Throughput-Optimal Multihop Broadcast on
Directed Acyclic Wireless Networks
We study the problem of efficiently broadcasting packets in multi-hop wireless networks. At each time slot the network controller activates a set of non-interfering links and forwards selected copies of packets on each activated link. A packet is considered jointly received only when all nodes in the network have obtained a copy of it. The maximum rate of jointly received packets is referred to as the broadcast capacity of the network. Existing policies achieve the broadcast capacity by balancing traffic over a set of spanning trees, which are difficult to maintain in a large and time-varying wireless network. We propose a new dynamic algorithm that achieves the broadcast capacity when the underlying network topology is a directed acyclic graph (DAG). This algorithm is decentralized, utilizes local queue-length information only and does not require the use of global topological structures such as spanning trees. The principal technical challenge inherent in the problem is the absence of work-conservation principle due to the duplication of packets, which renders traditional queuing modelling inapplicable. We overcome this difficulty by studying relative packet deficits and imposing in-order delivery constraints to every node in the network. Although in-order packet delivery, in general, leads to degraded throughput in graphs containing cycles, we show that it is throughput optimal in DAGs and can be exploited to simplify the design and analysis of optimal algorithms. Our characterization leads to a polynomial time algorithm for computing the broadcast capacity of any wireless DAG under the primary interference constraints. Additionally, we propose a multiclass extension of our algorithm which can be effectively used for broadcasting in any network with arbitrary topology. Simulation results show that the our algorithm has superior delay performance as compared to the tree-based approaches.
I Introduction and Related Work
Broadcast refers to the fundamental network functionality of delivering data from a source node to all other nodes. For efficient broadcasting, we need to use appropriate packet replication and forwarding to eliminate unnecessary packet retransmissions. This is especially important in power-constrained wireless systems which suffer from interference and collisions. Broadcast applications include mission-critical military communications , live video streaming , and data dissemination in sensor networks .
The design of efficient wireless broadcast algorithms faces several challenges. Wireless channels suffer from interference, and a broadcast policy needs to activate non-interfering links at every time slot. Wireless network topologies undergo frequent changes, so that packet forwarding decisions must be made in an adaptive fashion. Existing dynamic multicast algorithms that balance traffic over spanning trees  may be used for broadcasting, since broadcast is a special case of multicast. These algorithms, however, are not suitable for wireless networks because enumerating all spanning trees is computationally prohibitive that needs to be performed repeatedly when the network topology changes with time.
In this paper, we study the fundamental problem of throughput optimal broadcasting in wireless networks. We consider a time-slotted system. At every slot, a scheduler decides which non-interfering wireless links to activate and which set of packets to forward over the activated links, so that all nodes receive packets at a common rate. The broadcast capacity is the maximum common reception rate of distinct packets over all scheduling policies. To the best of our knowledge, there does not exist any capacity-achieving scheduling policy for wireless broadcast without the use of spanning trees 111Note that we exclude network-coding operations throughout the paper.. The main contribution of this paper is to design provably optimal wireless broadcast algorithms that does not use spanning trees when the underlying topology is a DAG.
We start out with considering a rich class of scheduling policies that perform arbitrary link activations and packet forwarding. We define the broadcast capacity as the maximum common rate achievable over this policy class . We next enforce two constraints that lead to a smaller set of policies. First, we consider the subclass of policies that enforce the in-order delivery of packets. Second, we focus on the subset of policies that allows the reception of a packet by a node only if all its incoming neighbours have received the packet. It is intuitively clear that the policies in the more structured class are easier to describe and analyze, but may yield degraded throughput performance. We show the surprising result that when the underlying network topology is a directed acyclic graph (DAG), there is a control policy that achieves the broadcast capacity. In contrast, we prove the existence of a network containing a cycle in which no control policy in the policy-space can achieve the broadcast capacity.
To enable the design of the optimal broadcast policy, we establish a queue-like dynamics for the system-state, represented by relative packet deficits. This is non-trivial for the broadcast problem because explicit queueing structure is difficult to define in the network due to packet replications. We subsequently show that, the problem of achieving the broadcast capacity reduces to finding a scheduling policy stabilizing the system, which can be accomplished by stochastic Lyapunov drift analysis techniques [5, 6].
