Interference Alignment in Dense Wireless Networks

Interference Alignment in Dense Wireless Networks

Urs Niesen U. Niesen is with the Mathematics of Networks and Communications Research Department, Bell Labs, Alcatel-Lucent. Email: urs.niesen@alcatel-lucent.com
Abstract

We consider arbitrary dense wireless networks, in which nodes are placed in an arbitrary (deterministic) manner on a square region of unit area and communicate with each other over Gaussian fading channels. We provide inner and outer bounds for the -dimensional unicast and the -dimensional multicast capacity regions of such a wireless network. These inner and outer bounds differ only by a factor , yielding a fairly tight scaling characterization of the entire regions. The communication schemes achieving the inner bounds use interference alignment as a central technique and are, at least conceptually, surprisingly simple.

Capacity scaling, interference alignment, multicast, multicommodity flow, opportunistic communication, wireless networks.

I Introduction

Interference alignment is a recently introduced technique to cope with the transmissions of interfering users in wireless systems see [1, 2, 3]. In this paper, we apply this technique to obtain fairly precise (up to factor) information-theoretic scaling results for the unicast and multicast capacity regions of dense wireless networks.

I-a Related Work

The study of scaling laws for wireless networks, describing the system performance in the limit of large number of users, was initiated by Gupta and Kumar in [4]. They analyzed a network scenario in which nodes are placed uniformly at random on a square of area one (called a dense network in the following) and are randomly paired into source-destination pairs with uniform rate requirement. Under a so-called protocol channel model, in which only point-to-point communication is allowed and interference is treated as noise, they showed that the largest uniformly achievable per-node rate scales as up to a polylogarithmic factor in . Achievability was shown using a multi-hop communication scheme combined with straight-line routing. Different constructions achieving slightly better scaling laws, i.e., improving the polylogarithmic factor in , were subsequently presented in [5, 6].

These results are in some sense negative, in that they show that with current technology, captured by the protocol channel model assumption, the per-node rate in large wireless networks decreases with increasing network size even if the deployment area is kept constant. An immediate question is therefore if this negative result is due to the protocol channel model assumption or if there is a more fundamental reason for it. To address this question, several authors have considered an information-theoretic approach to the problem, in which the channel is simply assumed to be a Gaussian fading channel without any restrictions on the communication scheme [7, 8, 9, 10, 11]. We shall refer to this as the Gaussian fading channel model in the following. These works construct cooperative communication schemes and show that they can significantly outperform multi-hop communication in dense networks. In particular, Özgür et al. showed in [11] that in Gaussian fading dense wireless networks with randomly deployed nodes and random source-destination pairing, the maximal uniformly achievable per-node rate scales like111The notation is used to indicate that the maximal uniformly achievable per-node rate is upper bounded by and lower bounded by . Similar expressions will be used throughout this section. for any . In other words, in dense networks222We point out that the situation is quite different in extended networks, in which nodes are placed on a square of area . Here network performance depends on the path-loss exponent , governing the speed of decay of signal power as a function of distance. For small , cooperative communication is order optimal, whereas for large , multi-hop communication is order optimal [12, 13, 14, 15, 16, 17, 18, 11, 19]., cooperative communication can increase achievable rates to almost constant scaling in —significantly improving the scaling resulting from the protocol channel model assumption. The scaling law was subsequently tightened to in [20, 21].

While these results removed the protocol channel model assumption made in [4], they kept the assumptions of random node placement and random source-destination pairing with uniform rate. Wireless networks with random node placement and arbitrary traffic pattern have been analyzed in [22, 23] for the protocol channel model and in [24] for the Gaussian fading channel model. On the other hand, wireless networks with arbitrary node placement and random source-destination pairing with uniform rate have been investigated in [25] for the protocol channel model and in [20] for the Gaussian fading channel model. While methods similar to the ones developed in [25] can also be used to analyze wireless networks with arbitrary node placement and arbitrary traffic pattern under the protocol channel model, the performance of such general networks under a Gaussian channel model (i.e., an information-theoretic characterization of achievable rates) is unknown.

