- Research Article
- Open Access
Lower Bounds on the Maximum Energy Benefit of Network Coding for Wireless Multiple Unicast
© Jasper Goseling et al. 2010
- Received: 28 September 2009
- Accepted: 21 April 2010
- Published: 26 May 2010
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 . 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 decode-and-recombine strategy are not always optimal regarding energy efficiency.
- Wireless Network
- Time Slot
- Transmission Range
- Network Code
- Reduce Energy Consumption
Emerging applications in wireless networks, like environment monitoring in rural areas by ad hoc networks, require more and more resources. One of the most important limitations is formed by battery life. Since battery technology is not keeping up with the increasing demand from resource-consuming applications, it is imperative that more efficient use is made of the available energy. There has been significant recent attention to the problem of minimizing energy consumption in networks. Some of the topics considered are minimum cost routing [1–3], power control algorithms [4–6], and cross-layer protocol design for energy minimization . In this work, we are interested in the use of network coding [8–14] for reducing the energy consumption in wireless networks. We compare the reduction with traditional routing solutions. The contributions of this work are lower bounds on the energy reduction that can be achieved by using network coding for multiple unicast problems in wireless networks.
In recent years, there has been significant interest in network coding with the aim of reducing energy consumption in networks. More generally, network coding with a cost criterion has been considered. Much progress has been made in understanding the case of multicast traffic. In fact, it has been shown by Lun et al. that a minimum-cost network coding solution can be found in a distributed fashion in polynomial time . The fact that the complexity of finding this solution is polynomial in time is surprising, since the corresponding routing problem is a Steiner tree problem that is known to be NP-complete .
Besides constructing minimum-cost coding solutions, it is also of interest to know what the benefits of network coding are compared to routing. In this work we, are interested in the energy benefit of network coding, which is the ratio of the minimum energy solution in a routing solution compared to the minimum energy network coding solution, maximized over all configurations. It has been shown by Goel and Khanna  that the energy benefit of network coding for multicast problems in wireless networks is upper bounded by a constant. The problem of reducing energy consumption for many-to-many broadcast traffic in wireless networks has been studied by Fragouli et al. in  and Widmer and Le Boudec in , providing lower bounds on the energy benefit of network coding for specific topologies. More importantly, algorithms have been presented in [18, 19] that allow to exploit these benefits in practical scenarios, that is, in a distributed fashion.
The above demonstrates that for multicast traffic and for many-to-many broadcast traffic, there is some understanding of the energy benefits of network coding and how to exploit them. In order to reduce energy consumption in practical networks, however, it is important to consider also multiple unicast traffic. Indeed, in practice a large part of the data will be generated by unicast sessions. For the case of multiple unicast traffic, contrary to multicast and broadcast, not much is known. This paper deals with the energy benefits of network coding for wireless multiple unicast. Remember from the above that for multicast, the problem of minimum-cost routing is hard, whereas minimum-cost network coding is easy. In stark contrast, the problem of minimum-cost multiple unicast routing is easy. One constructs the minimum-cost solution, that is, the shortest path, for each session individually. The minimum-cost multiple unicast network coding problem, however, seems hard and in general very little is known.
Network coding for the multiple unicast problem was first studied by Wu et al. in , in which it was shown that in the information exchange problem on the line network, the energy saving achieved by network coding is a factor two. The network codes that we construct in this work are in a sense a generalization of the results on one-dimensional networks , to higher-dimensional networks. The networks considered in this work are lattices. More specifically, the hexagonal lattice and the rectangular lattice. Effros et al.  and Kim et al.  have considered energy-efficient network codes on the hexagonal lattice. We improve the lower bounds on the energy savings of network on the hexagonal lattice given in . More precisely, we improve the previously known bound of and obtain a new bound of .
Kramer and Savari have developed techniques that can be used to upper bound the achievable throughputs in a multiple unicast problem . No methods are known, however, to lower bound the cost of network coding solutions for a configuration. A lower bound to the ratio of the minimum energy consumption of routing and coding solutions for a given multiple unicast configuration was provided by Keshavarz-Haddad and Riedi in . For the type of configurations used in this paper, however, the results from  give the trivial lower bound of one. We will see, however, that network coding has large energy savings for these configurations.
An important class of network codes operates according to a principle that we will refer to in the remainder as decode-and-recombine. These codes satisfy the constraint that each symbol in each linear combination that is transmitted is explicitly known by the node transmitting that linear combination. Note, that this is a restriction from the general linear coding strategy, in which linear combinations of coded messages can be retransmitted. The motivation behind using decode-and-recombine codes is that it prevents information from spreading too much in the network, away from the path between source and destination, a heuristic introduced by Katti et al. . The use of a decode-and-recombine strategy results in reduced complexity. However, an important question that has to be addressed is whether the use of decode-and-recombine codes leads to a higher energy consumption than is strictly necessary. We answer this question affirmatively. An upper bound of three on the energy benefit of decode-and-recombine codes has been given by Liu et al. . One of the contributions of this work is to show that larger energy benefits can be obtained by considering also other types of codes.
