Wireless Network Coding with Local Network Views: Coded Layer Scheduling

Wireless Network Coding with Local Network Views: Coded Layer Scheduling

Alireza Vahid Vaneet Aggarwal A. Salman Avestimehr and Ashutosh Sabharwal A. Vahid and A. S. Avestimehr are with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY (email: av292@cornell.edu, avestimehr@ece.cornell.edu). V. Aggarwal is with AT&T Labs - Research, Florham Park, NJ 07932 (email: vaneet@research.att.com). A. Sabharwal is with Department of Electrical and Computer Engineering, Rice University, Houston, TX (email: ashu@rice.edu). The results in this paper were presented in part at the Allerton Conference [1].
Abstract

One of the fundamental challenges in the design of distributed wireless networks is the large dynamic range of network state. Since continuous tracking of global network state at all nodes is practically impossible, nodes can only acquire limited local views of the whole network to design their transmission strategies. In this paper, we study multi-layer wireless networks and assume that each node has only a limited knowledge, namely 1-local view, where each S-D pair has enough information to perform optimally when other pairs do not interfere, along with connectivity information for rest of the network. We investigate the information-theoretic limits of communication with such limited knowledge at the nodes. We develop a novel transmission strategy, namely Coded Layer Scheduling, that solely relies on 1-local view at the nodes and incorporates three different techniques: (1) per layer interference avoidance, (2) repetition coding to allow overhearing of the interference, and (3) network coding to allow interference neutralization. We show that our proposed scheme can provide a significant throughput gain compared with the conventional interference avoidance strategies. Furthermore, we show that our strategy maximizes the achievable normalized sum-rate for some classes of networks, hence, characterizing the normalized sum-capacity of those networks with 1-local view.

I Introduction

In dynamic wireless networks, optimizing system efficiency requires information about the state of the network in order to determine what resources are actually available. However, in large wireless networks, keeping track of the state for making optimal decisions is typically infeasible. Thus, in the absence of centralization of network state information, nodes have limited local views of the network and make decentralized decisions based on their own local view of the network. The key question then is, how do optimal decentralized decisions perform in comparison to the optimal centralized decisions which rely on full network state information.

In this paper, we consider multi-source multi-destination multi-layer wireless networks and seek sum-rate optimal transmission strategies when sources have only limited local view of the network. To model local views at the nodes, we use a generalization of the hop-count based model that we introduced in  [2] for single-layer networks. In the hop-count based model each source knows the channel gains of those links that are up to certain number of hops away and beyond that it only knows whether a link exists or not. The hop-count based model was appropriate for single-layer networks where all destinations are within one-hop from their respective sources. For multi-layer networks, a more scalable approach is to model local views based on the knowledge about source-destination (S-D) routes in the network (instead of source-destination links). The motivation for the route-based model stems from coordination protocols like routing which are often employed in multi-hop networks to discover S-D routes in the network. Hence, a reasonable quanta for network state information is the number of such end-to-end routes that are known at the source nodes. In this paper, we consider the case where each S-D pair has enough information to perform optimally when other pairs do not interfere. Beyond that, the only other information available at each node is the global network connectivity. We refer to this model of local network knowledge as -local view.

Since each channel gain can range from zero to a maximum value, our formulation is similar to compound channels [3, 4] with one major difference. In the multi-terminal compound network formulations, all nodes are missing identical information about the channels in the network, whereas, in our formulation, the -local view results in asymmetric information about channels at different nodes.

In this paper, our metric to measure the performance of transmission strategies is normalized sum-capacity as defined in [2], which represents the maximum fraction of the sum-capacity with full knowledge that can be always achieved when nodes only have partial knowledge about the network.

I-a Contributions

Our main contribution is a new transmission scheme, named Coded Layer (CL) Scheduling, which only requires -local view at the nodes and combines coding with interference avoidance scheduling. Developed as a graph coloring algorithm on a route-extended graph, coded layer scheduling is a combination of three main techniques: (1) per layer interference avoidance, (2) repetition coding to allow overhearing of the interference, and (3) network coding to allow interference neutralization.