Finally, it is worth mentioning [26, 27], which derive scaling laws for large dense interference networks. In particular, [27] considers a dense random node placement with random source-destination pairing. However, the model there is an interference channel as opposed to a wireless network as modeled in the works mentioned above. In other words, the source nodes cannot communicate with each other, and similarly the destination nodes cannot communicate with each other. This differs from the model adopted in this paper and the works surveyed so far, in which no such restrictions are imposed. For such interference networks, [27] derives the asymptotic sum-rate as the number of nodes in the network increases.

I-B Summary of Results

In this paper, we consider the general problem of determining achievable rates in dense wireless networks with arbitrary node placement and arbitrary traffic pattern. We assume a Gaussian fading channel model, i.e., the analysis is information-theoretic, imposing no restrictions on the nature of communication schemes used. We analyze the -dimensional unicast capacity region , and the -dimensional multicast capacity region of an arbitrary dense wireless network. describes the collection of all achievable unicast traffic patterns (in which each message is to be sent to only one destination node), while describes the collection of all achievable multicast traffic patterns (in which each message is to be sent to a set of destination nodes). We provide explicit approximations and of and in the sense that

 ^Λ\textupUC(n) ⊂Λ\textupUC(n)⊂K1log(n)^Λ\textupUC(n), ^Λ\textupMC(n) ⊂Λ\textupMC(n)⊂K2log(n)^Λ\textupMC(n),

for constants not depending on . In other words, and approximate the unicast and multicast capacity regions and up to a factor . This provides tight scaling results for arbitrary traffic pattern and arbitrary node placement.

The results presented in this paper improve the known results in several respects. First, as already pointed out, they require no probabilistic modeling of the node placement or traffic pattern, but rather are valid for any node placement and any traffic pattern and include the results for random node placement and random source-destination pairing with uniform rate as a special case. Second, they provide information-theoretic scaling results that are considerably tighter than the best previously known, namely up to a factor here as compared to in [11] and in [20, 21]. Moreover, the results in this paper provide an explicit expression for the pre-constant in the term that is quite small, and hence these bounds yield good results also for small and moderate sized wireless networks. Third, the achievable scheme used to prove the inner bound in this paper is, at least conceptually, quite simple, in that the only cooperation needed between users is to perform interference alignment. This contrasts with the communication schemes achieving near linear scaling presented so far in the literature, which require hierarchical cooperation and are harder to analyze.

I-C Organization

The remainder of this paper is organized as follows. Section II introduces the network model and notation. Section III presents the main results of this paper. Section IV describes the communication schemes used to prove achievability. Section V contains proofs, and Sections VI and VII contain discussions and concluding remarks.

Ii Network Model and Notation

Let

 A≜[0,1]2

be a square of area one, and consider nodes (with ) placed in an arbitrary manner on . Let be the Euclidean distance between nodes and , and define

 rmin(n)≜n1/2minu≠vru,v.

The minimum separation between nodes in the node placement is then . Note that for a grid graph, and with high probability for nodes placed uniformly and independently at random on . In general, we have

 rmin(n)≤4/√π<3, (1)

and, while the results presented in this paper hold for any , the case of interest is when decays at most polynomially with , i.e., for some constant . Note that we do not make any probabilistic assumptions on the node placement, but rather allow an arbitrary (deterministic) placement of nodes on . In particular, the arbitrary node placement model adopted here contains the random node placement model as a special case. The arbitrary node placement model is, however, considerably more general since it allows for classes of node placements that only appear with vanishing probability under random node placement (e.g., node placements with large gaps or isolated nodes).

