Degrees of Freedom for Multiple-Multicast TrafficShaileshh Bojja Venkatakrishnan

University of Illinois, Urbana-ChampaignEmail: bjjvnkt2@illinois.edu

Pramod ViswanathUniversity of Illinois, Urbana-Champaign

Email: pramodv@illinois.edu

Sreeram KannanUniversity of California, Berkeley

Email: ksreeram@eecs.berkeley.edu

Abstract—We propose a new coding scheme for interferencealignment in a single hop fast fading wireless network withgeneral message demands. For the X-Channel, the Degrees ofFreedom (DoF) region achievable by the scheme is shown to toucha known outer-bound at several points. For multiple-multicastdemands we show that the achievable region is at least halfof the cut-set bound region. The key innovation in our schemeis the reduction of the vector space alignment problem to acombinatorial arrangement problem. Finally, we use the schemeto give a poly-logarithmic bound for the flow-cut gap in fastfading Gaussian wireless networks with multiple multicasts.

I. INTRODUCTION

Interference is a first order issue in wireless networks. Tra-ditional methods to handle interference such as power controland scheduling can be suboptimal and limit the maximumthroughput in the network. Interference alignment (IA) is arelatively recent technique to mitigate this problem in whichthe transmitted signals are beam-formed in such a way that theinterference is restricted to only a subspace at each receiver.The DoF of the K-user interference channel with ergodicfading has been analyzed in [1] where they show that aDoF per message of up to 1

2 is achievable. [2] looks at aninterference network in which every transmitter has a singlemulticast message. The DoF region for networks with moregeneral message demands has not yet been studied. In thispaper, we present an IA scheme that generalizes the schemein [2] for (i) multiple-unicasts (also referred to as the X-Channel), in which each transmitter can have an independentmessage to each receiver and (ii) multiple-multicasts, in whicheach transmitter can have an independent multicast message toeach subset of receivers. For the case of multiple-unicasts, weshow that the DoF outer bound in [3] can be achieved undera symmetry condition. We also show that our scheme canachieve a DoF of within 1

2 of the cut-set bound in both cases.Finally we present an approximate DoF region of general fastfading Gaussian networks under a conjecture on the flow-cutgap of node-capacitated undirected graphs.

II. NETWORK MODEL

Following the model presented in [3], we consider a singlehop wireless network with K transmitters and K receivers,each having a single antenna. Let Hji be a generic channelmatrix between transmitter i and receiver j. Assume that thecausal channel state information is known globally. Then, theinput-output relation is given by:

Yj(t) =∑

i∈1,...,K

Hji(t)Xi(t) + Zj(t), j = 1, 2, . . . ,K

where t represents the channel use index, which can beinterpreted as time or frequency index. Xi(t) is the signaltransmitted by the transmitter i, Yj(t) is the signal receivedby receiver j and Zj(t) represents the additive white Gaussiannoise at receiver j at time k. [3] has given an outer bound forthe DoF region of the X-network as:

Dout =

[dij ]K×K : ∀(m,n) ∈ 1, 2, . . . ,K×1, 2, . . . ,K,

K∑q=1

dmq +

K∑p=1

dpn − dmn ≤ 1

(1)

where dij refers to the DoF of the message from transmitteri to receiver j. Throughout this paper we restrict ourselvesto the single antenna case. Notation: For any k ∈ N, let [k]denote the set 1, . . . , k. For any two matrices U and V welet U ⊂ V mean the (column) subspace spanned by U iscontained in the subspace spanned by V. Similarly U ≡ Vmeans the subspace spanned by both the matrices coincide.

III. X NETWORK: INTERFERENCE ALIGNMENT SCHEME

In the alignment scheme discussed in [1] for the K-userinterference channel, a key problem is to construct a matrixV such that, for generic matrices T1, . . . ,Tn, the spacesT1V, . . . ,TnV are aligned, i.e., T1V ≡ . . . ≡ TnV. Oneapproximate method that guarantees asymptotic alignment isdone as follows [4, section 4.6.2]:

V =[(T1)α1(T2)α2 . . . (Tn)αnw,

s.t.n∑i=1

αi ≤ λ− 1, α1, . . . , αn ∈ Z+],

where w is a generic vector used to evolve the space andλ a parameter. Let us call such a vector w used to evolvesubspaces in the above fashion as a base-vector for thesubspace.