In this paper, we make the following contributions:
We define the broadcast capacity of a wireless network and show that it is characterized by an edge-capacitated graph that arises from optimizing the time-averages of link activations. For integral-capacitated DAGs, the broadcast capacity is determined by the minimum in-degree of the graph , which is equal to the maximal number of edge-disjoint spanning trees.
We design a dynamic algorithm that utilizes local queue-length information to achieve the broadcast capacity of a wireless DAG network. This algorithm does not rely on spanning trees, has small computational complexity and is suitable for mobile networks with time-varying topology. This algorithm also yields a constructive proof of a version of Edmonds’ disjoint tree-packing theorem  which is generalized to wireless activations but specialized to DAG topology.
Based on our characterization of the broadcast capacity, we derive a polynomial-time algorithm to compute the broadcast capacity of any wireless DAG under primary interference constraints.
We propose a randomized multiclass extension of our algorithm, which can be effectively used to do broadcast on wireless networks with arbitrary underlying topology.
We demonstrate the superior delay performance of our DAG-policy, as compared to centralized tree-based algorithm , via numerical simulations. We also explore the efficiency/complexity trade-off of our proposed multiclass extension through extensive simulations.
In the literature, a simple method for wireless broadcast is to use packet flooding . The flooding approach, however, leads to redundant transmissions and collisions, known as broadcast storm . In the wired domain, it has been shown that forwarding useful packets at random is optimal for broadcast ; this approach does not extend to the wireless setting due to interference and the need for scheduling . Broadcast on wired networks can also be done using network coding [12, 13]. However, efficient link activation under network coding remains an open problem.
The rest of the paper is organized as follows. Section II introduces the wireless network model. In Section III, we define the broadcast capacity of a wireless network and provide a useful upper bound from a cut-set consideration. In Section IV, we propose a dynamic broadcast policy that achieves the broadcast capacity in a DAG. In section V, we propose an efficient algorithm for computing the broadcast capacity of any wireless DAG under primary interference constraints. Our DAG-broadcast algorithm is extended to networks with arbitrary topology in section VI. Illustrative simulation results are presented in Section VII. Finally, we conclude our paper in section VIII.
Ii The Wireless Network Model
We consider a time-slotted wireless network model represented by the tuple , where is the set of nodes, is the set of directed links, is the capacity-vector of the links and is the set of all feasible link-activation vectors, whose elements are binary vectors such that the links with can be activated simultaneously. The structure of the activation set depends on the underlying interference model. For example, under the primary interference constraint (also known as node-exclusive interference constraint ), the set consists of all binary vectors corresponding to matchings of the underlying graph , see Fig. 1. In the case of a wired network, is the set of all binary vectors since there is no interference. In this paper we allow an arbitrary link-activation set , which captures different wireless interference models. Let be the source node at which stochastic broadcast traffic is generated (or arrives externally). The number of packets generated at the node at slot is given by the random variable , which is i.i.d. over slots with mean . These packets are to be delivered efficiently to all other nodes in the network.
Iii Wireless Broadcast Capacity
Intuitively, the network supports a broadcast rate if there exists a scheduling policy under which all network nodes can receive distinct packets at rate . The broadcast capacity is the maximally supportable broadcast rate in the network. Formally, we consider a class of scheduling policies where each policy consists of a sequence of actions executed at every slot . Each action comprises of two operations: (i) the scheduler activates a subset of links by choosing a feasible activation vector ; (ii) each node forwards a subset of packets (possibly empty) to node over an activated link , subject to the link capacity constraint. The class includes policies that use all past and future information, and may forward any subset of packets over a link.
Let be the number of distinct packets received by node from the beginning of time up to time , under a policy . The time average is the rate of distinct packets received at node .
A policy is called a “broadcast policy of rate ” if all nodes receive distinct packets at rate , i.e.,
where is the packet arrival rate at the source node .
The broadcast capacity of a wireless network is the supremum of all arrival rates for which there exists a broadcast policy of rate .
Iii-a An upper bound on broadcast capacity
We characterize the broadcast capacity of a wireless network by proving a useful upper bound. This upper bound is understood as a necessary cut-set bound of an associated edge-capacitated graph that reflects the time-averaged behaviour of the wireless network. We first give an intuitive explanation of the bound, assuming that the involved limits exist. Then in the proof of Theorem 1 we rigorously prove the bound by relaxing this assumption.
