node is equal to. We are now ready to deine a linear code multicast. Notation: Let be the maximum o over all. Throuhout Sections II V, the symbol will denote a ixed -dimensional vector space over a suiciently lare base ield. Convention: The inormation unit is taken as a symbol in the base ield. In other words, symbol in the base ield can be transmitted on a channel every unit time. Deinition: A linear-code multicast (LCM) on a communication network, is an assinment o a vector space to every node a vector to every channel such that 1) ; 2) or every channel ; 3) or any collection o nonsource nodes in the network The notation is or linear span. Condition 3) says that the vector spaces on all nodes inside toether have the same linear span as the vectors on all channels to nodes in rom outside o. Conditions 2) 3) show that an LCM abides by the law o inormation low stated in Section I. The vector assined to every channel may be identiied with a -dimensional column vector over the base ield o by choosin a basis or. Applyin Condition 3) to the collection o a sinle nonsource node, the space is linearly spanned by vectors

3 LI et al.: LINEAR NETWORK CODING 373 on all incomin channels to. This shows that an LCM on a communication network is completely determined by the vectors it assins to the channels. Toether with Condition 2), we have 4) The vector assined to an outoin channel rom must be a linear combination o the vectors assined to the incomin channels o. Condition 4) may be rearded as the law o inormation low at a node. However, unlike in network low theory, the law o inormation low bein observed or every sinle node does not necessarily imply it bein observed or every set o nodes when the network contains a directed cycle. A counterexample will appear in Section IV. An LCM speciies a mechanism or data transmission over the network as ollows. We encode the inormation to be transmitted rom as a -dimensional row vector, which we shall call the inormation vector. Under the transmission mechanism prescribed by the LCM, the data lowin on a channel is the matrix product o the inormation (row) vector with the (column) vector. In this way, the vector acts as the kernel in the linear encoder or the channel. As a direct consequence o the deinition o an LCM, the vector assined to an outoin channel rom a node is a linear combination o the vectors assined to the incomin channels to. Consequently, the data sent on an outoin channel rom a node is a linear combination o the data sent on the incomin channels to. Under this mechanism, the amount o inormation reachin a node is iven by the dimension o the vector space when the LCM is used. The physical realization o this mechanism will be discussed in Section IV. Example 2.2: Consider the multicast o two bits,, rom to in the communication network in Fi. 1(b). This is achieved with the LCM speciied by Hence,, which is at most equal to the value o the cut. In particular, is upper-bounded by the minimum value o a cut between, which by the Max-Flow Min-Cut Theorem is equal to. This corollary says that is an upper bound on the amount o inormation received by when an LCM is used. III. ACHIEVING THE MAX-FLOW BOUND THROUGH A GENERIC LCM In this section, we derive a suicient condition or an LCM to achieve the max-low bound on in Proposition 2.3. Deinition: An LCM on a communication network is said to be eneric i the ollowin condition holds or any collection o channels or : or i only i the vectors,,, are linearly independent. I are linearly independent, then since. A eneric LCM requires that the converse is also true. In this sense, a eneric LCM assins vectors which are as linearly independent as possible to the channels. Example 3.1: With respect to the communication network in Fi. 1(b), the LCM in Example 2.2 is a eneric LCM. However, the LCM deined by is not eneric. This is seen by considerin the set o channels where The data sent on a channel is the matrix product o the row vector with the column vector assined to that channel by. For instance, the data sent on the channel is. Note that, in the special case when the base ield o is GF, the vector reduces to the exclusive-or in an earlier example. Proposition 2.3: For every LCM on a network, or all nodes Then,but are not linearly independent. Thereore, is not eneric. Lemma 3.2: Let be a eneric LCM. Any collection o channels rom a node with must be assined linearly independent vectors by. Theorem 3.3: I is a eneric LCM on a communication network, then or all nodes Proo: Fix a nonsource node any cut between the source Proo: Consider a node not equal to. Let be the common value o the minimum value o a cut between. The inequality

