The advent of network coding presents promising opportunities in many areas of communication and networking. It has been recently shown that network coding technique can significantly increase the overall throughput of wireless networks by taking advantage of their broadcast nature. In wireless networks, each transmitted packet is broadcasted within a certain area and can be overheard by the neighboring nodes. When a node needs to transmit packets, it employs the opportunistic coding approach that uses the knowledge of what the node's neighbors have heard in order to reduce the number of transmissions. With this approach, each transmitted packet is a linear combination of the original packets over a certain finite field. In this paper, we focus on the fundamental problem of finding the optimal encoding for the broadcasted packets that minimizes the overall number of transmissions. We show that this problem is NP-complete over GF(2) and establish several fundamental properties of the optimal solution. We also propose a simple heuristic solution for the problem based on graph coloring and present some empirical results for random settings.
In recent years, there has been an enormous interest in the design and deployment of wireless networks. Such networks are indispensable for providing ubiquitous network coverage and have many applications in both civil and military areas.
Recently, it was observed that the broadcast nature of wireless networks can be exploited in order to increase throughput and reduce energy consumption. In a wireless environment, each packet is broadcasted within a small neighborhood, which allows the neighboring nodes to overhear packets sent by their neighbors. When a node needs to transmit packets, it can employ the opportunistic coding [1], [2] approach that uses the knowledge of what the node’s neighbors have heard in order to reduce the number of transmissions. With this approach, each transmitted packet is a linear combination of the original packets over a certain finite field.
Example 1: Consider the network depicted in Figure 1. In this example, the central node, referred to as a server, needs to deliver four packets p 1 , . . . , p 4 to four clients c 1 , . . . , c 4 ; packet p i needs to be received by client c i . Each client c i has an access to some of the packets overheard from prior transmissions. This set is referred to as its “has” set. It is easy to verify that Since without network coding all packets p 1 , . . . , p 4 are needed to be transmitted, network coding allows to reduce the number of transmissions by 50%.
In this paper, we focus on the single hop wireless setting and consider the problem of minimizing the number of broadcast transmissions necessary to satisfy all the clients. Our contributions can be summarized as follows. First, we prove that the problem of determining the minimum number of transmissions over GF (2) is NP-complete. Next, we show that the number of transmissions may depend on the size of the finite field, and that such a dependence is not necessarily monotonic. Further, we prove that the problem of finding the size of the finite field which results in the minimum number of transmissions is an NP-hard problem. Next, we establish lower and upper bounds on the coding advantage, i.e., the ratio between the total number of packets and the minimum number of transmissions that can be achieved by using network coding. In particular, we show that the coding advantage depends on the size of the"has" sets. Next, we evaluative the value of coding advantage in random settings. Finally, we present a heuristic solution based on graph coloring and verify its performance through simulations.
The considered problem is a special case of the general network coding [3] problem for non-multicast networks. The general network coding problem has recently attracted a large body of research (see e.g., [4], [5] and references therein), however, many of the results (such as NP-hardness) cannot be immediately extended to our problem.
While we present our results in the context of wireless data transmission, the considered problem is very general and can arise in many other practical settings. For example, consider a content distribution network that needs to deliver a set of large files (such as video clips) to different clients. In this setting, if some of the files are already available for some clients, the distribution can be efficiently implemented by multicasting a (small) set of linear combinations of the original files.
We consider a one-hop wireless channel with a single server s and a set of m clients C = {c 1 , . . . , c m }. The server needs to transmit a set P = {p 1 , p 2 , . . . , p n } of packets to the clients. Each client requires a certain subset of packets in P , while some packets in P are already available to it. Specifically, each client c i ∈ C is associated with two sets:
• W (c i ) ⊆ P -the set of packets required by c i .
• H(c i ) ⊆ P -the set of packets available at c i ; We refer to W (c i ) and H(c i ) as the “wants” and “has” sets of c i , respectively. The server can transmit any packet from P as well as linear combinations (over GF (q)) of packets in P . Each transmission i is specified by an encoding vector g i = {g j i } ∈ GF (q) n such that the packet x i transmitted in communication round i is equal to x i = n j=1 g j i • p j . The practical issues related to this model are discussed in [2].
Our goal is to find the set of encoding vectors Φ = {g i } of minimum cardinality that allow each client to decode the packets it requested. We refer to this problem as Problem MIN-T-q.
Problem MIN-T-q: Find the minimum number of transmissions and the corresponding set Φ of encoding vectors {g i }, g i = {g j i } ∈ GF (q) n , that allow each client c i ∈ C to decode all the packets in its “wants” set W (c i ).
We assume, without loss of generality, that for each packet p i ∈ P , there exists at least one client c j ∈ C such that p i belongs to the “wants” set W (c j ) of c j . We also assume that for each client
Observation 2: Without loss of generality, we can assume
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