We characterize the achievable normalized sum-rate of the CL scheduling as the solution to a new graph coloring problem and analyze its optimality for some classes of networks. In particular, we show that coded layer scheduling achieves the normalized sum-capacity in single-layer and two-layer -folded chain networks (defined in Section V). Furthermore, by considering -nested folded-chain networks (defined in Section V), we show that the gain from CL scheduling over interference avoidance scheduling can be unbounded.

We also investigate network topologies in which with -local view at the nodes, interference avoidance scheduling is information-theoretically optimal. More specifically, we consider another class of networks, i.e. networks, which is a -flow network where all intermediate layers have only relays. We show that for this class, a simpler scheme based on only interference avoidance techniques, named Independent Layer (IL) scheduling, is optimal and coding is not required to achieve normalized sum-capacity with -local view. In our limited experience, coding across time-slots can provide gains when there is some regular topological structure in the network and/or there is dense connecvitity. However, the general connection between network topology, partial information and optimal schemes remains a largely open problem.

I-B Related Work

In any network state learning algorithm, network state information is obtained via a form of message passing between the nodes. Since the channels through which the communication takes place are noisy and have delay, imprecise network information at the nodes becomes an important problem. Many models for imprecise network information have been considered for interference networks. These models range from having no channel state information at the sources [5, 6, 7, 8, 9], delayed channel state information [10, 11, 12, 13] or analog feedback of channel state for fully-connected interference channels [14]. Most of these works assume fully connected network or a small number of users. A study to understand the role of limited network knowledge, was first initiated in [15, 16] for general single-layer networks with arbitrary connectivity, where the authors used a message-passing abstraction of network protocols to formalize the notion of local view of the network at each node, such that the view at different nodes are mismatched from each others’. The key result was that local-view-based (decentralized) decisions can be either sum-rate optimal or can be arbitrarily worse than the global-view (centralized) sum-capacity.

The initial work in [15, 16] was strengthened for arbitrary -user single-layer interference network in [2, 17, 18], where the authors proposed a new metric, normalized sum-capacity, to measure the performance of distributed decisions. Further, the authors computed the normalized sum-capacity of distributed decisions for several network topologies with one-hop, two-hop and three-hop local-view information at each source. In this paper, we investigate the performance of decentralized decisions for multi-layer wireless networks.

The rest of the paper is organized as follows. In section II, we will introduce our network model and the new model to capture partial network knowledge and we define the notion of normalized sum-capacity. In Section III, via a number of examples, we motivate our transmission strategies. In Section IV, we present our main result, i.e. coded layer scheduling, and we charaterize its performance for multi-layer networks. In Section V, we prove the optimality of our strategies (in terms of achieving normalized sum-capacity) for some networks. Finally, Section VI concludes the paper and presents some future directions.

Ii Problem Formulation

In this section, we introduce our models for channel, network, and network knowledge at the nodes. We further define the notions of normalized sum-capacity introduced in [2], which will be used to measure the performance of the strategies with partial network knowledge.

Ii-a Network Model and Notations

In this subsection, we will describe two channel models that will be studied in the paper, namely the linear deterministic model [19], and the Gaussian model. In both models, a network is represented by a directed graph , where is the set of vertices representing nodes in the network, is the set of directed edges representing links among the nodes, and represents the channel gains associated with the edges.

We consider a layered network in this paper, i.e. the nodes in this network can be partitioned into subsets . Out of nodes in the network, are denoted as sources and are destinations. We label these source and destination nodes by and respectively, . We set and . The remaining nodes are relay nodes which facilitate the communication between sources and destinations. We denote a specific relay in by , and . Without loss of generality, we can also refer to a node in simply as , .

The layered structure of the network imposes the following constraint on the edges in the network,

(1)

The two channel models used in this paper are as follows.

  1. The Linear Deterministic Model [19]: In this model, there is a non-negative integer, , associated with each link , which represents its gain. Let be the maximum of all the channel gains in this network. In the linear deterministic model, the channel input at node at time is denoted by . The received signal at node at time is denoted by , and is given by

    (2)

    where is the shift matrix and the operations are in . If a link between and does not exist, we set to be zero.