4 374 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 2, FEBRUARY 2003 ollows rom Proposition 2.3. Thus, we only have to show that. Let or any cut between. We will show that by contradiction. Assume let be the collection o cuts between such that. Since implies, where is the set o all the nodes in, is nonempty. By the assumption that is a eneric LCM, the number o edes out o is at least,. Thereore,. Then there must exist a minimal member in the sense that or any,. Clearly, because. Let be the set o channels in cut be the set o boundary nodes o, i.e., i only i there is a channel, such that. Then or all, which can be seen as ollows. The set o channels in cut but not in is iven by. Since is an LCM I, then, is con- the subspace spanned by the channels in cut tained by. This implies Proo: By removin an ede, the value o a cut between the source node (respectively, node ) is reduced by i ede is in, otherwise, the value o is unchaned. By the Max-Flow Min-Cut Theorem, we see that are reduced by at most when ede is removed rom the raph. Now consider the value o a cut between the source node.i contains node, then ede is not in,, thereore, the value o remains unchaned upon the removal o ede.i does not contain node, then is a cut between the source node. By the Max-Flow Min-Cut Theorem, the value o is at least. Then upon the removal o ede, the value o is lower-bounded by. Hence, by the Max-Flow Min-Cut Theorem, remains to be upon the removal o ede. The lemma is proved. Example 3.5: Consider a communication network or which,,or or nodes in the network. The source is to broadcast 12 symbols,, taken rom a suiciently lare base ield. (Note that is the least common multiple o,,.) Deine the set or For simplicity, we use the second as the time unit. We now describe how,, can be broadcast to the nodes in,, in 3, 4, 12 s, respectively, assumin the existence o an LCM on the network or. a) Let be an LCM on the network with. Let a contradiction. Thereore, or all,. For all, since implies Then, by the deinition o a eneric LCM, is a collection o vectors such that. Finally, by the Max-Flow Min-Cut Theorem,, since,. This is a contradiction to the assumption that. The theorem is proved. An LCM or which or all provides a way or broadcastin a messae enerated at the source or which every nonsource node receives the messae at rate equal to. This is illustrated by the next example, which is based upon the assumption that the base ield o is an ininite ield or a suiciently lare inite ield. In this example, we employ a technique which is justiied by the ollowin lemma. Lemma 3.4: Let, be nodes such that,,, where. By removin any ede in the raph, are reduced by at most, remains unchaned. In the irst second, transmit as the inormation vector usin, in the second second, transmit, in the third second, transmit. Ater 3 s, all the nodes in can recover,,. Throuhout this example, we assume that all transmissions computations are instantaneous. b) Let be a vector in such that intersects trivially with or all in, i.e., or all in. Such a vector can be ound when is suiciently lare because there are a inite number o nodes in. Deine or. Now remove incomin edes o nodes in, i necessary, so that becomes i is in, otherwise, remains unchaned. This is possible by virtue o Lemma 3.4. Let be an LCM on the resultin network with. Let transmit as the inormation vector usin in the ourth second. Then all the nodes in can recover hence,,. c) Let be two vectors in such that intersects with trivially or all in, i.e.,,, or all in. Deine or. Now remove incomin edes o nodes in, i necessary, so that becomes

5 LI et al.: LINEAR NETWORK CODING 375 i is in or, otherwise, remains unchaned. Aain, this is possible by virtue o Lemma 3.4. Now let be an LCM on the resultin network with. In the ith the sixth seconds, transmit as the inormation vectors usin. Then all the nodes in can recover. d) Let be two vectors in such that, intersects with trivially or all in, i.e.,,,, or all in. Deine. In the seventh eihth seconds, transmit as the inormation vectors usin. Since all the nodes in already knows, upon receivin, can then be recovered. e) Deine. In the nineth tenth seconds, transmit as the inormation vectors usin. Then can be recovered by all the nodes in. ) Deine. In the eleventh twelveth seconds, transmit as the inormation vectors usin. Then can be recovered by all the nodes in. Let us now ive a summary o the precedin scheme. In the th second or via the eneric LCM, each node in receives all our dimensions o, each node in receives three dimensions o, each node in receives one dimension o. In the ourth second, via the eneric LCM, each node in receives the vector, which provides the three missin dimensions o,, (one dimension or each) durin the irst 3 s o multicast by. At the same time, each node in receives one dimension o. Now, in order to recover, each node in needs to receive the two missin dimensions o durin the ourth second. This is achieved by the eneric LCM in the ith sixth seconds. So ar, each node in has received one dimension o or via durin the irst 3 s, one dimension o or rom via durin the ourth to sixth seconds. Thus, it remains to provide the six missin dimensions o,, (two dimensions or each) to each node in, this is achieved in the seventh to the twelveth seconds via the eneric LCM. Remark: The scheme in Example 3.5 can readily be eneralized to arbitrary sets o max-low values. The details are omitted here. In the scheme in Example 3.5, at the end o the 12-s session, each node