A. Relative arrangement problem

In the case of a K-user X-channel each transmitter can havea message to each receiver. Let X[1]

i , . . . ,X[N ]i denote a set

of N message vectors (of equal dimension) from transmitter i.Each message is meant for a particular receiver in [K] denotedby label(X[l]

i ). Suppose we use N base-vectors w1, . . . ,wN toevolve the beamforming matrices V

[1]i , . . . ,V

[N ]i respectively

for each transmitter i ∈ [K]. Now, let (X[1]i , . . . , X

[N ]i ) denote

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(a) DoF matrix d (b) Message demand matrix n (c) Arrangement of the messages as an array B

Fig. 1. An example showing the optimal arrangement of messages for the X channel with the given message demand. The red dotted lines show the interferingmessages at receiver 1. Notice that all the interfering messages are contained under (in the same columns as) the largest interference (at all receivers)

an ordering of the set of messages of transmitter i, such thatthe message X

[l]i is beamformed using V

[l]i , i.e.,

Xi(t) =

N∑l=1

V[l]i X

[l]i (t)

is the signal transmitted by the ith user. A beam-forming ma-trix V

[l]i undergoes the linear transformation V

[l]i → HjiV

[l]i

when subjected to the channel matrix Hji between transmitteri and receiver j. Now, suppose further that V

[l]i ’s have the

following property:

Property 1. At each receiver j, the interference subspaces inthe set HjiV

[l]i : i ∈ 1, . . . ,K, label(X[l]

i ) 6= j alignwith each other ∀ l ∈ [N ], while all the message subspaces inHjiV

[l]i : i ∈ 1, . . . ,K, label(X[l]

i ) = j become linearlyindependent to the interference spaces and to each other.

With this property the ordering of the messages becomeimportant. For example, consider a 2-user X-channel wheretransmitter 1 has the messages X[1]

1 ,X[2]1 for receiver 1,

message X[3]1 for receiver 2, while transmitter 2 has the

message X[1]2 for receiver 1 and messages X[2]

2 ,X[3]2 for

receiver 2. Suppose user 1 transmits it’s messages as:

X1(t) = V[1]1 X

[1]1 (t) + V

[2]1 X

[2]1 (t) + V

[3]1 X

[3]1 (t).

Now, consider the following two ways by which user 2 cantransmit:

(i) X2(t) =V[1]2 X

[1]2 (t) + V

[2]2 X

[2]2 (t) + V

[3]2 X

[3]2 (t)

(ii) X2(t) =V[1]2 X

[3]2 (t) + V

[2]2 X

[2]2 (t) + V

[3]2 X

[1]2 (t).

If the V[l]i s satisfy Property 1, then the interference subspace

takes a total of 3 signalling dimensions in case (i), but only 2in case (ii) at the receivers. The messages themselves occupy3 dimensions. Therefore, the messages have a DoF of 3/6in case (i) but only 3/5 in case (ii), at the receivers. Clearlythe way we start out with message arrangements dictates theefficiency of the communication scheme. This is precisely theproblem we address. In the sections that follow we describehow best to permute the message tuples at each transmitter inorder to achieve the optimal DoF.

B. Alignment scheme

X networks represent the most general class of non-multicast communication scenario possible in a single-hopwireless network. Let d = [dij ]K×K ∈ [0, 1]K×K denote theDoF matrix, where the (i, j)th entry dij refers to the DoF ofthe message from transmitter i to receiver j. Since the verticesof the DoF region given in (1) are rational, we consider onlythe achievability of points with rational coordinates. Let usfirst consider those points d on the outer bound for which∑Kj=1 d

ij = D ∀i ∈ [K] for some D ∈ R, i.e., the sum-

DoF of the messages from each transmitter are equal. We laterconsider the scheme for a general DoF point. For any rationalpoint d, let κ ∈ Z+ be such that n , κd has all integralentries, i.e., nij ∈ Z ∀i, j ∈ [K] where nij = κdij . We interpretnij as the number of messages from transmitter i to receiver j.Therefore, every transmitter has a total of N = κD messages.The following sections III-C and III-D are the two key stepsinvolved in our scheme.