  2. The Gaussian Model: In this model, the channel gain is denoted by . The channel input at node at time is denoted by , and the received signal at node at time is denoted by given by

    (3)

    where is the additive white complex Gaussian noise with unit variance. We also assume a power constraint of at all nodes, i.e. .

A route from a source to a destination is a set of nodes such that there exists an ordering of these nodes where the first one is , last one is , and any two consecutive nodes in this ordering are connected by an edge in the graph.

Definition 1.

An induced subgraph is a subgraph of with its vertex set being the union of all routes from source to a destination , and its edge set being the subset of all edges in between the vertices of .

We say that S-D pair and S-D are non-interfering if and are two disjoint induced subgraphs of .

The in-degree function , is the number of in-coming edges connected to node . Similarly, the out-degree function , is the number of out-going edges connected to node . Note that the in-degree of a source and the out-degree of a destination are both equal to . The maximum degree of the nodes in is defined as

(4)

We also need the following definitions that will be used later in this paper.

Definition 2.

At any node , we define the index set as follows

(5)

In other words, is the set of indices of those S-D pairs that have on a route between them.

Definition 3.

The route-expanded graph associated with a layered network with sources in and destinations in is constructed by replacing each node with nodes represented by where , and connect them according to

We define , (i.e. all the duplicates of node ) and we refer to it as a super-node (or equivalently a super-relay if is a relay).

For an illustration of the route-expanded graph see Figure 1. For simplicity, we have represented each pair with a shape, i.e. , , and for S-D pairs , , and respectively. The route-expanded graph of the network in Figure 1(a) is illustrated in Figure 1(b). Each relay is on a route for two S-D pairs, hence each super-relay contains two nodes.

Fig. 1: (a) A -layer network, and (b) its route-expanded graph.

Ii-B Model of Partial Network Knowledge

In this subsection, we describe the model of [1] for partial network information that will be used in this paper. We first define the route-adjacency graph of , which is an undirected bipartite graph consisting of all sources on one side and all destinations on the other side; see Figure 2 for an example. A source and a destination are connected in , if there exists a route between them in . More formally, where and .

We now define the model for partial network knowledge that will be used in the paper, namely -local view, as the following:

  • All nodes have full knowledge of the network topology, , i.e., which links are in , but not their channel gains. The network topology knowledge is denoted by side information .

  • Each source, , knows the gains of all those channels that are in a route from source to destination , such that and are at most hops away from in . The -hop channel knowledge at a source is denoted by .

  • Each node (which is not a source) has the union of the information of all those sources that have a route to it, and this knowledge at node is denoted by .

Note that this model is a generalization of the hop-based model for partial network knowledge in single layer networks [2]. While the partial information model is general, we will focus on the case where . In other words, each S-D pair has enough information to perform optimally when other pairs do not interfere (i.e., it knows the channel gains of all links that are in a route to its own destination). However beyond that, each pair only knows the connectivity in the network (structure of interference). We are interested to find if one can outperform interference avoidance techniques with such limited knowledge. In the following subsection, we define the metrics we use to measure the performance of transmission strategies with -local view.

Fig. 2: (a) A multi-layer network, and (b) its route-adjacency graph.

Ii-C Normalized Sum-Capacity

We now define the notion of normalized sum-capacity, which is our metric for evaluating network capacity with partial network knowledge [2, 18]. Normalized sum-capacity represents the maximum fraction of the sum-capacity with full knowledge that can be always achieved when nodes only have partial knowledge about the network, and is defined as follows.

Consider the scenario in which source wishes to reliably communicate message to destination during uses of the channel, . We assume that the messages are independent and chosen uniformly. For each source , let message be encoded as using the encoding function , which depends on the available local network knowledge, , and the global side information, .

Each relay in the network creates its input to the channel , using the encoding function , which depends on the available network knowledge, , and the side information, , and all the previous received signals at the relay . A relay strategy is defined as the union of of all encoding functions used by the relays, , and .

Destination is only interested in decoding and it will decode the message using the decoding function , where is the destination ’s network knowledge. Note that the local view can be different from node to node.

Definition 4.

