On the Energy Benefit of Network Coding for Wireless Multiple Unicast

We consider the energy savings that can be obtained by employing network coding instead of plain routing in wireless multiple unicast problems. We establish lower bounds on the benefit of network coding, defined as the maximum of the ratio of the minimum energy required by routing and network coding solutions, where the maximum is over all configurations. It is shown that if coding and routing solutions are using the same transmission range, the benefit in $d$-dimensional networks is at least $2d/\lfloor\sqrt{d}\rfloor$. Moreover, it is shown that if the transmission range can be optimized for routing and coding individually, the benefit in 2-dimensional networks is at least 3. Our results imply that codes following a \emph{decode-and-recombine} strategy are not always optimal regarding energy efficiency.


I. INTRODUCTION
It has been shown that network coding has the potential of reducing energy consumption in wireless networks [1]- [4]. In [1] it is shown that network coding can reduce energy consumption in wireless networks for many-to-many communication. In [2] and [3] multiple unicast traffic is considered. Some design principles and a lower bound on the maximum energy savings of network codes are presented in [2]. In [3] the focus is on reducing the number of transmissions in order to reduce interference and hence increase throughput. Some of the considerations of [3], however, apply to energy savings of network coding as well.
In this paper we are interested in the energy savings network coding can offer for wireless multiple unicast problems. More precisely, we want to find bounds on the maximum ratio of the energy consumption of routing to the energy consumption of network coding, where the maximum is over all possible multiple unicast configurations. We call this ratio the energy benefit of network coding. The best known lower bound on the energy benefit of network coding is 3 for two dimensional networks [4]. Our main result is a new lower of 2d/⌊ √ d⌋ for d-dimensional networks.
For 2-dimensional networks our lower bound equals 4, in 3 dimensions it equals 6. It is interesting to compare this with the upper bound of 3 presented in [5]. This upper bound is obtained under the restriction that only the type of network codes introduced in [3] are allowed. These codes follow a decode-and-recombine strategy, i.e. nodes transmit linear combinations of only those symbols they have successfully decoded themselves. Note, that in general, it is also possible to retransmit linear combinations of coded symbols without decoding the corresponding source symbols. Our lower bound shows that it can be beneficial to consider also these types of coding operations. This paper is organized as follows. In Section II the model is defined more precisely. The main results of the work are presented in Section III. The network code that achieves a high benefit is constructed in Section IV. Section V, finally, provides a discussion of the work.

II. MODEL AND NOTATION
Let V ⊂ R d be the nodes of a d-dimensional wireless network. We consider a wireless network model with broadcast, where all nodes within range r of a transmitting node can receive, and nodes outside this range cannot. The energy required to transmit one unit of information to all other nodes within range r equals cr α , where α is the path loss exponent and c some constant. We will fix the transmission range r and compare network coding and routing solutions on the resulting toplogy, i.e. a node v is broadcasting to all nodes in the set The traffic pattern that we consider is multiple unicast. All symbols are from the field F 2 , i.e. they are bits and addition corresponds to the xor operation. The source of each unicast session has a sequence of source symbols that need to be delivered to the corresponding receiver. Let M be the set unicast session. We will call C = {V, r, M } a wireless multiple unicast configuration.
We measure energy consumption by the total energy required to deliver one symbol for each unicast session. Our goal is to establish lower bounds on energy benefit = max C minimum energy consumption of any routing solution on C minimum energy consumption of any network coding solution on C , where the maximum is over all wireless multiple unicast configurations. Since r is fixed, the energy per transmission is a constant and the benefit is equivalent to the ratio of the number of transmissions required in routing and network coding solutions.

III. RESULTS
First we construct a multiple unicast configuration that will be used in the remainder of the paper. Let d ≥ 1 and K > 1. Now, take r = √ d and V = Z d K . The set of unicast sessions M is defined as follows. There are 2d(K + 1) d−1 sessions in total. We have sessions x(i, v) and We consider configurations C(d, K) = {V, r, M }. Note, that in general, we will omit dependence on d and K from the notation. Figures 1(a)-1(c) give an example of such a configuration for d = 2 and K = 3.

Lemma 1. The optimal routing solution on
Proof: The optimal routing solution on C(d, K) takes the shortest paths for all sessions. For each session, the shortest path takes ⌈K/⌊ √ d⌋⌉ hops, hence ⌈K/⌊ √ d⌋⌉2d(K + 1) d−1 transmissions are required in total.
In Section IV we will prove the following result.
Our main result is the following.
Theorem 1. The energy benefit of network coding in d dimensional wireless networks is at least 2d/⌊ √ d⌋.
Proof: From Lemmas 1 and 2 it follows that In two dimensions this gives a new lower bound of 4. For three dimensions it is 6.
Note that we have defined the energy benefit of network coding by fixing both the node positions and the transmission range. Alternatively, we could have optimized the transmission range independently for routing and network coding solutions. In this case, one can observe that an optimal routing solution uses transmission range 1. This increases the number of hops per session to K, but the cost per transmission reduces from cd α/2 to c. The energy benefit of the proposed network coding solution (still with r = √ d) would hence be which equals 2d 1−α/2 . Therefore, under this model, since α ≥ 2, the benefit of our coding solution reduces to at most 2.
The benefit of network coding on the configuration constructed in [4] is 3 under both models, since the transmission range that is used for the network coding solution is the minimum required for connectivity. Also, the lower bound of 2.4 obtained in [2] holds under both models. The codes that are constructed in [2] follow the decode-and-recombine strategy.