Fix a policy . Let be the fraction of time link is activated under ; that is, we define the vector
where is the link-activation vector under policy in slot . The average flow rate over a link under the policy is upper bounded by the product of the link capacity and the fraction of time the link is activated, i.e., . Hence, we can define an edge-capacitated graph associated with policy , where each directed link has capacity ; see Fig. 2 for an example of such an edge-capacitated graph. Next, we provide a bound on the broadcast capacity by maximizing the broadcast capacity on the ensemble of graphs over all feasible vectors .
We define a proper cut of the network graph as a proper subset of the node set that contains the source node . Define the link subset
Since , there exists a node . Consider the throughput of node under policy . The max-flow min-cut theorem shows that the throughput of node cannot exceed the total link capacity across the cut . Since the achievable broadcast rate of policy is an upper-bound on the throughput of all nodes, we have . This inequality holds for all proper cuts and we have
Equation (4) holds for any policy . Thus, the broadcast capacity of the wireless network satisfies
where the last inequality holds because the vector associated with any policy lies in the convex hull of the activation set . Our first theorem formalizes the above intuitive characterization of the broadcast capacity of a wireless network.
The broadcast capacity of a wireless network with activation set is upper bounded as follows:
See Appendix -A. ∎
Iii-B In-order packet delivery
Studying the performance of any arbitrary broadcast policy is formidable because packets are replicated across the network and may be received out of order. To avoid unnecessary re-transmissions, the nodes must keep track of the identity of the received set of packets, which complicates the system state; because instead of the number of packets received, the system state is properly described here by the subset of packets received at each of the nodes.
To simplify the system state, we focus on the subset of policies that enforce the following constraint:
Constraint 1 (In-order packet delivery).
A network node is allowed to receive a packet only if all packets have been received by that node.
In-order packet delivery is useful in live media streaming applications , where buffering out-of-order packets incurs increased delay that degrades video quality. In-order packet delivery greatly simplifies the network state space. Let be the number of distinct packets received by node by time . For policies in , the set of received packets by time at node is . Therefore, the network state in slot is given by the vector .
In section IV we will prove that there exists a throughput-optimal broadcast policy in the space when the underlying network topology is a DAG. Ironically, Lemma (1) shows that there exists a network containing a cycle in which any broadcast policy in the space is not throughput optimal. Hence the space can not, in general, be extended beyond DAGs while preserving throughput optimality.
Let be the broadcast capacity of the policy subclass that enforces in-order packet delivery. There exists a network topology containing a directed cycle such that .
See Appendix -B. ∎
We will return to the problem of broadcasting in networks with arbitrary topology in Section VI.
Iii-C Achieving the broadcast capacity in a DAG
At this point we concentrate our attention to Directed Acyclic Graphs (DAGs). Graphs in this class are appealing for our analysis because they possess well-known topological ordering of the nodes . For DAGs, the upper bound (5) on the broadcast capacity in Theorem 1 will be simplified further. For each receiver node , consider the proper cut that separates the network from node :
Using these cuts , we define another upper bound on the broadcast capacity as:
where the first inequality uses the subset relation and the second inequality follows from Theorem 1. In Section IV, we will propose a dynamic policy that belongs to the policy class and achieves the broadcast rate . Combining this result with (7), we establish that the broadcast capacity of a DAG is given by
This is achieved by a broadcast policy that uses in-order packet delivery. In other words, we show that imposing the in-order packet delivery constraint does not reduce the broadcast capacity when the underlying topology is a DAG.
From a computational point of view, the equality in Eqn. (8) is attractive, because it implies that for computing the broadcast capacity of any wireless DAG, it is enough to consider only those cuts that separate a single (non-source) node from the source-side. Note that, there are only of such cuts, in contrast with the total number of cuts, which is exponential in the size of the network. This fact will be exploited in section V to develop a strongly poly-time algorithm for computing the broadcast capacity of any DAG under the primary interference constraints.
Iv DAG Broadcast Algorithm
In this section we design an optimal broadcast policy for wireless DAGs. We start by imposing an additional constraint that leads to a new subclass of policies . As we will see, policies in can be described in terms of relative packet deficits which constitute a simple dynamics. We analyze the dynamics of the minimum relative packet deficit at each node , where the minimization is over all incoming neighbours of . This quantity plays the role of virtual queues in the system and we design a dynamic control policy that stabilizes them. The main result of this section is to show that this control policy achieves the broadcast capacity whenever the network topology is a DAG.
Iv-a System-state by means of packet deficits
We showed in Section III-B that, constrained to the policy-space , the system-state is completely represented by the vector .