A Strategy is defined as the set of: (1) all encoding functions at the sources; (2) all decoding functions at the destinations; and (3) the relay strategy for , i.e.

(6)

An error occurs when and we define the decoding error probability, , to be equal to . A rate tuple is said to be achievable, if there exists a set of strategies such that the decoding error probabilities go to zero as for all network states consistent with the side information. Moreover, for any S-D pair , denote the maximum achievable rate with full network knowledge by . The sum-capacity , is the supremum of over all possible encoding and decoding functions with full network knowledge.

We will now define the normalized sum-rate and the normalized sum-capacity.

Definition 5 ([2]).

Normalized sum-rate of is said to be achievable, if there exists a set of strategies such that following holds. As goes to infinity, strategy yields a sequence of codes having rates at the source , , such that the error probabilities at the destinations, , go to zero, satisfying

for all the network states consistent with the side information, and for a constant that is independent of the channel gains.

Definition 6 ([2]).

Normalized sum-capacity , is defined as the supremum of all achievable normalized sum-rates . Note that .

Iii Motivating Examples

Before diving into the main results in Section IV, we will use a sequence of examples to arrive at the main ingredients of the proposed coded layer scheduling. The key point of the discussion is to understand the mechanisms that allow outperforming interference avoidance with only -local view.

As defined earlier, -local view means that each S-D pair has enough information to perform optimally when other pairs do not interfere. However, beyond 1-local view, each node only knows the connectivity in the network (structure of interference). So, at first glance it seems that the optimal strategy is to avoid interference between the S-D pairs and at each time, schedule as many non-interfering pairs as possible. Through an example, we investigate the performance of the above strategy which maximizes spatial reuse while avoiding interference at each node.

Fig. 3: (a) A network where interference avoidance between S-D pairs is optimal, and (b) its route-adjacency graph.

Consider the network depicted in Figure 3(a) with -local view. From the route-adjacency graph of this network depicted in Figure 3(b), we can see that S-D pairs and are non-interfering. We implement an achievability strategy described as follows. We split the communication block into two time-slots of equal length and represent each time-slot with a color, namely black and white. S-D pairs and communicate over time-slot black, whereas, S-D pair communicate over time-slot white. With this coloring, we have effectively seperated induced subgraphs of interfering pairs, see Figure 4. Now, since each pair can communicate interference-free over half of the communication block length, it can achieve half of its capacity with full network knowledge. Hence, we achieve a normalized sum-rate of .

Fig. 4: (a) S-D pairs and can simultaneously communicate interference-free over their induced subgraphs in the first time-slot (black time-slot), and (b) S-D pair can communicate interference-free over its induced subgraph in the second time-slot (white time-slot).

This scheduling strategy can be viewed as a specific coloring of nodes in the route-expanded graph (defined in Section II). Consider the route-expanded graph of this example, as shown in Figure 5. The aforementioned scheduling strategy can be viewed as a coloring of nodes in the route-extended graph, such that (1) all nodes of the same shape, i.e. pair ID, receive the same color, and (2) any two nodes with different shapes, i.e. different pair IDs, that are connected to each other should have different colors. Note that since the nodes inside the same super-node are connected to the same nodes in and they have different shapes, they will be assigned different colors. Figure 5 illustrates such coloring of nodes in this example by using only two colors, B and W. In other words, we have assigned different colors to the induced subgraphs of interfering S-D pairs. Therefore, each S-D pair gets a chance to communicate over its induced subgraph interference-free during the time-slot associated with its color.

Fig. 5: Route-expanded graph, , of the example in Figure 3(a).

As we will see in the next lemma, which is proved in Appendix A, is also an upper bound on the normalized sum-capacity of this network. Hence, scheduling non-interfering pairs performs optimally in this example. More generally, the following upper bound on exists for a general class of multi-layer networks.

Lemma 1.

In a -user multi-layer network (linear deterministic or Gaussian) with -local view, if there exists a path from to , for some , then the normalized sum-capacity is upper-bounded by .