IV. NETWORK CODE CONSTRUCTION
In this section we prove Lemma 2 by constructing a network code using the indicated number of transmissions. In Section IV-A we specify the coding operations performed by nodes at the border of the network. In Section IV-B we specify the coding operation of internal nodes and show that the linear combinations transmitted are all of a particular form. In Section IV-C we specify how receivers can decode the required source symbols. Finally, in Section IV-D we connect the parts and prove Lemma 2.
We assume that for t ≤ 0, for all i = 1, . . . , d and v ∈ Z d−1 K , source symbols x t (i, v),x t (i, v) and all transmitted data symbols are zero. The code that we construct is such that at the end of time slot t − 1, receivers are able to decode the source symbols that have been generated by the sources at time t − K.

A. Operation at the Border
Nodes that are at the border of the network transmit 2d symbols each time slot. where For notational convenience,

Lemma 3. Assume that for all
then for all i = 1, . . . , d and any For the other cases the result follows directly from (1) and (2).

B. Operation of Internal Nodes
Internal nodes in the network transmit only once in each time slot. In order to describe the coding operation performed by internal nodes we introduce some notation. Symbols from only a subset of the neighbors are used, i.e. we consider Finally, we introduce sets Θ δ ⊂ {1, . . . , 2d}, 0 ≤ δ ≤ d. Let I δ ∈ F 2d 2 be the indicator vector corresponding to Θ δ . Let where in the shift operations, symbols that are shifted out are discarded and zeros are shifted in. In the remainder of the paper we will repeatedly make use of the fact that τ ∈Θ δ+1 for 0 < δ < d and We show that all symbols transmitted by v are linear combinations of exactly one source symbol from each of the connections for which v is on its shortest path.

Lemma 4.
Assume that for all t ′ < t and u ∈ V , u t ′ satisfies then, for any v ∈ Proof: By the assumption in the lemma and (6) we have

We rewrite this as
where and . This shows that v t satisfies (8).

C. Decoding
In this section we present the decoding operations that need to be performed at the receivers. First we consider decoding of the x t−K (i, v \i ), v such that v i = K, at the end of time slot t−1. We will see that if v is located at a rib of the convex hull of V , i.e. if ∃j = i such that v j ∈ {0, K}, the required symbol is simply transmitted by one of the neighbors. Otherwise, a more complicated decoding procedure is required. This procedure is based on the assumption that symbols transmitted by neighbors satisfy the relations given in Lemmas 3 and 4. In the Section IV-D we will finalize the proof of Lemma 2 by showing that the conditions for Lemmas 3-6 are satisfied for all time slots.
Assume that for all t ′ < t and u ∈ • V , u t ′ satisfies (7), and, that for all t ′ < t, Otherwise take (11) For the other case, we first observe that in (11) all terms correspond to symbols that have been received by v in time slots before t. Now denote the rhs of (11) asx t−K (i, v \i ). By the assumption in the lemma this can be rewritten aŝ We will show that in (12), v B = x t−K (i, v \i ) and that v A (j) = vĀ(j) = vB = 0 for all j.
For v A (j), j = i, following the proof of Lemma 4, we have Therefore, v A (j) = 0. Similarly one can show that vĀ(j) = 0 For v B it follows from (10) that v B = x t−K (i, u \i ). The last equality in (10) follows from (4) and the fact that u i = K. Similarly, we have vB = 0. Therefore, The decoding procedure for thex t−K (i, v \i ), v such that v i = 0, can be obtained by considering the symmetry of the network topology and the coding operations. The procedure is mostly mechanical, details are omitted due to space constraints.
Assume that for all t ′ < t and u ∈ • V , u t satisfies (7), and, (3). At the end of time slot t − 1, node v can decodē x t−K (i, v \i ).
Proof: Follows from Lemma 5 by considering the symmetry of the configuration and the coding operations (1), (2) and (6).

D. Proof of Lemma 2
For t ≤ 0 all symbols are assumed zero and therefore satisfy (3) and (7). Also, at t = 1, no non-zero decoded symbols are required in (1) and (2). The conditions to Lemmas 3-6 for t = 1 are, therefore, satisfied. By using induction over time, it follows that in all time slots, the source symbols required in (1) and (2) have been successfully decoded and that all transmitted symbols satisfy (3) and (7).
There are (K + 1) d nodes in total, of which the (K − 1) d internal ones transmit once. The remaining nodes transmit 2d times in each time slot. Moreover, one source symbol for each unicast session is decoded in each time slot.

V. DISCUSSION
We have obtained a lower bound on the energy benefit of network coding for multiple unicast in d-dimensional wireless networks. For 2 and 3 dimensional networks these results improve existing work. For higher dimensions our results might lead to a better insight in the energy benefit of network coding for wireless networks. These insights could in turn lead to new results for lower dimensions.
In the network code that has been constructed, nodes retransmit linear combination of symbols they have not been able to decode themselves. The code, therefore, does not qualify as decode-and-recombine [3], [5]. The point to note is that the energy benefit obtained is larger than is possible with decodeand-recombine codes [5].