On the Minimum Number of Transmissions in SingleHop Wireless Coding Networks
Abstract
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 NPcomplete over 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.
I Introduction
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 to four clients ; packet needs to be received by client . Each client 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 all clients can be satisfied by broadcasting two packets and (all additions are over . Since without network coding all packets are needed to be transmitted, network coding allows to reduce the number of transmissions by .
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 is NPcomplete. 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 NPhard 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 nonmulticast 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 NPhardness) 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.
Ii Model
We consider a onehop wireless channel with a single server and a set of clients . The server needs to transmit a set of packets to the clients. Each client requires a certain subset of packets in , while some packets in are already available to it. Specifically, each client is associated with two sets:

 the set of packets required by .

 the set of packets available at ;
We refer to and as the “wants” and “has” sets of , respectively. The server can transmit any packet from as well as linear combinations (over ) of packets in . Each transmission is specified by an encoding vector such that the packet transmitted in communication round is equal to . The practical issues related to this model are discussed in [2].
Our goal is to find the set of encoding vectors of minimum cardinality that allow each client to decode the packets it requested. We refer to this problem as Problem MINTq.
Problem MINTq
Find the minimum number of transmissions and the corresponding set of encoding vectors , , that allow each client to decode all the packets in its “wants” set .
We assume, without loss of generality, that for each packet , there exists at least one client such that belongs to the “wants” set of . We also assume that for each client it holds that .
Observation 2
Without loss of generality, we can assume that the “wants” set of each client contains exactly one packet. Indeed, we can substitute each client whose “wants” set includes more than one packet by multiple clients such that the “has” sets of all clients in are equivalent to that of and each client in requests one of the packets in . It is easy to verify that the resulting instance of Problem MINTq is equivalent to the original one.
Iii Hardness results
In this section we focus on the case in which the encoding is performed over and prove that Problem MINT2, which is a special case of Problem MINTq for , is NPcomplete.
Theorem 3
Problem MINT2 is NPcomplete.
It is easy to verify that the problem belongs to . To prove that the problem is NPcomplete we show a reduction from the minimum vertex cover problem. In this problem we are given a graph and need to find a subset of , of minimum cardinality, such that each edge is incident to at least one of the nodes in . We denote by the size of the optimal solution for the vertex cover problem.
Given an instance to the vertex cover problem we build the following instance for Problem MINT2. The packet set includes a packet for any node in and a packet for any edge in . We denote by the subset of packets in that correspond to nodes in and by the subset of packets in that correspond to edges in .
For each edge we define two clients and such that:

and ;

and .
We denote by the size of the optimal solution for this instance of Problem MINT2, i.e., the minimum number of transmissions necessary to satisfy all clients. In the following two lemmas we prove that .
Lemma 4
.
Let be the optimal solution to the vertex cover problem. Then, all clients can be satisfied by transmitting the following set of packets of size :

For each node we transmit the corresponding packet ;

For each edge we transmit the packet , where , , and are packets that correspond to nodes , , and edge , respectively.
It is easy to verify that the set of corresponding encoding vectors is a feasible solution to Problem MINT2. Since the total number of transmitted packets is it follows that .
Lemma 5
.
Consider an optimal solution to Problem MINT2, where
With this solution, the packet transmitted at round is equal to
We denote by the linear subspace of dimension of generated by the vectors in .
We show that there exist two sets and of vectors in and a vertex cover such that the following three conditions holds:

For any edge , there exists an encoding vector such that , and for any edge ;

For each it holds that for any edge ;

For each there exists an encoding vector such that and for any node .
Note that all encoding vectors in are linearly independent, , and .
First, we show how to construct the set . Let be an edge in and let and be the two clients that correspond to . We note that in order to satisfy , must contain at least one vector for which it holds and for any edge . Thus, we can form by including, for each , the vector that corresponds to .
Second, we show how to construct set and the vertex cover . Again, let be an edge in and let and be the two clients that correspond to . Note that, in order to satisfy the client , the set must contain a vector for which it holds that for all , for all , and either or (or both) are nonzero. Let be a set that contains such vectors for all . Let . It follows from linear algebra that there exists an matrix over that satisfies the following conditions:

The rows of span ;