receives a messae o 12 symbols taken rom the base ield. Thus, the averae inormation rate arrivin at each node over the whole session is 1 symbol/s. The result in [1] asserts that this rate, which is the minimum o the individual max-low bounds o all the nodes in the network, can be achieved. However, it is seen in our scheme that the nodes in,, can actually receive the whole messae in the irst 3, 4, 12 s, respectively. In this sense, each node in our scheme can receive the messae at rate equal to its individual max-low bound. Thus, our result is somewhat stroner than that in [1]. However, our result does not mean that inormation can be multicast continually rom the source to each node at rate equal to its individual max-low bound. I it is not necessary or the nodes in to receive the messae, i.e., the messae is multicast to the nodes in only, the session can be terminated ater 4 s. Then the averae inormation rate arrivin at each node in over the truncated session is 3 symbols/s, with the nodes in receivin the whole messae in the irst 3 4 s, respectively. IV. THE TRANSMISSION SCHEME ASSOCIATED WITH AN LCM Let be an LCM on a communication network,, where the vectors assined to outoin channels linearly span a -dimensional space. As beore, the vector assined to a channel is identiied with a -dimensional column vector over the base ield o by means o the choice o a basis. On the other h, the total inormation to be transmitted rom the source to the rest o the network is represented by a -dimensional row vector, called the inormation vector. Under the transmission scheme prescribed by the LCM, the data lowin over a channel is the matrix product o the inormation vector with the column vector.wenow consider the physical realization o this transmission scheme associated with an LCM. Deinition: A communication network, is said to be acyclic i the directed multiraph does not contain a directed cycle. Lemma 4.1: An LCM on an acyclic network, is an assinment o a vector space to every node a vector to every channel such that 1) ; 2) or every channel ; 3) or every nonsource node, the space is the linear span o vectors on all incomin channels to. In other words, or an acyclic network, the law o inormation low bein observed or every sinle node implies it bein observed or every set o nodes. Proo: Let be a set o nonsource nodes on an acyclic network. Let an ede be called an internal ede o when. Similarly, let an ede be called an incomin ede o when. We need to show that, or every internal ede o, is enerated by is an incomin ede o Because the network is acyclic, there exists a node in without any ede, where. Thus, 1) every incomin ede o is an incomin ede o. By induction on, we may assume that or every internal ede o 2) is enerated by : is an incomin ede o. Given an internal ede o, we need to show that is enerated by : is an incomin ede o. Since is enerated by is an ede

6 376 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 2, FEBRUARY 2003 it suices to show that is enerated by : is an incomin ede o. I the ede is incomin o, then it is incomin to by ). Otherwise, is enerated by is an incomin ede o accordin to 2), thereore, is also enerated by is an incomin ede o because o 1). Lemma 4.2: The nodes on an acyclic communication network can be sequentially indexed such that every channel is rom a smaller indexed node to a larer indexed node. Lemma 4.3: Assume that nodes in a communication network are sequentially indexed as such that every channel is rom a smaller indexed node to a larer indexed node. Then, every LCM on the network can be constructed by the ollowin procedure: or (j =0; j n; j++) arrane all outoin channels X jy rom X j in an arbitrary order; take one outoin channel rom X j at a time let the channel taken be X jy ; assin v(x jy ) to be a vector in the space v(x j ); v(x j+1) = linear span by vectors v(xx j+1) on all incomin channels XX j+1 to X j+1; On an acyclic network, a straihtorward realization o the above transmission scheme is as ollows. Take one node at a time accordin to the sequential indexin. For each node, wait until data is received rom every incomin channel beore perormin the linear encodin. Then send the appropriate data on each outoin channel. This physical realization o an LCM over an acyclic network, however, does not apply to a network that contains a directed cycle. This is illustrated by the ollowin example. Example 4.4: Let be vectors in, where are linearly independent. Deine This speciies an LCM on the network illustrated in Fi. 2 i the vector is a linear combination o. Otherwise, the unction ives an example in which the law o inormation low is observed or every sinle node but not observed or Fi. 2. An LCM on a cyclic network. every set o nodes. Speciically, the law o inormation low is observed or each o the nodes,,, but not or the set o nodes,,. Now, assume that Then, is an LCM. Write the inormation vector as, where belon to the base ield o. Accordin to the transmission scheme associated with the LCM, all three channels on the directed cycle transmit the same data. This leads to the loical