To simplify the system dynamics further, we restrict further as follows.
We say that node is an in-neighbor of node iff there exists a directed link in the underlying graph .
A packet is eligible for transmission to node at a slot only if all the in-neighbours of have received packet in some previous slot.
We denote this new policy-class by . We will soon show that it contains an optimal policy. Fig. 3 shows the relationship among different policy classes222We note that, if the network contains a directed cycle, then a deadlock might occur under a policy in and may yield zero broadcast throughput. However, this problem does not arise when the underlying topology is a DAG..
Following properties of the system-states under a policy will be useful.
For , let denote the set of in-neighbors of a node in the network. Under any policy , we have:
The indices of packets that are eligible to be transmitted to the node at slot is given by
We define the packet deficit over a directed link by . Under a policy in , is always non-negative because, by part (1) of Lemma 2, we have
The quantity denotes the number of packets received by node but not by node , upto time . Intuitively, if all packet deficits are bounded asymptotically, the total number of packets received by any node is not lagging far from the total number of packets generated at the source; hence, the broadcast throughput will be equal to the packet generation rate.
To analyze the system dynamics under a policy in , it is useful to define the minimum packet deficit at node by
From part (2) of Lemma 2, is the maximum number of packets that node is allowed to receive from its in-neighbors at slot . As an example, Fig. 4 shows that the packet deficits at node , relative to the upstream nodes , , and , are , , and , respectively. Thus and node is only allowed to receive four packets in slot due to Constraint 2. We can rewrite as
and the node is the in-neighbor of node from which node has the smallest packet deficit in slot ; ties are broken arbitrarily in deciding .333We note that the minimizer is a function of the node and the time slot ; we slightly abuse the notation by neglecting to avoid clutter. Our optimal broadcast policy will be described in terms of the minimum packet deficits .
Iv-B The dynamics of the system variable
We now analyze the dynamics of the system variables
under a policy . Define the service rate vector by
Equivalently, we may write , and the number of packets forwarded over a link is constrained by the choice of the link-activation vector . At node , the increase in the value of depends on the identity of the received packets; in particular, node must receive distinct packets. Next, we clarify which packets are to be received by node at time .
The number of available packets for reception at node is , because: (i) is the maximum number of packets node can receive from its in-neighbours subject to the Constraint 2; (ii) is the total incoming transmission rate at node under a given link-activation decision. To correctly derive the dynamics of , we consider the following efficiency requirement on policies in :
Constraint 3 (Efficient forwarding).
Given a service rate vector , node pulls from the activated incoming links the following subset of packets (denoted by their indices)
The specific subset of packets that are pulled over each incoming link are disjoint but otherwise arbitrary.444Due to Constraints 1 and 2, the packets in (12) have been received by all in-neighbors of node .
Constraint 3 requires that scheduling policies must avoid forwarding the same packet to a node over two different incoming links. Under certain interference models such as the primary interference model, at most one incoming link is activated at a node in a slot and Constraint 3 is redundant.
In Eqn. (11), the packet deficit increases with and decreases with , where and are both non-decreasing. Hence, we can upper-bound the increment of by the total capacity of the activated incoming links at node . Also, we can express the decrement of by the exact number of distinct packets received by node from its in-neighbours, and it is given by by Constraint 3. Consequently, the one-slot evolution of the variable is given by555We emphasize that the node is defined in (10), depends on the particular node and time , and may be different from the node .
where and we recall that . It follows that evolves over slot according to
where the equality (a) follows the definition of , equality (b) follows because node and equality (c) follows from Eqn. (13). In Eqn. (IV-B), if , we abuse the notation to define for the source node , where is the number of exogenous packet generated at slot .
Iv-C The optimal broadcast policy
Our broadcast policy is designed to keep the minimum deficit process stable. For this, we regard the variables as virtual queues that follow the dynamics (IV-B). By performing drift analysis on the virtual queues , we propose the following max-weight-type broadcast policy , described in Algorithm 1. We have and it enforces the constraints 1, 2, and 3. We will show that this policy achieves the broadcast capacity of a wireless network over the general policy class when the underlying topology is a DAG.
At each slot , the network-controller observes the state-variables and executes the following actions
The next theorem demonstrates the optimality of the broadcast policy .