While the aforementioned interference avoidance strategy performed optimally in the network depicted in Figure 3(a), we now illustrate an example where it is not optimal. A key observation is that the scheduling described above, ignores the available knowledge of interference structure in each layer of the network and it only schedules pairs that are non-interfering over all layers. To see how this knowledge of interference structure can be exploited, consider the network depicted in Figure 6(a) with -local view. Applying the previous scheduling to this network, we achieve a normalized sum-rate of . However, we show that it is possible to go beyond and achieve a normalized sum-rate of for Figure 6 example.

Fig. 6: (a) A network where end-to-end interference avoidance is not optimal, and (b) its route-adjacency graph.

We implement an achievability strategy described as follows. Similar to the previous example, we split the communication block into two time-slots of equal length and represent each time-slot with a color, namely black and white. Unlike the previous case where we assigned a color to each S-D pair, in this example, the color assignment is carried on in each layer seperately. We let sources and to communicate in the first layer over time-slot black and source over time-slot white. However, in the second layer, relays communicate to destinations and in time-slot black and to destination in time-slot white, see Figures 7(a) and 7(b). With this strategy each S-D pair can communicate over its induced subgraph interference-free during half of the communication block length, see Figures 7(c), 7(d) and 7(e). Hence, we achieve a normalized sum-rate of . By Lemma 1, we know that is also an upper-bound on the normalized sum-rate of this network, hence, we have achieved it normalized sum-capacity.

Fig. 7: (a) Sources and can simultaneously communicate interference-free over time-slot black, and relays can communicate with destinations and interference-free over time-slot black, (b) Source can communicate interference-free over time-slot white, and destination can receive its signal interference-free over the same time-slot, and (b), (c), and (d) the interfernce-free induced subgraphs of S-D pair , , and respectively.

This strategy can be viewed as a modification of our previous coloring of the nodes in the route-expanded graph as follows. Nodes with the same shape can be assigned different colors at different layers, however, still any two nodes with different shapes that are connected to each other should have different colors. In other words, we assign colors such that the induced subgraphs of different S-D pairs have different colors in each layer only if they are interfering at that layer. Figure 8 illustrates such coloring of nodes in this example by using only two colors, B and W. Since the induced subgraphs of interfering pairs have different colors in each layer, each S-D pair has a chance to communicate over its induced subgraph interference-free during half of the communication block.

Fig. 8: (a) Route-expanded graph, , of the network depicted in Figure 6(a).

The scheduling developed for the network depicted in Figure 6, illustrates a major deficiency of the scheduling developed for the network depicted in Figure 3, which is the restriction of applying the same scheduling to all nodes on a route between and . By exploiting the available information of interference structure and scheduling nodes in different layers separately, we outperformed the first scheduling. In Section IV, we will formally define this new scheme and refer to it as Maximal Independent Layer (MIL) scheduling.

So far, our proposed transmission strategies are based on interference avoidance either in an end-to-end manner or in a per-layer manner (MIL scheduling). But, can we go beyond interference avoidance with such limited knowledge at the nodes? To answer this question, first consider the single-layer network depicted in Figure 9(a). Since the conflict graph of this network is fully connected, using MIL scheduling we can only achieve . However, we now show that it is possible to achieve by employing a coding strategy that only requires -local view.

Fig. 9: (a) A network in which coding is required to achieve normalized sum-capacity, and (b) the induced subgraphs.

Consider the linear deterministic model. By using repetition coding at the sources (as in [2]), we show that it is possible to achieve . Consider the induced subgraphs of all three S-D pairs, as shown in Figures 9(b). We show that any transmission strategy over these three induced subgraphs can be implemented in the original network by using only two time-slots, such that all nodes receive the same signal as if they were in the induced subgraphs. This would immediately imply that a normalized sum-rate of is achievable.111Since any transmission strategy for the diamond networks can be implemented in the original network by using only two time-slots, we can implement the strategies that achieve the capacity for any S-D pair with full network knowledge, i.e. , over two time-slots. Hence, we can achieve . On the other hand, we have . As a result, we can achieve a set of rates such that , and by the definition of normalized sum-rate, we achieve .