problem o how any o these cyclic channels acquires the data in the irst place. In order to realize the transmission scheme associated with an LCM over a network containin a directed cycle, we shall introduce the parameter o time into the scheme. Instead o transmittin a sinle data symbol (i.e., an element o the base ield o ) throuh each channel, we shall transmit a time-parameterized stream o symbols. In other words, the channel will be time-slotted. Concomitantly, the operation o codin at a node will be time-slotted, as well. Deinition: Given a communication network, a positive inteer, the associated memoryless communication network denoted as, is deined as ollows. The set o nodes in includes the node all the pairs o the type,, where is a nonsource node in ranes throuh inteers to. The channels in the network, belon to one o the three types listed below. For any nonsource nodes in, 1) or, the multiplicity o the channel rom to, is the same as that o the channel in the network, ; 2) or, the multiplicity o the channel rom, to, is the same as that o the channel in the network, ; 3) or, the multiplicity o the channel rom, to, is equal to. Lemma 4.5: The memoryless communication network, is acyclic. Lemma 4.6: There exists a ixed number, independent o, such that or all nonsource nodes in,, the maximum volume o a low rom to the node, in, is at least times.

7 LI et al.: LINEAR NETWORK CODING 377 Proo: Given a nonsource node in, consider a maximum low rom to in the network,. This maximum low may be partitioned into directed simple paths rom to. Let be the maximum lenth amon these directed simple paths. Then, there exists a low rom to the node, in, that is isomorphic to the maximum low in,. Moreover, time-shited isomorphs o this low in, are lows rom to, or. Moreover, the isomorphs are ede-disjoint lows in the network, due to the acyclic nature in the deinition o a low. The union o these lows, toether with edes rom, to, or in,, constitute a low rom to, with volume times. The proo is completed by choosin to be the larest minus amon all. Transmission o data symbols over the network, may be interpreted as memoryless transmission o data streams over the network, as ollows. 1) A symbol sent rom to, in, corresponds to the symbol sent on the channel in, durin the time slot. 2) A symbol sent rom, to, in, corresponds to the symbol sent on the channel in, durin the time slot. This symbol is a linear combination o symbols received by durin the time slot is unrelated to symbols received earlier by. 3) The channels rom, to, or siniy the accumulation o received inormation by the node in, over time. Now suppose an LCM has been constructed on the network,. Since this is an acyclic network, the LCM can be physically realized in the way mentioned above. The physical realization can then be interpreted as a memoryless transmission o data streams over the oriinal network,. Transmission, with memory, o data streams over the oriinal network, is associated with the acyclic network deined below, which is just a sliht modiication rom,. Deinition: Given a communication network, a positive inteer, the associated communication network with memory, denoted as,, is deined as ollows. The set o nodes in includes the node all pairs o the type,, where is a nonsource node in ranes throuh inteers to. Channels in the network, belon to one o the three types listed below. For any nonsource nodes in, 1) or, the multiplicity o the channel rom to, is the same as that o the channel in the network ; 2) or, the multiplicity o the channel rom, to, is the same as that o the channel in the network ; 3) or, the multiplicity o channels rom, to is equal to times. Lemma 4.7: The communication network, is acyclic. Lemma 4.8: Every low rom the source to the node in the network, corresponds to a low with the same volume rom the source to the node, in the network,. Lemma 4.9: Every LCM on the network, corresponds to an LCM on the network, such that or all nodes in Communication networks with or without memory both have been deined in order to compensate or the lack o a direct physical realization o an LCM on a network that may contain a directed cycle. In Section VI, we shall present another way to make the compensation, which will oer a physical realization by time-invariant linear codin. V. CONSTRUCTION OF A GENERIC LCM ON AN ACYCLIC COMMUNICATION NETWORK We have proved that or a eneric LCM, the dimension o the vector space at each node is equal to the maximum low between. However, we have not shown how one can construct a eneric LCM or a iven communication network. In the next theorem, we present a procedure which constructs a eneric LCM or any acyclic communication network. Theorem 5.1: A eneric LCM exists on every acyclic communication network, provided that the base ield o is an ininite ield or a lare enouh inite ield. Proo: Let the nodes in the acyclic network be sequentially indexed as, such that every channel is rom a smaller indexed node to a larer indexed node. The ollowin procedure constructs an LCM by assinin a vector to each channel, one channel at a time. or all channels XY v(xy )= the zero vector; // initialization or (j =0; j n; j++) arrane all outoin channels X jy rom X j in an arbitrary order; take one outoin channel rom X j at a time let the channel taken be X j Y ; choose a vector w in the space v(x j ) such that w =2 hv(uz): UZ 2 i or any collection o at most d 0 1 channels with v(x j) 6 hv(uz): UZ 2 i; v(x j Y )=w; v(x j+1) = the linear span by vectors v(xx j+1) on all incomin channels XX j+1 to X j+1 ;