There are linearly independent columns in such that each column contains exactly one nonzero element.
Indeed, we can first construct an matrix whose rows span . Such matrix is of rank , hence it contains at least nonzero columns which are linearly independent. The matrix can be constructed form by performing Gaussian elimination. We denote by the subset of that corresponds to linearly independent columns of , each column contains exactly one nonzero element. Then, we set to be the set of row vectors of . Note that has elements.
We proceed to show that is a vertex cover in . We note that the structure of implies that for any nonzero vector in the row span of , and, in turn, in it must hold that for some . For each edge let be the vector that correspond to in . Recall has one or two nonzero components, which are either or , or both. This implies that either or , or both belong to .
We proved that there exist two sets and of independent vectors in such that , and . We conclude that
Iv Dependence on the field size
In this section we consider a variant of Problem MINTq which allows flexibility in choosing the underlying finite filed. . Specifically, for each instance of the problem, we can choose the finite field that minimizes the required number of transmissions.
We denote by the minimum required number of transmissions over . We also denote by the minimum number of transmissions that can be achieved over any finite field.
We begin by observing that the minimum number of transmissions may depend on the size of the finite field . For example, consider the problem described in Table I, where , and for every client , . We prove that in this problem .
First, we show that . Suppose, by way of contradiction, that there is a solution to this problem with two transmissions:
To satisfy all clients, the vectors should be all distinct and different from , which is not possible over . Since the set of transmissions satisfies all clients, it follows that . We note that is at least two, since . We also observe that over only two transmissions are sufficient, hence .
The following lemma shows that is not necessarily a monotonic function of .
Lemma 6
There exists an instance of Problem MINTq for which it holds that for fields with odd characteristic, such as and for fields with even characteristic.
Proof (sketch): Consider the problem instance described in Table II, where , and .
For fields with odd characteristic, the transmission sequence
satisfies all clients, hence .
We observe that for fields with even characteristic () it holds that . Indeed, for any solution with three transmissions, consider the matrix whose row vectors are , , and . To satisfy the demands of all the clients the vector matroid of should be isomorphic to the Fano matroid [6]. But, the Fano matroid is only representable over fields with odd characteristics. Therefore, there are no solutions to above problem with three transmissions over . \QED
The next lemma shows that deciding whether the optimal number of transmissions can be achieved for a given field is an NPhard problem.
Lemma 7
For given a prime power , it is an NPhard problem to decide whether .
Proof (sketch): Similar to [5], we use a reduction from the problem of graph coloring. Given an undirected graph , we construct the following instance to the broadcast problem. For each node , the set includes a packet . For each edge , the set includes a client such that and . It is easy to verify that for this problem it holds that .
We show the problem can be solved with two transmissions over if and only if is colorable. First, suppose that can be colored with colors. Let be the color of vertex . As shown in [5], there exists pairwise independent vectors over . For each node we set . It is easy to verify that the two encoding vectors and constitute a feasible solution for the broadcast problem.
Second, suppose that there exists a solution for the broadcast problem with two transmissions. We show that this implies that there exists a coloring of graph . For each vertex , the vector determines the coefficients for packet for the first and the second transmissions in . The set of such vectors can be partitioned into equivalence classes, such that any two linearly dependent vectors are placed into the same equivalence class. Next, for each equivalence class we assign one of the colors. Next, for each vertex we assign the color that corresponds to the equivalence class of . It is easy to verify that this will result in a valid coloring of that requires at most colors. \QED
V Bounds on coding advantage
Given a onehop transmission problem with packets and clients , we define the coding gain as the ratio between the minimum number of transmissions without coding and the minimum number of transmissions with coding, i.e.,
where is the minimum number of transmissions achievable over any finite field .
Let and . The following theorem establishes lower and upper bounds on .
Theorem 8
The coding gain is bounded by
(1) 
We assume, without loss of generality, that for each client . Let be the field that requires transmissions. Consider an optimum solution that includes encoding vectors .
Let , , be the unit vector whose components are all zeros except for the th one which is 1. For , we define if client wants packet . Also, we define
To guaranty that each client is able to decode the packet in its “wants” list, there must be a vector such that , , where and are the linear subspaces generated by the vectors in and , respectively. We note that the Hamming weight of is upper bounded by .
Let . By the optimality of the solution the dimension of the linear subspace generated by is equal to that of . Let be the matrix whose row vectors belong to and form the basis of . We note that must satisfy the following two conditions:

Each row of contains at most nonzero elements.