To achieve , we split the communication block into two time-slots of equal length and represent each time-slot with a color, namely black and white. Sources and transmit the same codewords as if they are in the induced subgraphs over time-slot black. Destination will receive the same signal as if it is only in the induced subgraph without any interference and destination receives interference from source . Over time-slot white, source transmits the same codewords as if they are in the induced subgraphs, and source repeats its transmitted signal from time-slot black. Destination will receive its signal interference-free. Now, if destination adds its received signals over two time-slots, it recovers its intended signal interference-free, see Figure 10. In other words, we have used interference cancellation at destination . Therefore, all S-D pairs can effectively communicate interference-free over two time-slots.

Fig. 10: Achievability strategy for the network depicted in Figure 9.

Again, we can view this strategy as a modification of the previous colorings of the nodes in the route-expanded graph as follows. Each shape, i.e. pair ID, can be assigned a subset of colors such that any two nodes with different shapes that are connected, have either different colors, or if they share a color, one them has a different color in its subset. Figure 11 illustrates such coloring of nodes in this example by using only two colors, and . The subset assigned to source represents repetition coding, i.e. the transmitted signal in time-slot white is the same as the one transmitted in time-slot black. Since the interference can be cancelled out as described before, each S-D pair has a chance to communicate over its induced subgraph interference-free during half of the communication block.

Fig. 11: Route-expanded graph of the network depicted in Figure 9 and a coloring that yields .

This example illustrated that with only 1-local view it is still possible to take advantage of (repetition) coding at the sources and go beyond interference avoidance. This raises a natural question: can we also exploit network coding at the relays with only -local view? If so, what is a systematic procedure for doing that?

To shed light on the aforementioned questions, consider a multi-layer network as depicted in Figure 12(a). Assume linear deterministic model for the channels. It is straightforward to see that by using interference avoidance, we can at-most achieve normalized sum-rate of . We now show that by using repetition coding at the sources and linear coding at the relays, it is possible to achieve . Consider the induced subgraphs of all three S-D pairs, as shown in Figures 12(b), 12(c), and 12(d). We now show that any transmission strategy over these three induced subgraphs can be implemented in the original network by using only two time-slots, such that all nodes receive the same signal as if they were in the diamond network. Therefore, a normalized sum-rate of is achievable.

Fig. 12: (a) a two-layer network in which we need to incorporate network coding to achieve the normalized sum-capacity, (b), (c) and (d) the induced subgraphs of S-D pairs 1,2,and 3 respectively.

Consider any strategy for S-D pairs , , and as illustrated in Figures 12(b), 12(c), and 12(d). In the first layer, we implement the achievability strategy of Figure 10 and we have illustrated it in Figure 13. As it can be seen in this figure, at the end of the second time-slot, each relay has access to the same received signal as if it was in the diamond networks of Figures 12(b), 12(c), and 12(d).

Fig. 13: Achievability strategy for the first layer of the network in Figure 12(a).

In the second layer, during time-slot black, relays and transmit and respectively, whereas, relay transmits , see Figure 14. Destination receives the same signal as in Figure 12(b). During time-slot white, relays and transmit and respectively, whereas, relay transmits . Destination receives the same signal as as in Figure 12(c). If destination adds its received signals over the two time-slots, it recovers the same signal as in Figure 12(d). Therefore, each destination receives the same signal as if it was only in its corresponding diamond network, over two time-slots. Hence, the normalized sum-rate of is achievable. By Lemma 1, we know that is also an upper-bound on the normalized sum-rate of this network, hence, we have achieved it normalized sum-capacity.

Fig. 14: Achievability strategy for the second layer of the network in Figure 12(a).
Fig. 15: The route-expanded graph for the the two-layer folded-chain network.

This strategy can be viewed as a new coloring of the nodes in the route-expanded graph as follows. Each shape, i.e. pair ID, can be assigned two subsets of colors. Figure 15 illustrates such coloring of nodes in this example by using only two colors, and . The subset assigned to source represents repetition coding as before. The second subset of colors can be interpreted as the time instants from which we can add (or subtract for the Gaussian model) the codewords to perform network coding. To clarify, consider the first super-relay in Figure 15, node circle communicates the codeword of S-D pair over time-slot . Over time-slot , node square within this super-relay adds the codeword of S-D pair transmitted by the other node in the same super-node over time-slot to the codeword of S-D pair it has to send. Similar interpretation can be used for the other tuple of colors in this route-expanded graph. Since the interference can be cancelled out as described before, each S-D pair has a chance to communicate over its induced subgraph interference-free during half of the communication block.