C. Step 1: Combinatorial message alignment

Since each transmitter has N messages in total, let us use aset of N base-vectors W = wi, 1 ≤ i ≤ N (where wi’s aregeneric) in order to evolve the beamforming matrices. Noticethat the same set of base-vectors is used in all the transmitters.As discussed previously in section III-A, the ultimate aim hereis to optimally assign each message to one of the beamformingmatrices.

We view such an assignment of messages to beamformingmatrices or equivalently, to base-vectors, as an array B whereB(i, j) denotes the message from transmitter i listed underbase vector j. Fig. 1(c) shows an example assignment for K =3, N = 10. We have listed only the message labels and notthe actual messages itself in the assignment B because for anyassignment of messages, exchanging of messages having thesame label does not change the performance.

Let us call a column a j-block (for j ∈ [K]) if all the entriesof that column have the label j. A column which is not a j-block for any j is called a φ-block. Let Nj denote the numberof j-blocks in B for j ∈ [K] and Ij , N −Nj .

Proposition 1. For any point d in the outer bound region (1)such that the sum-DoF of messages from all transmitters are

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(a) Gn(V,E) (b) Time layered version (c) Steady state

Fig. 2. An example showing a node capacitated graph Gn(V,E) with a single source, node 1, having a multicast message to nodes 2 and 3. In Fig. 2(a),two possible Steiner trees are shown in red and blue dotted lines. Fig. 2(b) shows the passing of messages with time. At each time instant, the interior nodesforward the message received in the previous time slot while the source generates a new message and in Fig. 2(c), we have shown the steady state traffic inthe network

equal, there exists an arrangement B such that:

K∑i′=1

ni′

j + Ij ≤ κ ∀ j ∈ [K] (2)

where n, κ, Ij are as was defined above.

Proof: Since d is within the outer bound region (1), wemust have:

K∑i′=1

ni′

j +∑

j′∈[K]:j′ 6=j

nij′ ≤ κ ∀ j ∈ [K]. (3)

Let δ1 = minini1 : 1 ≤ i ≤ K be the smallest messageto receiver 1 from a transmitter. We first arrange δ1 of the ni1messages from each transmitter i together in δ1 columns. Thisimplies I1 = N−δ1. In fact, I1 =

∑j′∈[K]:j′ 6=1 n

i∗

j′ where i∗ isa transmitter having the largest number of interfering messagesor equivalently the smallest number of non-interfering mes-sages for 1. As such, we have I1 ≤

∑j′∈[K]:j′ 6=1 n

ij′ ∀i ∈ [K].

Therefore, from equation (3) we get:

K∑i′=1

ni′

1 + I1 ≤ κ. (4)

Now, we sequentially perform this operation for each re-ceiver so that equation (2) holds. This is possible because,∑Kj=1 δj ≤

∑Kj=1 n

ij ≤ N since δj ≤ nij ∀i ∈ [K] and hence

the above operations require no more than N = |W | columns.The remaining messages can be arbitrarily assigned to anyunassigned columns. The proposition follows.

Fig. 1 shows an illustration of a K = 3 case with κ = 20.

D. Step 2: Evolution of beamforming matrices

The next step is to generate the beamforming matricesfrom the base-vectors. The subspaces are created such thatthey satisfy Property 1, i.e., an interfering message from asubspace remains within the subspace while a non-interferingmessage becomes linearly independent to the subspace at thereceiver. This property, together with the previous combi-natorial arrangement of the messages allows us to achieveinterference alignment. Let us use a symbol expansion ofτ = κt(λ + 1)K

2−K where κt = κ. Let V[l]i denote

the beamforming matrix associated with base-vector wl forl ∈ [N ] at transmitter i. Now, the evolution is done as:

V[l]i =

∏

(m,n)∈[K]2

B(m,l)6=n

Hαnmnm

wl :

αnm ∈ 0, . . . , λ if m 6= i,αnm ∈ 0, . . . , λ− 1 if m = i

, (5)

i.e., for each V[l]i we use all the K2 channel matrices except

those associated with the messages listed under the base vectorwl in B. In the following we show that contructing thebeamforming matrices this way guarantees alignment. Thearguments we use in this section are based on the results in [3]and [2].