We assume the following complex baseband-equivalent channel model. The received signal at node at time is given by

 yv[t]≜∑u≠vhu,v[t]xu[t]+zv[t],

where is the channel gain from node to node , is the signal sent by node , and is additive receiver noise at node , all at time . The additive noise components are assumed to be independent and identically distributed (i.i.d.) circularly-symmetric complex Gaussian random variables with mean zero and variance one. The channel gain has the form

 hu,v[t]≜r−α/2u,vexp(√−1θu,v[t]), (2)

where is the path-loss exponent. As a function of and , the phase shifts are assumed to be i.i.d. uniformly distributed over . As a function of time , we only assume that varies in a stationary ergodic manner as a function of for every . Note that the distances between the nodes do not change as a function of time and are assumed to be known throughout the network. The phase shifts are assumed to be known at time at every node in the network. Together with the knowledge of the distances , this implies that full causal channel state information (CSI) is available throughout the network. We impose a unit average power constraint on the transmitted signal at every node in the network.

The phase-fading model (2) is adopted here for consistency with the capacity-scaling literature. All results presented in this paper can be extended to Rayleigh fading, see Section VI-C.

A unicast traffic matrix associates with every node pair the rate at which node wants to transmit a message to node . The messages corresponding to distinct pairs are assumed to be independent. Note that we allow the same node to be source for several destinations , and the same node to be destination for several sources . The unicast capacity region is the closure of the collection of all achievable unicast traffic matrices . Knowledge of the unicast capacity region provides hence information about the achievability of any unicast traffic matrix .

A multicast traffic matrix associates with every pair of node and subset the rate at which node wants to multicast a message to the nodes in , i.e., every node wants to receive the same message from . The messages corresponding to distinct pairs are again assumed to be independent. Note that we allow the same node to be source for several multicast groups , and the same subset of nodes to be multicast group for several sources . The multicast capacity region is the closure of the collection of all achievable multicast traffic matrices . Observe that unicast traffic is a special case of multicast traffic, and hence is a -dimensional “slice” of the -dimensional region .

The next example illustrates the definitions of unicast and multicast traffic.

Example 1.

Consider and . Assume node wants to transmit a message to node at a rate of bit per second, and a message to node at rate bits per second. Node wants to transmit a message at rate bits per second to node . The messages are assumed to be independent. This traffic requirement can be described by a unicast traffic matrix with , , , and for all other pairs. Note that node is source for and , and that node is destination for and . Note also that node is neither a source nor a destination for any communication pair, and can hence be understood as a helper node.

Assume now node wants to transmit the same message to both and at rate bit per second, and a private message to only node at rate bits per second. Moreover, node wants to transmit the same message to both and at rate bits per second. The messages are assumed to be independent. This traffic requirement can be described by a multicast traffic matrix with , , , and for all other pairs. Note that is source for two multicast groups and , and that is multicast group for two sources and . ∎

Throughout, we denote by and the logarithms with respect to base and , respectively. To simplify notation, we suppress the dependence on within proofs whenever this dependence is clear from the context.

Iii Main Results

We now present the main results of this paper. Section III-A provides a scaling characterization of the unicast capacity region , and Section III-B provides a scaling characterization of the multicast capacity region of a dense wireless network. Section III-C contains example scenarios illustrating applications of the main theorems.

Iii-a Unicast Traffic

Define

 ^Λ\textupUC(n)≜{λ\textupUC∈\mathdsRn×n+:∑w≠uλ\textupUCu,w≤1 ∀u∈V(n), ∑u≠wλ\textupUCu,w≤1 ∀w∈V(n)}.

is the collection of all unicast traffic matrices such that for every node in the network the total traffic

 ∑w≠uλ\textupUCu,w

from is less than one, and such that for every node in the network the total traffic

 ∑u≠wλ\textupUCu,w

to is less than one.

The next theorem shows that is a tight approximation of the unicast capacity region of the wireless network.

Theorem 1.

For all , , and node placement with minimum node separation ,

 2−α/2^Λ\textupUC(n)⊂Λ\textupUC(n)⊂log(n2+α/2r−αmin(n))^Λ\textupUC(n).