In the following section, we incorporate all the ideas developed for the examples in this section to define a transmission strategy, i.e. coded layer scheduling, which outperforms interference avoidance techniques with -local view. We also characterize its performance and later in Section V, we evaluate its performance for some network topologies.

Iv Coded Layer Scheduling

Via the examples presented in Section III, we saw that multiple ideas can be incorporated to enhance the achievablility scheme in multi-layer networks with -local view: (1) per layer interference avoidance, (2) repetition coding to allow overhearing of the interference, and (3) network coding to allow interference neutralization. In this section, we define a general transmission strategy, named coded layer scheduling to incorporate all the aforementioned ideas. This scheduling can be represented by a specific coloring of nodes in the route-expanded graph (defined in Section II). We refer to this coloring as the Coded Layer coloring and it is defined as follows.

Consider a multi-layer wireless network , and its corresponding route-expanded graph . A Coded Layer coloring of with distinct colors assigns to any node ,

  1. a transmit color set, denoted by , which represents the time instants in which will be transmitting for S-D pair using repetition coding,

  2. a coding color set, denoted by , which represents the time instants from which node will use the transmit signal to perform network coding for S-D pair ,

  3. a receive color set, denoted by , which represents the time instants in which it is listening.222We refer to the transmit color set, the coding color set, and the receive color set of source , , by , , and respectively, similar notations hold for destinations, i.e. , , and for destination , .

To describe the conditions that these color assignments should satisfy, we need a few definitions.

Definition 7.

At any node , a node is called an interferer if and

  1. , i.e. an interferer should have a S-D pair ID different from ,

  2. , i.e. the colors used by an interferer are not used by any node in the same super-node that has S-D pair ID and performs network coding, otherwise its transmit signal will be neutralized.

  3. , i.e. an interferer transmits during a time instant that some node with S-D pair ID is transmitting to and is listening, the set of all such time instants is denoted by .

Definition 8.

We define as the set of all nodes in that have pair ID and are connected to , i.e.

(7)

The conditions on the assignment of , and of a coded layer coloring are as follows.

C.1: The transmit color sets assigned to the nodes that belong to the same super-node are disjoint, i.e.

(8)

C.2: If a node is performing network coding, it only transmits once, i.e. if , then .

C.3: The coding color set includes at most one color from each transmit color set of a node within the same super-node who is not performing network coding, i.e.

(9)

C.4: The receive color set includes at least one color from each such that , i.e.

(10)

C.5: The receive color set includes each such that , i.e.

(11)

C.6: If , then .

C.7: At each node either there are no interferers, or all interferers share a common color in their transmit color sets, which is in , i.e.

(12)

moreover, the color that the interferers share should be exclusive to them, i.e., for ,

(13)

Based on the coded layer coloring of nodes in , we now define the coded layer scheduling of nodes in as follows:

The transmission is broken into blocks of size time instants. At the beginning of the block, ,

  • Source , , creates a signal , which is a function of its message ( for the linear deterministic model and for the Gaussian model such that it satisfies the average power constraint at transmit nodes). The choice of this function depends on the specific strategy that each source picks,

  • Each relay node creates a signal for each S-D pair , which is a function of its received signals and the global side information. The choice of this function depends on the specific strategy that each relay picks.

During the block,

  • Source , , will transmit

    (14)

    where .

  • Each relay node will transmit

    (15)

    where . Note that subtraction in is the same as XOR operation.

Finally, each destination , , will decode based on its received signals and the global side information.

We next state our main result for the coded layer scheduling.

Theorem 1.

For a multi-layer network (linear deterministic or Gaussian) with -local view, if there exists a coded layer coloring of with colors as defined above, then a normalized sum-rate of is achievable by coded layer scheduling.

Proof.