Proposition 2. The set of beamforming matrices generatedaccording to Equation (5) satisfies Property 1.

Proof: The proof is as follows:1) Alignment of interfering messages: We first show that

at every receiver all the interfering vectors are aligned withina subspace. Let:

V[l] ,

∏

(m,n)∈[K]2

B(m,l)6=n

Hαnmnm

wl : αnm ∈ 0, . . . , λ

,

(6)and let V(j) , [V[l] : l ∈ [N ],∃ i s.t. B(i, l) 6= j] where formatrices A,B, [A B] stands for the augmented matrix. Nowfor any interference message coded via the matrix V

[l]i it is

easy to see that the received signal space:

HjiV[l]i ⊂ V[l] ⊂ V(j)

since left multiplication by the channel matrix only increasesthe exponent αji in the vectors of V

[l]i by 1, and such vectors

are already included in V[l]. In other words, our constructionensures that all the interfering messages in column l of Bare aligned at the receivers for each l. Such a columnwisealignment implies a global alignment of all the interferingmessages within a subspace. Hence we conclude that for any

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receiver j ∈ [K], the interference component is containedwithin the subspace V(j). We now proceed to show linearindependence between the signal and interference spaces atthe receiver.

2) Linear Independence of Signal and Interference: LetS(j) denote the subspace formed by the beamforming matricescorresponding to the message signals at receiver j:

S(j) = [HjiV[l]i : B(i, l) = j,∀i ∈ [K],∀l ∈ [N ]] (7)

so that if S(j) and V(j) are linearly independent, then V(j)can be zero-forced to retrieve the information of the messagesin S(j). Let Λ(j) denote the combined space:

Λ(j) = [S(j) V(j)]. (8)

Note that V[l]i is of dimension κ(λ+ 1)K

2−K by κλK−1(λ+

1)K2−2K+1 while V[l] is κ(λ + 1)K

2−K by κ(λ + 1)K2−K .

Using Proposition 1 we see that Λ(j) is a tall matrix. Since allchannel matrices are assumed to be diagonal, each element ofΛ(j) is a monomial term of several random variables. Henceby the argument used in [2] and [3, Lemma 1] we concludethat Λ(j) has full-rank almost surely. This proves the linearindependence of the message subspaces with the interferencesubspace and with each other, as claimed in Property 1.

With Property 1 being true, looking back at the messageassignment algorithm of section III-C, we see that a j-blockis an interference block for all receivers except j, a φ-blockis an interference block for all the receivers and Ij counts thenumber of interference blocks for receiver j. What we haveensured is that all the interfering messages at receiver j arealigned under the largest interfering message for receiver j.

Since the message blocks in Λ(j), j ∈ [K] with a dimensionof κ ∗ (λ + 1)K

2−K by (λ)K−1(λ + 1)K2−2K+1 have full-

column rank, in the asymptotic case [4] we can expect a DoFof:

|HjiV[l]i |

τ= limλ→∞

(λ)K−1(λ+ 1)K2−2K+1

κ ∗ (l + 1)K2−K =1

κ

corresponding to each message X[l]i (t) or equivalently a sum-

DoF of nij/κ = dij for each source-destination pair. It iscrucial to note that we are able to use a time expansionwith a scaling of κt = κ because our arrangement satisfiedProposition 1. For any other arrangement of B, it is possiblethat maxj

∑Ki′=1 n

i′

j +Ij is larger than κ in which case Λ(j)would no longer be full-rank for all j. Therefore any DoF pointin the outer-bound 1 with equal sum-DoF is achievable. Weconclude this section by stating the achievable DoF region forthe scheme.