8 378 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 2, FEBRUARY 2003 The essence o the above procedure is to construct the eneric LCM iteratively make sure that in each step the partially constructed LCM is eneric. One point in the above construction procedure o needs to be clariied. Given a node, there can be only initely many collections o cardinality at most such that :. When the base ield o is lare enouh where the union is over all such. Hence, the choice o the vector in the above procedure has been possible. Let be any collection o channels such that or. In order to assert that the LCM is eneric, we need to prove the linear independence amon the vectors,,,. The proo is by induction on. Without loss o enerality, we may assume that is the last amon the channels to be assined a vector in the above construction procedure o. Since the construction procedure ives On the other h, the induction hypothesis asserts the linear independence amon. Thus, the vectors are linearly independent. The theorem is proved. Given a communication network, a positive inteer, there exists a eneric LCM on the associated memoryless communication network, by Lemma 4.5 Theorem 5.1. From Theorem 3.3, or every node in,, the dimension o or the node, in, is equal to the maximum volume o a low rom the source to,. This maximum volume, accordin to Lemma 4.6, is at least times, where is a ixed inteer. In view o Lemmas , we have the ollowin similar conclusion about the adapted communication network. For every node in,, the dimension o the space or the correspondin node, in, is equal to the maximum volume o a low rom the source to, in,, this maximum volume is at least times or some ixed number. We now translate this conclusion about linear-code multicast over the memoryless network back to the oriinal network,. Let be the minimum o over all nonsource nodes in. Then or a suiciently lare inteer, usin the technique in Example 3.5, it is possible to desin a broadcast session o lenth equal to time units which broadcasts a messae o approximately symbols rom the source. Moreover, the whole messae can be recovered at each nonsource node ater approximately time units. VI. TIME-INVARIANT LCM AND HEURISTIC CONSTRUCTION Let be a inite ield. Let denote the rin o ormal power series over with the variable. is a principal ideal domain, every ideal in it is enerated by or some. Alebraic properties o vector spaces over a ield can oten be eneralized to modules over a principle ideal domain. In act, our previous results on LCM eneric LCM on vector spaces can readily be eneralized to modules over. The concept o the dimension o a vector space is eneralized to the rank o a module. To acilitate our discussion, we shall reer to the rank o a module over as its dimension. The elements o the module will be called vectors. Throuhout this section, we assume that is a module over with a inite but very lare dimension (say, larer than the number o channels times the lenth o transmission stream in the problem instance). Physically, an element o is the -transorm o a stream o symbols,,,,, that are sent on a channel, one symbol at a time. The ormal variable is interpreted as a unit-time shit. Deinition: A time-invariant linear-code multicast (TILCM) on a communication network, is an assinment o a module over to every node a vector to every channel such that 1) ; 2) or every channel ; 3) or any collection o nonsource nodes in the network In particular, the vector assined to an outoin channel rom a node is times a linear combination o the vectors assined to incomin channels to the same node. Hence a TILCM is completely determined by the vectors that it assins to channels. The adoption o times a linear combination, instead o simply any linear combination, allows a unit-time delay or transmission/codin. This allows a physical realization or the transmission scheme speciied by a TILCM. I a certain physical system requires nonuniorm duration o delay on dierent channels, then artiicial nodes need to be inserted. For instance, i the delay on a channel is three unit times, then two serial nodes should be added on the channel so that the channel becomes a tem o three channels. Example 6.1: On the communication network illustrated in Fi. 2, deine the TILCM by One can readily check that is in act a TILCM. For example Thus, is equal to times the linear combination o with coeicients, respectively. This speciies an encodin process or the channel that does not chane with time. It can be seen that the same is true or the encodin process o every other channel in the network. This explains the terminoloy time-invariant or an LCM.

