A Practical Perspective of Wireless Network Coding Marium Jalal Chaudhry1,2 , Timo H¨am¨al¨ainen1 , Jyrki Joutsensalo1 , Farhat Saleemi2 , Kari Luostarinen3 1 Department of Information Technology, University of Jyv¨askyl¨a, Finland 2 Department of Electronic Engineering, Lahore College For Women University, Lahore, Pakistan 3 Metso Paper Inc. Jyv¨askyl¨a, Finland
Abstract—Wireless networks are one of the most essential components of the communication networks. In contrast to the wired networks, the inherent broadcast nature of wireless networks provides a breeding ground for both opportunities and challenges ranging from security to reliability. Moreover, energy is a fundamental design constraint in wireless networks. The boom of wireless network is closely coupled with the schemes that can reduce energy consumption. Network coding for the wireless networks is seen as a potential candidate scheme that can help overcome the energy and security challenges while providing significant benefits. This paper presents a basic model for formulating network coding problem in wireless setting. Since the optimal solution to wireless network coding is NP (nondeterministic polynomial-time)-hard, we intend to explore the impact of different parameters with random and non-random solutions. We present extensive simulation to show the strength of random network coding scheme in general wireless network scenarios. This study provide a comprehensive insight into the limits of wireless network coding.
Required: P1 Overheared: P2
Required: P2 Overheared: P1
Required: P3 Overheared: P4
Required: P4 Overheared: P3
I. I NTRODUCTION Wireless networks are one of the most essential components of the communication networks. In contrast to the wired networks, the inherent broadcast nature of wireless networks provides a breeding ground for both opportunities and challenges ranging from security to reliability. Moreover, energy is a fundamental design constraint in wireless networks. The boom of wireless network is closely coupled with the schemes that can reduce energy consumption. Network coding for the wireless networks is seen as a potential candidate scheme that can help overcome the energy and security challenges while providing significant benefits. In reference to its applications in wireless network, the typical setting of the network coding problem of consist of a server and a set of clients. Server has set of all packets needed by clients. Each client needs a packet( or packets) called required packets and might have overheard packets due to broadcast nature of wireless networks during some previous transmission. Server can make use of the overheard packets at the clients to reduce the overall number of transmissions. The goal of the wireless network coding problem is to reduce the number of transmissions as explained in the following example. Consider an example of a simple wireless network shown in Fig 1 consisting of a server and four clients. In this example a server(wireless tower) needs to satisfy 4 wireless clients by transmitting 4 packets p1 , p2 , p3 , p4 if it does not use network coding but using network coding the transmissions is reduced to half i.e. p1 + p2 and p3 + p4 . Unfortunately optimal solution to network coding problem for wireless network is NP-Complete i.e. a polynomial time solution does not exist. Therefore finding the minimum number
Fig. 1.
An example of network coding in wireless networks
of transmissions and the coded packet combinations that can help in reducing the number of transmissions is not possible in a realistic time for any general input. Due to hardness of finding exact solution to the problem, a set of algorithms is developed to address the issue which, in most of the practical situations, guarantees a lesser number of transmissions as compared to traditional schemes although not the least one. This paper presents a basic model for formulating network coding problem in wireless setting. Since the optimal solution to wireless network coding is NP-hard [1], we intend to explore the impact of different parameters with random and nonrandom solutions. We present extensive simulation to show the strength of random network coding scheme in general wireless network scenarios. This study provide a comprehensive insight into the limits of wireless network coding. II. P REVIOUS W ORK The network coding was firstly introduced by Ahlswede et al. [2] in 2000. Figure 2 represents the famous example given by Ahlswede et al. showing the necessity of using network coding i.e., coding at the intermediate nodes of the network in order to achieve multicast capacity. The main result by Ahlswede et al. [2] was to show that network coding can help to send the data traffic at the same rate as the min-cut between the sender and the receiver. The work on network coding was further explored by Koetter and Medard [3] giving Corresponding author Marium Jalal Chaudhry
[email protected]
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Fig. 2.
packets to the clients such that all clients receive whatever they require. We define two sets for each client ti ∈ T : 1) Overheard(ti ) ⊆ P - the set of packets overheard by terminal ti ; 2) Required(ti ) ⊆ P - the set of packets required by terminal ti . The coefficient of each packet is an element of a finite field GF (q). We assume packets of one bit each without loss of generality as large packets can always be divided into smaller packets of one bit size that can be send separately with the network coding applied to each of the smaller packets individually [13], [11]. The server knows the packets in Overheard(ti ) and Required(ti ) set for all ti ∈ T the can transmit any packet from pool P as well as can transmit any linear combinations (over GF (q)) of packets in that P . Each transmission i is specified by an encoding vector xi . The problem is to find the set of encoding vectors Φ = {gi } of minimum rank that allows each terminal to decode the packets requested. We assume, without loss of generality, that for each terminal ti ∈ T , there exists at least one packet pi ∈ P such that pi belongs to the Required set Required(ti ) of terminal ti . We also assume that for each client Required(ti ) ∩ Overheard(ti ) = ∅. To show advantage of network coding over traditional broadcast scheme two parameters of performance is define for any underlying network model.
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An example of a butterfly network [2]
the first algebraic framework for linear network coding over any given network. The results of the authors in [3] also ties network coding with the robustness. Koetter and Medard [3] also studied their proposed solutions for more practical cyclic networks that incorporate delays. Authors in [4] have also presented a sound mathematical model for finding the network coding solution by expressing the global coding coefficients as transfer matrices, which was also shown to have an interesting relationship with a mapping to the bipartite matching and then to network flows. Single-source multicast network was studied in detail by Sanders et al. [5] and Jaggi et al. [6]. They not only presented the bounds on the required field size but also presented the first deterministic algorithm for the network coding solutions. Fragouli et al. [7] presented an easier and efficient heuristic solution for the network coding problem by relating it to the graph coloring problem. Rasala Lehman and Lehman [8] and Feder et al. [9] studied how the characterization of any networks impacts the feasibility of the proposed solution. Network Coding problem over undirected networks was explored by Li and Li [10] bounding the network coding gain by a factor of 2.