Assuming that decays no faster than polynomial in (see the discussion in Section II), Theorem 1 states that approximates up to a factor . In other words, provides a scaling characterization of the unicast capacity region . This scaling characterization is considerably more general than the standard scaling results, in that it holds for any node placement and provides information on the entire -dimensional unicast capacity region (see Fig. 1). In particular, define

 ρ⋆λ\textupUC(n)≜max{ρ:ρλ\textupUC∈Λ\textupUC(n)}

to be the largest multiple such that is achievable. Then, for any arbitrary node placement and arbitrary unicast traffic matrix , Theorem 1 determines up to a multiplicative gap of order uniform in . This contrasts with the standard scaling results, which provide information on only for a uniform random node placement and a uniform random unicast traffic matrix (constructed by pairing nodes randomly into source-destination pairs with uniform rate).

Theorem 1 also reveals that the unicast capacity region of a dense wireless network has a rather simple structure in that it can be approximated up to a factor by an intersection of half-spaces. Each of these half-spaces corresponds to a cut in the wireless network, bounding the total rate across this cut. While there are such cuts in the network, Theorem 1 implies that only a small fraction of them are of asymptotic relevance. From the definition of , these are precisely the cuts involving just a single node (with traffic flowing either into or out of that node).

Iii-B Multicast Traffic

Let

 ^Λ\textupMC(n)≜{λ\textupMC∈\mathdsRn×2n+:∑W⊂V(n):W∖{u}≠∅λ\textupMCu,W≤1 ∀u∈V(n), ∑u≠w∑W⊂V(n):w∈Wλ\textupMCu,W≤1 ∀w∈V(n)}. (3)

Similarly to defined in Section III-A, the region is the collection of multicast traffic matrices such that for every node in the network the total traffic

 ∑W⊂V(n):W∖{u}≠∅λ\textupMCu,W

from is less than one, and such that for every node in the network the total traffic

 ∑u≠w∑W⊂V(n):w∈Wλ\textupMCu,W

to is less than one.

The next theorem shows that is a tight approximation of the multicast capacity region of the wireless network.

Theorem 2.

For all , , and node placement with minimum node separation ,

 2−1−α/2^Λ\textupMC(n)⊂Λ\textupMC(n)⊂log(n2+α/2r−αmin(n))^Λ\textupMC(n).

Assuming as before that decays no faster than polynomial in , Theorem 2 asserts that approximates up to a factor . In other words, as in the unicast case, we obtain a scaling characterization of the multicast capacity region . Again, this scaling characterization is considerably more general than standard scaling results, in that it holds for any node placement and provides information about the entire -dimensional multicast capacity region . Define, as for unicast traffic matrices,

 ρ⋆λ\textupMC(n)≜max{ρ:ρλ\textupMC∈Λ\textupMC(n)}

to be the largest multiple such that is achievable. Then Theorem 2 allows, for any arbitrary node placement and arbitrary multicast traffic matrix , to determine up to a multiplicative gap of order uniform in . In particular, no probabilistic assumptions about the structure of or are necessary.

As with , Theorem 2 implies that the multicast capacity region of a dense wireless network is approximated up to a factor by an intersection of half spaces. In other words, we are approximating a region of dimension (i.e., exponentially big in ) through only a linear number of inequalities. As in the case of unicast traffic, each of these inequalities corresponds to a cut in the wireless network, and it is again the cuts involving just a single node that are asymptotically relevant.

Iii-C Examples

This section contains several examples illustrating various aspects of the capacity regions and their approximations . Example 2 compares the scaling laws obtained in this paper with the ones obtained using hierarchical cooperation as proposed in [11]. Example 3 discusses symmetry properties of and . Example 4 provides a traffic pattern showing that the outer bounds in Theorems 1 and 2 are tight up to a constant factor.

Example 2.