9 LI et al.: LINEAR NETWORK CODING 379 Write the inormation vector as, where belon to. The product o the inormation (row) vector with the (column) vector assined to that channel represents the data stream transmitted over a channel Fi. 3. Redundant channels in a network. That is, the data symbol is sent on the channel at the time slot. This TILCM, besides bein time invariant in nature, is a memoryless one because the ollowin linear equations allows an encodin mechanism that requires no memory: Adopt the convention that or all. Then the data symbol lowin over the channel, or example, at the time slot is or all. Example 6.2: Another TILCM on the same communication network can be deined by The data stream transmitted over the channel is represented by, or instance, There are potentially various ways to deine a eneric TILCM, as an analo to Theorem 3.3, to establish desirable dimensions o the module assined to every node. Another desirable characteristic o a TILCM is to be memoryless. Empirically, it has been relatively easy to construct a TILCM that carries all conceivable desirable properties. The remainder o this section presents a heuristic construction procedure or a ood TILCM. The heuristic construction will ollow the raph-theoretical block decomposition o the network. For the sake o computational eiciency, the procedure will irst remove redundant channels rom the network beore identiyin the blocks so that the blocks are smaller. Deinition: A channel in a communication network is said to be irredundant i it is on a simple path startin at the source. Else, it is said to be redundant. Moreover, a communication network is said to be irredundant i it contains no redundant channels. Example 6.3: In the network illustrated in Fi. 3, the channels,, are redundant. Lemma 6.4: The deletion o a redundant channel rom a network results in a subnetwork with the same set o irredundant channels. Consequently, the irredundant channels in a network deine an irredundant subnetwork. Theorem 6.5: Let be an LCM (respectively, a TILCM) on a network. Then also deines an LCM (respectively, a TILCM) on the subnetwork that results rom the deletion o any redundant channel.

10 380 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 2, FEBRUARY 2003 Proo: We shall prove the case o LCM, the case o TILCM bein similar. Denote the deleted redundant channel as. Given a set o nonsource nodes with, we need to show Due to the redundancy o, any simple path rom to must o throuh the node. Let denote the set consistin o the node plus all those nodes such that any path rom to must o throuh. (I there is no simple path rom to a particular node, then that node ully qualiies as a node in.) Then, every channel rom outside o into must be toward the node. Thereore, This implies the desired equality. Corollary 6.6: Let be an LCM (respectively, a TILCM) on a network. Then deines an LCM (respectively, a TILCM) on the subnetwork ormed by irredundant channels. In a directed multiraph, an equivalence relationship (which by deinition is relexive, symmetric transitive) amon nodes can be deined as ollows. Two nodes are equivalent i there exists a directed path leadin rom one node to the other vice versa. An equivalence class under this relationship is called a block in the raph. The source node by itsel always orms a block. When every block contracts into a sinle node, the resultin raph is acyclic. In other words, the blocks can be sequentially indexed so that every interblock channel is rom a smaller indexed block to a larer indexed block. For the construction o a ood TILCM, smaller sizes o blocks tend to acilitate the computation. The extreme avorable case o the block decomposition o a network is when the network is acyclic, which implies that every block consists o a sinle node. The opposite extreme is when all nonsource nodes orm a sinle block exempliied by the network illustrated in Fi. 3. The removal o redundant channels sometimes serves or the purpose o breakin up a block into pieces. For the network illustrated in Fi. 3, the removal o the three redundant channels breaks the block,,,, into the three blocks,,,,. In the construction o a ood LCM on an acyclic network, the procedure inside the proo o Theorem 5.1 takes one node at a time accordin to the acyclic orderin o nodes assins vectors to outoin channels rom the taken node. For a eneral network, we can start with the trivial TILCM on the network consistin o just the source then exp it to a ood TILCM that covers one more block at a time. The sequential choices o blocks are accordin to the acyclic order in the block decomposition o the network. Thus, the expansion o the ood TILCM at