This paper is organized as follows. In Section 2 we specify our model and problem statement more precisely. Our main results are presented in Section 3. Constructions of configurations that allow a large energy benefit for network coding and proofs of our results are given in Sections 4 and 5. In Section 6, finally, we discuss our work.
where denotes the Euclidean norm of . The energy required to transmit one unit of information to all other nodes within range equals , where is the path loss exponent and is some constant. In analyzing the energy consumption of nodes, we will consider only the energy consumed by transmitting. Receiver energy consumption as well as energy consumed by processing are assumed to be negligible compared to transmitter energy consumption. In particular, note that little additional processing is required for network coding, compared to the processing that is performed in a traditional wireless protocol stack.
The traffic pattern that we consider is multiple unicast. All symbols are from the field , that is, 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 destination. Let be the set of unicast sessions. We call a wireless multiple unicast configuration.
We will compare energy consumption of routing and network coding. Our goal is to establish lower bounds on the maximum of the ratio of the minimum energy required by routing and network coding solutions, where the maximum is over all configurations. We will refer to this ratio as the energy benefit of network coding. Let and be the minimum energy required for network coding and routing solutions, respectively, for a configuration . The energy consumption of a coding or routing scheme is defined as the time-average of the total energy spent by all nodes in the network to deliver one symbol for each unicast session. In analyzing coding schemes, we will ignore the energy consumption in an initial startup phase and consider only steady-state behavior.
where the maximization is over all node locations and multiple unicast sessions , with the transmission range being optimized individually for the routing and network coding solutions. If no confusion can arise, we will omit dependency on in the notation for and .
Since in , is equal for and , the energy per transmission is equal in and and the benefit is equal to the ratio of the number of transmissions required in routing and network coding solutions.
Since we are interested in energy consumption only, we can assume that all transmissions are scheduled sequentially and/or that there is no interference. All coding and routing schemes that we consider proceed in time slots or rounds. In each time slot, all nodes are allowed to transmit one or more messages. We assume that the length of the time slot is large enough to accommodate sequential transmission of all messages in that round. Coding operations will be based on messages received in previous time slots only. Finally, we assume that all nodes have complete knowledge of the network topology and the network code that is being used.
To conclude this section, we introduce here some of the notation that will be used in the remainder of the paper. The symbol transmitted by a node in time slot is denoted by . If transmits more than one symbol in time slot , these will be distinguished by a superscript, giving, for instance, and . Nodes are represented by vectors. Given vectors and , let , , and .
Unicast sessions are denoted by , with being an integer and a vector. We will see in Sections 4 and 5 that defines the location of the source and the relative location of the destination, that is, the direction of the session. In some cases will be denoted as or similar forms. The th source symbol of a session is denoted by . The source and destination of session are denoted by and , respectively.
We provide lower bounds on and .
The result states that is at least , , and for -, - and -dimensional networks, respectively. The result that is at least in one-dimensional networks also follows from the results in . The lower bound for -dimensional networks exceeds the previously known bound of . This new lower bound is of particular interest, since it exceeds the upper bound of for decode-and-recombine type network codes . Indeed, the code that we construct does not follow a decode-and-recombine strategy. This shows that energy can be saved by considering strategies other than decode-and-recombine. No lower bounds for three-dimensional networks have been previously established.
Details of the configuration and a proof of Theorem 1 are given in Section 5.
By considering a different configuration, we show that .
For -dimensional wireless networks, the ratio of the minimum energy consumption of routing solutions and the minimum energy consumption of network coding solutions, maximized over all node locations and multiple unicast sessions, with the transmission range optimized individually for the routing and network coding solutions, is at least , that is,
Besides providing new lower bounds on the energy benefit of network, the network codes that are constructed in this paper are of interest by themselves. They might lead to insight in how to operate in networks with another structure. Finally, even though the case is not of any practical relevance, the bounds as well as the code constructions might lead to a better insight for lower-dimensional networks.
In this section, we present a multiple unicast configuration in which the nodes form a subset of the hexagonal lattice. It will be shown that the energy benefit on this configuration is , proving Theorem 2. Since the code construction used here is less involved then the construction used to prove Theorem 1, we start with the proof of Theorem 2. This section is organized as follows. In Section 4.1 we present the configuration in more detail after which we give the construction of the network code in Section 4.2. Section 4.3 is used to prove that the code is valid. Finally, in Section 4.4 we analyze the energy consumption of the network code and prove Theorem 2.