does not contain the allzero column vector.
The first condition follows from the upper bound on the Hamming weights of the vectors in . The second condition follows from the observation that for every packet at the source, there is at least one client that wants it. These two conditions imply that .
We proceed with the lower bound. Given an instance of Problem MINTq, we form another instance with where all the “has” sets have order , and where . Instance is formed by deleting arbitrary elements from the “has” sets of and expanding the “wants” sets of its elements. Note that any valid solution for instance is also a valid solution for instance .
For a field of large enough size (larger than the number of clients), we can always find a subspace of dimension in that is simultaneously orthogonal to all the subspaces corresponding to (Theorem 1 in [7]). Any basis of will constitute a solution for , and, in turn, for , which requires transmissions. Thus, the lower bound follows.
Vi Heuristic Approach and Numerical Results
Via Heuristic Approach
In Section III, we proved that Problem MINT2 is NPcomplete, hence finding an optimal solution for large instances of the problem can be impractical. In this section, we present a heuristic approach to solve this problem. Our heuristic solution employs memoryless decoding, i.e., each client uses exactly one of the transmitted packets to decode one of the packets in its “wants” list and never uses a linear combination of the transmitted packets. While memoryless decoding, in general, results in a suboptimal solution, our numerical results, presented below, show that in many cases the number of required packets is close to the optimum. We observe that the problem of finding the minimum number of transmissions with memoryless decoding is equivalent to the problem of finding the minimum chromatic number of an undirected graph.
Specifically, consider an instance problem of Problem MINTq, in which the “wants” set of each client is of cardinality one. Then, we construct an instance to graph coloring problem through the following procedure:

For each client there is a corresponding vertex in

Each two vertices and are connected by an edge if one of the following holds:

Clients and have identical “wants” sets;

and .

Let be a clique in , i.e., each two vertices of are connected by an edge in . Note that all clients that correspond to nodes in can be satisfied by one transmission, which includes a linear combination of all packets in their “wants” sets. Thus, the minimum number of transmissions with memoryless decoding can be found by solving a clique partition problem [8], i.e., partition of into disjoint subsets , such that for , the subgraph of induced by is a complete graph. This problem, in turn, corresponds to the minimum graph coloring problem of the complimentary graph. The latter problem is a wellstudied problem with a wealth of heuristic solutions developed in the recent years.
ViB Numerical results
We performed several numerical experiments in order to evaluate the coding gain as well as the performance of the heuristic solution in random settings. In all of our experiments described below, the “wants” set of each client is of cardinality one, and the number of clients is equal to the number of packets.
In the first experiment, we evaluated the coding gain of a singlehop wireless system with seven clients. Specifically, we generated 50 instances of Problem MINTq, in each setting the set “has” of each client is randomly selected. The results of the experiment are shown in Figure 2. The results show that in the majority of the experiments, there is a significant coding gain (more than 1.75).
The second experiment is similar to the first one, but the clients only employ memoryless decoding. The results of the experiment are shown in Figure 3. The results show that a significant coding gain (up to 2.5) can be achieved, while in the majority of the cases, the coding gain is at least 1.7.
In the third experiment, we studied the dependence of average coding gain on the cardinality of the “has” sets. In particular, we generated a problem instance in which the cardinality of the “has” set is equal for all clients, while the content of the “has” set was randomly selected. Figure 4 shows the average coding gains of the system with seven clients using optimal decoding as a function of cardinality of the ”has” sets, while Figure 5 shows the comparison of average coding gains as a function of cardinality of the ”has” sets, for three techniques i.e., optimal decoding, memoryless decoding and heuristic approach.The results show that the average coding gain increases with the size of the “has” sets, which confirms the intuition that coding is more beneficial if the clients have more packets in their “has” sets.
Finally, we evaluated the coding gain that can be obtained through the heuristic approach presented in Section VIA. The results of this experiment are depicted in Figure 6. The results show that the proposed heuristic approach allows to obtain a significant reduction in the number of transmitted packets.
Vii Conclusion
The paper focuses on minimizing the number of transmissions necessary for satisfying all clients in singlehop wireless settings. We employ the technique of network coding which allows to take advantage of the packets that were overheard from prior transmissions.
Our paper makes the following contributions. First, we proved that the problem of finding the minimum number of transmissions is NPcomplete over the binary field. Second, we analyzed an extended version of the problem in which the encoding can be performed over a larger finite field. Furthermore, we established lower and upper bounds on the value of the coding gain. Next, we presented a heuristic solution based on graph coloring. Finally, we conducted a simulation study that evaluates the coding gains in practical settings.
The considered problem presents significant challenges and provides a fertile ground for future research. In particular, we would like to prove the NPhardness and inapproximability for finite fields of larger size as well as for nonlinear network codes.
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