Theorem 1. The achievable DoF region of the scheme for theX-network is given by the set: d ∈ [0, 1]K×K such that,

(|s| − 1)D + maxi

∑j:j∈sc

dij

+∑i,j

j∈s,i∈[K]

dij ≤ |s| (9)

∀s ⊆ 1, . . . ,K, s 6= , where D = maxi∑j∈[K] d

ij .

The details are left to a longer version of this paper. Notethat the achievable region (9) touches the outer bound (1) atpoints where the sum-DoF of messages from the transmittersare equal.

IV. DOF INNER BOUND

In the previous section we presented a scheme for a multipleunicast traffic model. For a general multicast model we canessentially use the same two step procedure for alignment asoutlined in sections III-C and III-D. A technical difficultyin this model is that the message labels can now denotesets of receivers instead of just single receivers. One naturalimplication of our alignment framework is:

Proposition 3. In a K-user multiple multicast network, anyd such that:∑

s⊆[K]j∈s

K∑i=1

dis ≤1

2and

∑s⊆[K]

dis ≤1

2∀ i, j (10)

is achievable using our alignment scheme,

i.e., we can show that our scheme comes within 12 of the

cut-set bound. Since the X-channel and multiple multicastchannels are only a special case of the above generalizedmultiple multicast channel, this inner bound holds even inthose cases.

V. APPROXIMATE DOF OF GENERAL GAUSSIAN NETWORK

We now characterize flow on a wireless network withmulticast demands. In [5] the authors propose a novel layeringtechnique for a network with multiple unicast message de-mands. Such a scheme comes to within a logarithmic factor ofthe cut-set bound. In the following, we show that this methodcan be generalized to networks with multicast demands usingour IA scheme in the physical layer. Consider a wirelessnetwork G with vertex set V and an undirected edge set E.Assume flat-fading between each pair of nodes connected byan edge, with channel coefficients of unit variance. Let ussuppose there are k multicast demands with source node ρiand destination set Si corresponding to demand i ∈ [k]. LetGi = ρi ∪ Si and t = maxi |Gi|. Without loss of generalitywe assume the ρi’s are all distinct, i.e., each source node hasonly one multicast message.

A. Achievability

In order to obtain the flow-cut gap for the Gaussian net-work, we first consider an undirected node-capacitated graphGn(V,E), on the same vertex and edge sets, with an incom-ing and outgoing node capacity of C(P )

2 for all the nodes.Here the node capacity bounds the total multicast messagerate forwarded (resp. received) by a node, with the rate ofeach multicast message counted only once. Each Steiner treespanning Gi provides a dissemination path for the messagesfrom the source ρi to the destination set Si.

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Our main observation is that any multicasting over thenode-capacitated graph Gn can be emulated on the Gaussiannetwork, or in other words:

Rgach(P ) ⊇ Rnach(P ) (11)

where Rgach(P ) and Rnach(P ) stand for the achievable rateregions in the Gaussian and the node-capacitated networkrespectively. Consider the steady-state bipartite graph corre-sponding to the node-capacitated network as illustrated inFig. 2. This steady state network also has a node capacityof 1

2C(P ). From our interference alignment scheme we knowthat half of the cut-set bound is achievable at high SNR fora complete bipartite topology (10). It is easy to see that theachievability of 1

2C(P ) does not change for any other bipartitetopology (for feasible multicasts). Now, by routing informationin the Gaussian network via any set of Steiner trees that leadto that steady state rate, we can conclude that any feasible flowachievable in the node-capacitated network Gn(V,E) can beachieved on the Gaussian network G(V,E). This proves (11).

B. Cut-set bounds

Let us now relate the corresponding cut-set bounds.Let Rgcut(P ) and Rncut(P ) denote the Gaussian and node-capacitated cut-set regions respectively for power P . For anycut Ω ⊆ V in the Gaussian network G(V,E), let D(Ω) =1 ≤ i ≤ k : ρi ∈ Ω, Si ∩ Ωc 6= φ denote the set ofcommodities that are separated by cut Ω. We then have:∑

i∈D(Ω)

Ri ≤ cutg(Ω) = I(XΩ;YΩc |XΩc). (12)

The cut value above can be upper bounded by evaluating thecapacity of a ‘colored’ Gaussian network as explained in [5].