4.2. Network Code
The network code is such that in each time slot a new source symbol from each session is transmitted. Also, one symbol of each session is decoded by its destination in each time slot. After successfully decoding a symbol, it is retransmitted by the destination in the next time slot. Nodes at the border will, therefore, transmit twice in each time slot. Nodes in the interior of the network transmit only once. The symbol that they transmit is a linear combination of one symbol from each of the sessions for which the shortest path between source and destination includes that node.
This coding operation (i.e., in time slot , a node transmits the sum of what was transmitted by its top-left neighbour in time slot , by its top right-neighbour in time slot , and so forth, as visualized in Figure 5) is performed by all nodes that are in the interior of the network. The idea behind the coding operation is to cancel, by means of the XOR operation, all symbols that should not be retransmitted. In (12), for instance, we have . The exact operation of the network code is made more precise in the remainder of this subsection. The coding operation for interior nodes is given in exact form in (17).
Nodes at the border of the network operate as follows. Let . In time slot node transmits two symbols and , where
Since is the source of session it has source symbol , available. Also, is the destination for session . It remains to be shown that symbol can be decoded by using the information obtained from its neighbours up to time slot . For notational convenience, let
In a similar fashion, we have the following transmissions at the right and bottom borders of the network.
where , , and . Moreover, and are not symbols that are transmitted, but only notational shortcuts.
4.3. Validity of the Network Code
We need to show that destinations can decode in time in order to retransmit the required symbols according to (13), (15), and (16). In order to do so we first analyze how data propagates through the network. If we look at the nodes in the network that transmit linear combinations that contain a certain source symbol, we see that symbols propagate exactly along the shortest paths between source and destination. This is made more precise in the following two lemmas.
which by the induction hypothesis is equal to if and and zero otherwise.
From Lemma 1, the time-invariance of the system, and the symmetry of the coding operation (17) of the internal nodes.
We are now ready to prove that the destinations can correctly decode source symbols. We present the decoding procedure for nodes on the right border of the network. The decoding procedures at the other borders can be obtained by exploiting the symmetry of the system.
4.4. Energy Consumption
The energy consumption of the network coding scheme presented above is given in the following lemma.
Next, we give the minimum energy required by a routing solution.
Using the above two lemmas, we are able to prove Theorem 2.
Proof of Theorem 2.
In this section, we present a multiple unicast configuration in which the nodes are placed at integer coordinates in a -dimensional space, that is, at the rectangular lattice.
which corresponds to those nodes that are part of exactly one face of the network.
5.2. Network Code
where denotes symmetric difference and . Note that irrespective of we have . As an example for we have , and .
5.3. Validity of the Network Code
The following result follows directly from the definition of the sets , but is stated here as a lemma because of its importance in the remainder of the paper.
for all and .
which proves the lemma, since by the induction hypothesis if and zero otherwise.
where the second equality follows from Lemma 6, the third equality follows from (37)-(38), and the last equality holds because we work over .
By linearity, time-invariance and symmetry of (34) together with Lemma 7.
We are now ready to prove that the destinations can correctly decode source symbols. We present the decoding procedure for nodes on the right border of the network, that is, for nodes of type . The decoding procedures at the other borders can be obtained by exploiting the symmetry of the system.
First note that all terms in (45) correspond to symbols that have been received by before or in time slot .
The proof of the lemma follows by adding the final expressions from (46), (49) and (50) observing that the outcome is .
5.4. Energy Consumption
The energy consumption of the network coding scheme presented above provides an upper bound to .
All transmissions are over distance and cost . The nodes in are transmitting twice. On each of the sides of the network, there are nodes from ; hence . This gives transmissions. In addition, there are nodes in the interior, that are all transmitting once.
Next, we give the minimum energy required by a routing solution.
Since the transmission range is equal to , a routing solution requires transmissions per session. Moreover, there are sessions.
Using the above two lemmas, we are able to prove Theorem 1.
Proof of Theorem 1.
We have given several constructions of energy-efficient network codes. These constructions serve to show that compared to plain routing, network coding has the potential of reducing energy consumption in wireless networks. Since we have provided only codes that are based on a centralized design, it remains to be shown in future work if and how this potential can be exploited using practical codes. Moreover, it would also be of interest to consider the energy-benefit in topologies in which the nodes are not positioned at a lattice, for instance, random networks.
In this work we have provided lower bounds on the energy benefit of network coding for wireless multiple unicast. Another open problem is to find upper bounds on the benefit.
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