We first prove the theorem for the linear deterministic model. Assume that there exists a coded layer coloring of nodes with colors , denoted by , and . Suppose has S-D pairs and consider the induced subgraphs of all S-D pairs, i.e. , . We will show that by using the coded layer scheduling, any transmission snapshot over these induced subgraphs can be implemented in the original network over time instants, such that all nodes receive the same signal as if they were in the induced subgraphs.

Consider a transmission snapshot in the induced subgraphs where

  • Node in the induced subgraph transmits ,

  • Node in the induced subgraph receives

    (16)

Transmission strategy: At any time instant , node will choose and will transmit

(17)

and it will receive

(18)

where summation is carried on in .

Constructing the received signals: Based on the transmission strategy described above, we need to show that at any node , the received signal can be obtained. At any node , , we create as follows,

(19)

where is given by (18). We will show that . We have

(20)

Compairing (16) and (IV), we conclude that in order to show , it is sufficient to prove

(21)

Consider a node such that ; we face 2 cases:

Case 1: , then from condition C.1 and (17), we get

(22)

Then based on condition C.7, we have 2 sub-cases:

Case 1-a: There are no interferers. In this case since no interfering signal is received during time instants associated with colors in , hence,

(23)

Then, we have

(24)

Case 1-b: All interferers share a common color that is in but not in . In this case, the transmit signal of any interferer appears exactly twice in time instants associated with colors in , once during the time instant associated with the color that all interferers share, see (IV), and once during a different time instant, see Definition 7. Hence, when adding the received signals over all time instants associated with colors in , the transmit signal of any interferer gets canceled. Moreover, condition C.6 guarantees that the desired signal, i.e. , is transmitted exactly once during time instants associated with colors in . Hence, from (22) and the argument presented above, we have

(25)

Case 2: . Note that according to condition C.3 the colors in can only appear in where . As a result, each one of those codewords added to as in (17), are also transmitted during time instants corresponding to colors in . Hence, if ,

(26)

Again based on condition C.7, we have 2 sub-cases:

Case 2-a: There are no interferers. In this case since no interfering signal is received during time instants associated with colors in , hence,

(27)

Then, we have

(28)
(29)

Case 2-b: All interferers share a common color that is in but not in . The argument is similar to that of case 1-b, i.e. the transmit signal of any interferer appears exactly twice in time instants associated with colors in , once during the time instant associated with the color that all interferers share, see (IV), and once during a different time slot, see Definition 7. Hence, when adding the received signals over all time instants associated with colors in , the transmit signal of any interferer gets canceled. Moreover, condition C.6 guarantees that the desired signal, i.e. , is transmitted exactly once during time instants associated with colors in . Hence, from (26) and the argument presented above, we have

(30)

Therefore, we have shown that in all cases we have , which as described before proves that . As a result, at any node , the received signal can be obtained interference-free over time instants. This implies that the transmit signal , which is only a function of and the global side information, can be created over time instants as well. Hence, any transmission snapshot over the induced subgraphs can be implemented in the original network over time instants, such that all nodes receive the same signal as if they were in the induced subgraphs.

Moreover, any transmission strategy with block length for the induced subgraphs can be implemented in the original network over time instants in a similar manner. Hence, we can implement the strategies that achieve the capacity for any S-D pair with full network knowledge, i.e. , as over time instants. Therefore, by choosing ’s according to the optimal transmission strategies and creating as in (17), we can achieve . On the other hand, we have . As a result, we can achieve a set of rates such that , and by the definition of the normalized sum-rate, we achieve .

We will next prove the theorem for the Gaussian model. We will show that by using the coded layer scheduling, any transmission snapshot over the induced subgraphs , , can be implemented in the original network over time instants, such that all nodes receive the same signal as if they were in the induced subgraphs.

Consider a transmission snapshot in the induced subgraphs where

  • Node in the induced subgraph transmits ,

  • Node in the induced subgraph receives

    (31)

where is the additive white complex Gaussian noise with variance . We also assume a power constraint of at the transmit nodes in the induced subgraphs.

Transmission strategy: At any time instant , node will choose and will transmit

(32)

note that number of transmit signals at each time instant is less than and due to the power constraint of in the induced subgraphs, the power constraint in the original network is satisfied. At any time instant , node