each step involves only incomin channels to nodes in the new block. A heuristic alorithm or assinin vectors to such channels is or to be times an arbitrary convenient linear combination o vectors assined to incomin channels to. In this way, a system o linear equations o the orm o is set up, where is a square matrix with the dimension equal to the total number o channels in the network is the unknown column vector whose entries are or all channels. The elements o are polynomials in. In particular, the elements o are either or a polynomial in containin the actor. Thereore, the determinant o is a ormal power series with the constant term (the zeroth power o ) bein,, hence is invertible in. Accordin to Cramer s rule, a unique solution exists. (This is consistent with the physical intuition because the whole network is completely determined once the encodin process or each channel is speciied.) I this unique solution does not happen to satisy the requirement or bein a ood TILCM, then the heuristic alorithm calls or adjustments on the coeicients o the linear equations on the trial--error basis. Ater a ood TILCM is constructed on the subnetwork ormed by irredundant channels in a iven network, we may simply assin the zero vectors to all redundant channels. Example 6.7: Ater the removal o redundant channels, the network depicted by Fi. 3 consists o our blocks in the order o,,,,,. The subnetwork consistin o the irst two blocks is the same as the network in Fi. 2. When we exp the trivial TILCM on the network consistin o just the source to cover the block,,, a heuristic trial would be toether with the ollowin linear equations: The result is the memoryless TILCM in the precedin example. This TILCM can be urther exped to cover the block then the block. VII. CONCLUDING REMARKS In this paper, we have presented an explicit construction o a code or multicast in a network that achieves the max-low bound on the inormation transmission rate. An important aspect o our code is its linearity, which makes encodin decodin easy to implement in practice. Our reedy alorithm or code construction works or all networks, but the code

11 LI et al.: LINEAR NETWORK CODING 381 so constructed is not necessarily the simplest possible. In act, there oten exist optimal LCMs much simpler than the ones constructed accordin to the presented reedy alorithm. Thereore, there is much room or urther research in this direction. The code we have constructed or a cyclic network is time varyin, which makes it less appealin in practice. To our knowlede, there has not been a proo or the existence o an optimal time-invariant code or a cyclic network. However, examples o such a code have been iven in the present paper also in [1]. Thereore, provin the existence o such codes, in particular, constructin such codes in simple linear orm, is a challenin problem or uture research. Further research problems in network codin include code construction when two or more sources are simultaneously multicast in the network. This is the so-called multisource network codin problem in [1] which is yet unexplored (see also [5, Ch. 15]). When networkin codin is implemented in computer or satellite networks, synchronization is a problem that needs to be addressed. Since an encoder at an intermediate node may take more than one incomin data stream as input, it is necessary to acquire synchronization amon these data streams. This is not a problem or nonreal-time applications (e.., ile transer), but it can be a serious problem or real-time applications (e.., voice video transmission). On the other h, certain types o networks, or example switchin networks, are inherently ully synchronized. These networks are excellent cidates or network codin,, in act, the possible use o linear network codes in switchin networks has been investiated [6]. ACKNOWLEDGMENT The authors would like to thank Venkat Anantharam or pointin out an error in the oriinal proo o Theorem 3.3 or his very careul readin o the manuscript. They also thank the anonymous reviewers or their useul comments. REFERENCES [1] R. Alshwede, N. Cai, S.-Y. R. Li, R. W. Yeun, Network inormation low: Sinle source, IEEE Trans. Inorm. Theory, submitted or publication. [2] E. L. Lawler, Combinatorial Optimization: Network Matroid. Fort Worth, TX: Saunder Collee Pub., [3] S.-Y. R. Li R. W. Yeun, Network multicast low via linear codin, in Proc. Int. Symp. Operational Research Its Applications (ISORA 98), Kunmin, China, Au. 1998, pp [4] D. J. A. Welsh, Matroid Theory. New York: Academic, [5] R. W. Yeun, A First Course in Inormation Theory. Norwell, MA/New York: Kluwer/Plenum, [6] F. R. Kschischan, private communication.