If the underlying network graph is a DAG, then for any exogenous packet arrival rate , the broadcast policy yields
where is the upper bound on the broadcast capacity in the general policy class , as shown in (7). Consequently, the broadcast policy achieves the broadcast capacity for any Directed Acyclic Graphs.
See Appendix -C. ∎
Iv-D Number of disjoint spanning trees in a DAG
Theorem 2 provides an interesting combinatorial result that relates the number of disjoint spanning trees in a DAG to the in-degrees of its nodes.
Consider a directed acyclic graph that is rooted at a node , has unit-capacity links, and possibly contains parallel edges. The maximum number of edge-disjoint spanning trees in is given by
where denotes the in-degree of the node .
See Appendix -D. ∎
V Efficient Algorithm for Computing the Broadcast Capacity of a DAG
In this section we exploit Eqn. (8) and develop an LP to compute the broadcast capacity of any wireless DAG network under the primary interference constraints. Although this LP has exponentially many constraints, using a well-known separation oracle, it can be solved in strongly polynomial time via the ellipsoid algorithm .
Under the primary interference constraint, the set of feasible activations of the graphs are matchings . For a subset of edges , let where if and is zero otherwise. Let us define
We have the following classical result by Edmonds .
The set is characterized by the set of all such that :
Here is the set of edge (ignoring their directions) with both end points in U, () denotes the set of all incoming (outgoing) edges to (from) the vertex .
Hence following Eqn. (8), the broadcast capacity of a DAG can be obtained by the following LP :
From the equivalence of optimization and separation (via the ellipsoid method), it follows that the above LP is poly-time solvable if there exists an efficient separator oracle for the constraints (21), (22). Since there are only linearly many constraints (, to be precise) in (21), the above requirement reduces to an efficient separator for the matching polytope (22). We refer to a classic result from the combinatorial-optimization literature which shows the existence of such efficient separator for the matching polytope
This directly leads to the following theorem.
There exists a strongly poly-time algorithm to compute the broadcast capacity of any wireless DAG under the primary interference constraints.
The following corollary implies that, although there are exponentially many matchings in a DAG, to achieve the broadcast capacity, randomly activating (with appropriate probabilities) only matchings suffice.
The optimal broadcast capacity in a wireless DAG, under the primary interference constraints, can be achieved by randomly activating (with positive probability) at most matchings.
Vi Broadcasting on Networks with Arbitrary Topology
In this section we extend the broadcast policy for a DAG to networks containing cycles. From the negative result of Lemma 1, we know that any policy ensuring in-order packet delivery at every node cannot, in general, achieve the broadcast capacity of a network containing cycles. To get around this difficulty, we introduce the concept of broadcasting using multiple classes of packets. The idea is as follows: each class has a one-to-one correspondence with a specific permutation of the nodes; for an edge if the node appears prior to the node in the permutation (we denote this condition by ), then the edge is included in the class , otherwise the edge ignored by the class . The set of all edges included in the class is denoted by . It is clear that each class corresponds to a unique embedded DAG topology , which is a subgraph of the underlying graph .
A new incoming packet arriving at the source node is admitted to some class , according to some policy. All packets in a given class are broadcasted while maintaining in-order delivery property within the class , however packets from different classes do not need to respect this constraint. Hence the resulting policy does not belong to the class in but rather to the general class . This new policy keeps the best of both worlds: (a) its description-complexity is , where for each class we essentially have the same representations as in the in-order delivery constrained policies and (b) by relaxing the inter-class in-order delivery constraint it has the potential to achieve the full broadcast capacity of the underlying graph.
Hence the broadcast problem reduces to construction of multiple classes (which are permutations of the vertices ) out of the given directed graph such that it covers the graph efficiently, from a broadcast-capacity point of view. In Algorithm-8, we choose the permutations uniformly at random with the condition that the source always appears at the first position of the permutation.
The multiclass broadcast Algorithm-8 with classes supports a broadcast rate of
where we use the convention that if .
The right hand side of Eqn. (26) can be understood as follows. Consider a feasible stationary activation policy which activates class on the edge fraction of time. Since, by construction, each of the class follows a DAG, lemma (3) implies that the resulting averaged graph has a broadcast capacity of for the class . Thus the total broadcast rate achievable by this scheme is simply . Given the classes, following the same line of argument as in (20), we can develop a similar LP to compute the broadcast capacity (26) of all these -classes taken together in strongly poly-time.
We then compare the multiclass broadcast algorithm 8 with the stationary activation policy above to show that the Multiclass broadcast algorithm is stable under all arrival rates below . The details are omitted for brevity.