(Random source-destination pairing)

Consider a random node placement with every node placed independently and uniformly at random on . Assume we pair each node with a node chosen independently and uniformly at random. Denote by the resulting source-destination pairs. Note that each node is source exactly once and destination on average once. Each source wants to transmit an independent message to at rate (depending on , but not on ). The question is to determine , the largest achievable value of . This question was considered in [11], where it was shown that, with probability as and for every ,

 Ω(n−ε)≤ρ⋆(n)≤O(nε). (4)

The lower bound is achieved by a hierarchical cooperation scheme, and we denote its rate by .

We now show that using the results presented in this paper these bounds on can be significantly sharpened. Set for and for all other entries of . is then given by

 ρ⋆(n)=max{ρ:ρλ\textupUC∈Λ\textupUC(n)}.

Setting

 ^ρ⋆(n)≜max{^ρ:^ρλ\textupUC∈^Λ\textupUC(n)},

we obtain from Theorem 1 that

 2−α/2^ρ⋆(n)≤ρ⋆(n)≤log(n2+α/2r−αmin(n))^ρ⋆(n). (5)

It remains to evaluate . By construction of , we have

 maxu∈V(n)∑w≠uλ\textupUCu,w=1.

Moreover, by [28],

 \mathdsP(12≤lnln(n)ln(n)maxw∈V(n)∑u≠wλ\textupUCu,w≤2)≥1−o(1).

Using the definition of , this yields that

 lnln(n)2ln(n)≤^ρ⋆(n)≤2lnln(n)ln(n) (6)

with high probability.

Recall that the minimum distance between nodes is , and that, for a random node placement, with high probability as (see, e.g., [11, Theorem 3.1]). Hence (5) and (6) show that that for random node placement and random source-destination pairing

 2−1−α/2lnln(n)ln(n)≤ρ⋆(n)≤(4+3α)log(e)lnln(n) (7)

with probability as . The lower bound is achieved using a communication scheme presented in Section IV-B based on interference alignment, and we denote its rate by .

Comparing (7) and (4), we see that the scaling law obtained here is significantly sharper, namely up to a factor here as opposed to a factor for any in [11]. Moreover, (7) provides good estimates for any value of , whereas (4) is only valid for large values of , with a pre-constant in that increases rapidly as (see [21, 29] for a detailed discussion on the dependence of the pre-constant on ). For a numerical example, Table I compares per-node rates of the hierarchical cooperation scheme of [11] (more precisely, an upper bound to it, with optimized parameters as analyzed in [21]) with the per-node rates obtained through interference alignment as proposed in this paper. For the numerical example, we choose .

We point out that the per-node rate decreases as the number of nodes increases only because of the random source-destination pairing. In fact, if the nodes are paired such that each node is source and destination exactly once, then the interference alignment based scheme achieves a per-node rate , i.e., the per-node rate does not decay to zero as . ∎

Example 3.

(Symmetry of and )

Theorems 1 and 2 provide some insight into (approximate) symmetry properties of the unicast and multicast capacity regions and . Indeed, their approximations and are invariant with respect to node positions (and hence, in particular, also invariant under permutation of nodes).

More precisely, consider a unicast traffic matrix . For a permutation of the nodes set

 ~λ\textupUCu,w≜λ\textupUCπ(u),π(w).

Then if and only if . Hence Theorem 1 yields that if , then

 2−α/2log−1(n2+α/2r−αmin(n))~λ\textupUC∈Λ\textupUC(n).

Similarly, let be a multicast traffic matrix, and define

 ~λ\textupMCu,W≜λ\textupMCπ(u),π(W),

where, for , . Theorem 2 implies that if , then

 2−1−α/2log−1(n2+α/2r−αmin(n))~λ\textupMC∈Λ\textupMC(n).

In other words, the location of the nodes in a dense wireless network (with decaying at most polynomially in ) affects achievable rates at most up to a factor . This contrasts with the behavior of extended wireless networks, where node locations crucially affect achievable rates [20]. ∎

Example 4.