Now, in the node-capacitated network case, for any cut Ω ⊆V , let F = (uv) ∈ E : u ∈ Ω, v ∈ Ωc denote the edgescrossing the cut in Gn. We define the value of the cut asin the polymatroidal network case [5], i.e., for an assignmentfunction g that maps each edge in F to either of it’s end nodes,we have:

cutn(Ω) , ming|g(F )|C(P )

2(13)

where g(F ) is the range of g. For the concurrent flowproblem [6], where the message demand rates are given byRi = γDi ∀i ∈ [k] for fixed D1, . . . , Dk, we have:

γ ≤ minΩ

cutn(Ω)∑i:i∈D(Ω)Di

= ν∗ (14)

where ν∗ is called the sparsest cut. For the optimal γ∗ wedefine the flow-cut gap as a function f(k, t), such that γ∗ ≥

1f(k,t)ν

∗. If the flow-cut gap is also independent of the demandrates D1, . . . , Dk, we have

Rnach ⊇Rncut

f(k, t). (15)

Further, since the cut-set bound region for the node capacitatedgraph is exactly the same as the polymatroidal cut-set boundregion, using the result in [5] we have:

Rncut ⊇1

2Rgcut

(P

bd3

)⇒ Rgcut(P/bd

3)

2f(k, t)⊆ Rncut(P )

f(k, t)(16)

where d is the maximum degree of the graph and b =eE(log |h|

2)

2 . Now, it is shown in [6] that a flow-cut gap ofO(log3(kt)) is achievable for the setting of edge-capacitedundirected graphs. We conjecture that a similar result holdstrue even in our case:

Conjecture 1. In a node-capacitated undirected graph aflow-cut gap of f(k, t) = O(logc kt) is achievable for someconstant c > 0.

Under this conjecture, combining the results so far we get:

Theorem 2. The capacity of a Gaussian wireless network withk multicast demands as described above can be approximatedto within a poly-logarithmic factor by the cut-set bound:

Rgcut(P/bd3)

2O(logc kt)⊆ Rgach(P ). (17)

VI. CONCLUSION

We have presented a novel interference alignment schemethat simplifies the problem of interference alignment to a com-binatorial problem in networks with general message demands.This framework allows us to easily see the DoF region forsimpler networks like the K user interference network. Forthe X network we have shown that the achievable regionof our scheme touches a previously known outer bound. Wehave also extended the result of achievability of half of thecut-set bound to general multicast networks. And finally, wehave demonstrated a poly-logarithmic flow-cut gap in generalGaussian wireless networks with multicast demands.

REFERENCES

[1] V. R. Cadambe and S. A. Jafar, “Interference alignment and the degreesof freedom of the k-user interference channel,” IEEE transactions onInformation Theory, vol. 54, no. 8, pp. 3425–3441, August 2008.

[2] L. Ke, A. Ramamoorthy, Z. Wang, and H. Yin, “Degrees of freedomregion for an interference network with general message demands,” CoRR,vol. abs/1101.3068, 2011.

[3] V. R. Cadambe and S. A. Jafar, “Interference alignment and the degreesof freedom of wireless x networks,” IEEE transactions on informationtheory, vol. 55, no. 9, pp. 3893–3908, September 2009.

[4] S. A. Jafar, Interference Alignment - A New Look at Signal Dimensionsin a Communication Network, ser. Foundations and Trends in Com-munications and Information Theory. now, 2010, vol. 7, no. 1, dOI:10.1561/0100000047.

[5] S. Kannan and P. Viswanath, “Capacity of multiple unicast in wirelessnetworks: A polymatroidal approach,” IEEE Transactions on InformationTheory, accepted March 2014.

[6] P. N. Klein, S. A. Plotkin, S. Rao, and E. Tardos, “Approximationalgorithms for steiner and directed multicuts,” Journal of Algorithms,vol. 22, no. 2, pp. 241–269, 1997.

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