Since the broadcast-rate achievable by a collection of embedded DAGs in a graph is always upper-bounded by the actual broadcast capacity of , we have the following interesting combinatorial result from Theorem (6)
Consider a wired network, represented by the graph . For a given integer , consider classes as in Theorem (6), with being their corresponding edge-sets. Then, for any set of non-negative vectors with , the following lower-bound for the broadcast capacity holds:
where we use the convention that if .
The above corollary may be contrasted with Eqn. (7), which provides an upper bound to the broadcast capacity .
Vii Simulation Results
We present a number of simulation results concerning the delay performance of the optimal broadcast policy in wireless DAG networks with different topologies. For simplicity, we assume primary interference constraints throughout this section. Delay for a packet is defined as the number of slots required for it to reach all nodes in the network, after its arrival to the source r.
We first consider a -node diamond topology as shown Fig. 6. Link capacities are shown along with the links. The broadcast capacity of the network is upper bounded by the maximum throughput of node , which is because at most one of its incoming links can be activated at any time. To show that the broadcast capacity is indeed , we consider the three spanning trees rooted at the source node . By finding the optimal time-sharing of all feasible link activations over a subset of spanning trees using linear programming, we can show that the maximum broadcast throughput using only the spanning tree is . The maximum broadcast throughput over the two trees is , and that over all three trees is . Thus, the upper bound is achieved and the broadcast capacity is .
We compare our broadcast policy with the tree-based policy in . While the policy is originally proposed to transmit multicast traffic in a wired network by balancing traffic over multiple trees, we slightly modify the policy for broadcasting packets over spanning trees in the wireless setting; link activations are chosen according to the max-weight procedure. See Fig. 5 for a comparison of the average delay performance under the policy and the tree-based policy over different subset of trees. The simulation duration is slots. We observe that the policy achieves the broadcast capacity and is throughput optimal.
The broadcast policy does not rely on the limited tree structures and therefore has the potential to exploit all degrees of freedom in packet forwarding in the network; such freedom may lead to better delay performance as compared to the tree-based policy. To observe this effect, we consider the -node DAG network subject to the primary interference constraint in Fig. 8. For every pair of node , , the network has a directed link from to with capacity . By induction, we can calculate the number of spanning trees rooted at the source node to be . We choose five arbitrary spanning trees , over which the tree-based algorithm is simulated. Table I demonstrates the superior delay performance of the broadcast policy , as compared to that of the tree-based algorithm over different subsets of the spanning trees. It also shows that a tree-based algorithm that does not use enough trees would result in degraded throughput.
|tree-based policy over the spanning trees:||broadcast|
Multiclass Simulation for Arbitrary Topology
We randomly generate an ensemble of wired networks (not necessarily DAGs), each consisting of nodes and unit capacity links. By solving the LP corresponding to Eqn. (26), we compute the fraction of the total broadcast capacity achievable using randomly chosen classes by the Multiclass Algorithm 8 of section VI. The result is presented in Figure 9. It follows that a sizeable fraction of the optimal capacity may be achieved by using a moderate number of classes. However the number of required classes for achieving a certain fraction of the capacity increases as the broadcast capacity increases. This is because of the fact that increased broadcast capacity would warrant an increased number of DAGs to cover the graph efficiently.
We characterize the broadcast capacity of a wireless network under general interference constraints. When the underlying network topology is a DAG, we propose a dynamic algorithm that achieves the wireless broadcast capacity. Our novel design, based on packet deficits and the in-order packet delivery constraint, is promising for application to other systems with packet replicas, such as multicasting and caching systems. Future work involves the study of arbitrary networks, where optimal policies must be sought in the class .
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-a Proof of Theorem 1
Fix an . Consider a policy that achieves a broadcast rate of at least defined in (1); this policy exists by the definition of the broadcast capacity in Definition 2. Consider any proper cut of the network . By definition, there exists a node . Let be the link-activation vector chosen by policy in slot . The maximum number of packets that can be transmitted across the cut in slot is at most , which is the total capacity of all activated links across , and the link subset is given in (3). The number of distinct packets received by a node by time is upper bounded by the total available capacity across the cut up to time , subject to link-activation decisions of policy . That is, we have
where we define the vector , , and is the inner product of two vectors.666Note that (29) remains valid if network coding operations are allowed. Dividing both sides by yields