(Tightness of outer bounds)

We now argue that the outer bounds in Theorems 1 and 2 are tight up to a constant factor in the following sense. There exists a constant such that for every we can find traffic matrices and on the boundary of the outer bound in Theorems 1 and 2 such that and . Or, more succinctly, there exists a constant such that

 Λ\textupUC(n)∖Klog(n2+α/2r−αmin(n))^Λ\textupUC(n) ≠∅, Λ\textupMC(n)∖Klog(n2+α/2r−αmin(n))^Λ\textupMC(n) ≠∅.

This shows that the gap between the inner and outer bounds in Theorems 1 and 2 is due to the use of the interference alignment scheme to prove the inner bound, and that to further decrease this gap a different achievable scheme has to be considered. Throughout this example, we assume for some constant .

Choose a node , and let, for each ,

 λ\textupUCu,w≜{1n−1if w=w⋆,0otherwise.

Note that . Under this traffic matrix , each node has an independent message for a common destination node .

If we ignore the received signals at all nodes and transmit no signal at , we transform the wireless network into a multiple access channel with users. Since for any , each node can reduce its power such that the received power at node is equal to . In this symmetric setting, the equal rate point of the capacity region of the multiple access channel has maximal sum rate, and hence each node can reliably transmit its message to at a per-node rate of

 1n−1log(1+(n−1)2−α/2) ≥1n−1log(n2−α/2) =1n−1(1−α2log(n))log(n).

Thus, for ,

 12log(n)λ\textupUC∈Λ\textupUC(n). (8)

On the other hand, using the assumption ,

 log(n2+α/2r−αmin(n))<(2+α(1/2+κ))log(n),

and hence

 (9)

Therefore, setting

 K≜(4+α(1+2κ))−1>0,

we obtain from (8) and (9) that

 Λ\textupUC(n)∖Klog(n2+α/2r−αmin(n))^Λ\textupUC(n)≠∅.

In words, at least along one direction in , the outer bound in Theorem 1 is loose by at most a constant factor.

Since is a -dimensional “slice” of the -dimensional region , the same result follows for as well. ∎

Iv Communication Schemes

This section describes the communication schemes achieving the inner bounds in Theorems 1 and 2. Both schemes use the idea of interference alignment as a building block, which is recalled in Section IV-A. The communication scheme for unicast traffic is introduced in Section IV-B and the scheme for multicast traffic in Section IV-C.

Iv-a Interference Alignment

Interference alignment is a technique introduced recently in [1, 2]. The technique is best illustrated with an example taken from [3]. Assume we pair the nodes into source-destination pairs such that each node in is source and destination exactly once. Consider the channel gains and for two different times and . Assume we could choose and such that and for all . By adding up the received symbols and , destination node obtains

 ywi[t1]+ywi[t2]=hui,wi[t1](xui[t1]+xui[t2])+zwi[t1]+zwi[t2].

Thus, by sending the same symbol twice (i.e., ), every source node is able to communicate with its destination node at essentially half the rate possible without any interference from other nodes.

Using this idea and the symmetry and ergodicity of the distribution of the channel gains, the following result is shown in [3].

Theorem 3.

For any source-destination pairing such that and for , the rates

 λ\textupUCui,wj={12log(1+2|hui,wi|2)if i=j,0otherwise,

are achievable, i.e., .

For a source-destination pairing as in Theorem 3, construct a matrix such that

 Sui,wj={1if i=j,0otherwise.

Note that is a permutation matrix, and we will call such a traffic pattern a permutation traffic. Using and ,

 12log(1+2r−αui,wi)≥12log(1+21−α/2)≥2−α/2,

and hence Theorem 3 provides an achievable scheme showing that . In other words, Theorem 3 shows that, for every permutation traffic, a per-node rate of is achievable. In the next two sections, we will use this communication scheme for permutation traffic as a building block to construct communication schemes for general unicast and multicast traffic.

Iv-B Communication Scheme for Unicast Traffic

Consider a general unicast traffic matrix . If happens to be a scalar multiple of a permutation matrix, then Theorem 3 provides us with an achievable scheme to transmit according to . In order to apply Theorem 3 for general , we need to schedule transmissions into several slots such that in each slot transmission occurs according to a permutation traffic. This transforms the original problem of communicating over a wireless network into a problem of scheduling over a switch with input and output ports and traffic requirement .

This problem has been widely studied in the literature. In particular, using a result from von Neumann [30] and Birkhoff [31] (see also [32] for the application to switches) it can be shown that for any there exist a collection of schedules (essentially permutation matrices, see the proof in Section V-A for the details) and nonnegative weights summing to one such that

 ∑iωiSi=λ\textupUC.

This suggests the following communication scheme. Split time into slots according to the weights . In the slot corresponding to , send traffic over the wireless network using interference alignment for the schedule . In other words, we time share between the different schedules according to the weights .

We analyze this communication scheme in more detail in Section V-A. In particular, we show that it achieves any point in . Combined with a matching outer bound, we show that this scheme is optimal for any unicast traffic pattern up to a factor .

Recall from Example 3 that the capacity region is approximately symmetric with respect to permutation of the traffic matrix. This implies that the rate achievable for any permutation traffic is approximately the same. While the decomposition of the traffic matrix into schedules is not unique, this invariance suggests that it does not matter too much which decomposition is chosen. The situation is different for Rayleigh fading (as opposed to phase fading considered here), where different decompositions can be used for opportunistic communication. This approach is explored in detail in Section VI-C.

Iv-C Communication Scheme for Multicast Traffic

We now turn to multicast traffic. Given the achievable scheme presented for unicast traffic in Section IV-B reducing the problem of communication over a wireless network to that of scheduling over a switch, it is tempting to try the same approach for multicast traffic as well. Unfortunately, scheduling of multicast traffic over switches is considerably more difficult than the corresponding unicast version (see, for example, [33] for converse results showing the infeasibility of multicast scheduling over switches with finite speedup). We therefore adopt a different approach here. The proposed communication scheme is reminiscent of the two-phase routing scheme of Valiant and Brebner [34].

Consider a source node that wants to multicast a message to destination group . The proposed communication scheme operates in two phases. In the first phase, the node splits its message into parts of equal length. It then sends one (distinct) part over the wireless network to each node in . Thus, after the first phase, each node in has access to a distinct fraction of the original message. In the second phase, each node in sends its message parts to all the nodes in . Thus, at the end of the second phase, each node in can reconstruct the entire message. All pairs operate simultaneously within each phase, and contention within the phases is resolved by appropriate scheduling (see the proof in Section V-B for the details).

A different way to look at this proposed communication scheme is as follows. Consider the nodes in , and construct a graph with for some additional node and with if either or . In other words, is a “star” graph with central node (see Fig. 2). We assign to each edge an edge capacity of one. The proposed communication scheme for the wireless network can then be understood as a two layer architecture, consisting of a physical layer and a network layer. The physical layer implements the graph abstraction , and the network layer routes data over .

In Section V-B, we show that the set of rates that can be routed over contains . We then argue that if , then , i.e., if messages can be routed over the graph at rates , then almost the same rates are achievable in the wireless network. Combining this with a matching outer bound, we show that the proposed communication scheme is optimal for any multicast traffic pattern up to a factor .

V Proofs

This section contains the proofs of Theorem 1 (in Section V-A) and Theorem 2 (in Section V-B).

V-a Proof of Theorem 1

We start with the proof of the outer bound in Theorem 1. For subsets , , denote by the capacity of the multiple-input multiple-output (MIMO) channel between nodes in and nodes in . Applying the cut-set bound [35, Theorem 14.10.1] to the sets , , we obtain

 ∑u≠wλ\textupUCu,w≤C({w}