A. Coding Gain The gain in terms of number of ratio of minimum number of transmissions needed for a source to satisfy all terminals using simple broadcast over network coded broadcast is termed as coding gain G is defined as : G = n/m
(1)
where m is number of message sent using traditional method and n is number of messages sent using network coding. B. Problem Statement For a wireless network with a single server s and a set of terminals T and a pool of packets P and each client has at least one packet in its Required set and Overheard set, minimize the number of transmissions to satisfy each and every client.
III. O RGANIZATION This paper presents our basic model for formulating network coding problem into a graph theoretic problem in Section IV followed by the suggested performance metric. Then we formally define the problem statement for wireless network. We describe two algorithms in Section V. One is random network coding and other is non-random network coding. We then present extensive simulation studies in Section V-E. Finally we conclude in Section VI.
V. P RACTICAL A SPECTS OF W IRELESS N ETWORK C ODING Network Coding is proved to be NP-complete in wireless network scenarios i.e. finding solution in polynomial time is not possible. Working in wireless setting is constrained by a energy efficient scheme therefore the parameter of concern is to reduce the number of transmissions required to fulfill demand of all users. Wireless network have inherent overhearing setup that lays the basics of network coding and helps in achieving much better results than serving each user separately. We study two basic type of algorithms for network coding in wireless networks in terms of the several important parameters. • Random network coding • Non random network coding
IV. G ENERIC N ETWORK M ODEL F OR W IRELESS N ETWORKS The basic model presented in this chapter is specific for single-hop wireless channel with one server s and a set of x terminals T = {t1 , . . . , tx } [1], [11], [12]. The server has a pool P = {p1 , p2 , . . . , pn } of n packets that are required by clients and server has to transmit either all or a subset of these 2
A. Random Network Coding This is a very simple technique for solving the network coding problem for minimizing the number of transmissions required to satisfy all terminals. Simple and easy but proved to be much efficient in most of the practical scenarios which are studied during simulations and presented in Section V-E. In random network coding Server send out random linear combinations of memory contents or the packets form the pool P at every transmission opportunity. Terminal nodes can decode the desired packets if : • Received Packet contains the desired packet • Received Packet is a coded packet of the desired packet with packets in overheard packet of the client. Random network coding technique is different from the traditional networking approaches in which server need to broadcast each and every packet separately.
Fig. 3. For 5 clients and with random cardinality of required and Overheard sets, uniform distribution of packets and with non-random Coding technique.
B. Non-Random Network Coding In this scheme server transmit packets that are coded deterministically from the Required(ti ) and Overheard(ti ) sets of each terminal. C. Simulation Setup For the purpose of each experiment in the Section V-E we randomly generate an instance of the wireless network coding as follows. We generate a server with 100 packets p1 , p2 , · · · , p100 . For some experiments we generate n clients with a randomly generated Required-set and Overheard-set, where n is an integer and varies from 1 to 100. The distribution of packets in Required-set and Overheard-set for each client is selected by a probabilistic distribution which is either uniform or Gaussian distribution. Moreover the cardinality of Requiredset and Overheard-set is also selected with uniform random distribution ranging from 1 to n.
Fig. 4. For 5 clients and with random cardinality of required and Overheard sets, uniform distribution of packets and Random Coding technique.
D. Simulation Parameters We have studied the effect of following parameters on the wireless network: • Cardinality of Required-set: Number of packets Required by a client. • Cardinality of Overheard-set: Number of packets overheard by a client • Number of clients • Distribution of packets in both Required and Overheard sets: Packet required by each client can be chosen uniformly or according to Gaussian distribution.
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E. Results and Observations We have extensively studied the wireless network under different settings of the simulation parameters defined in Section V-D. The results are as follows: • Fig 3 shows the results for 5 clients and with random cardinality of required and Overheard sets, uniform distribution of packets and with non-random Coding technique. • Observation:Coding gain can go as high as 5 for 15% of cases studied and on average gain is about 2.25. • Fig 4 shows the results for 5 clients and with random cardinality of required and Overheard sets, uniform distribution of packets and Random Coding technique.
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Observation:Coding gain can go as high as 5 for 2% of cases studied and on average gain is about 2. Fig 5 shows the results for 10 clients and with random cardinality of required and Overheard sets and distribution of packets chosen is uniform. Coding technique used is non-random Observation: Coding gain can go as high as 3.25 for 5% of cases studied and on average gain is about 1.75. Fig 6 shows the results for 10 clients and with random cardinality of required and Overheard sets and distribution of packets chosen is uniform. Coding technique used is random Observation:Coding gain can go as high as 1.6 for 2% of cases studied and on average gain is about 1.2. Fig 7 shows the results for 20 clients and with random cardinality of required and Overheard sets and distribution of packets chosen is uniform. Coding technique used is non-random Observation: Coding gain can go as high as 1.8 for 3% of cases studied and on average gain is about 1.4. Fig 8 shows the results for 20 clients and with random cardinality of required and Overheard sets and distribution of packets chosen is uniform. Coding technique used
Fig. 8. For 20 clients and with random cardinality of required and Overheard sets and distribution of packets chosen is uniform. Coding technique used is random.
Fig. 5. For 10 clients and with random cardinality of required and Overheard sets and distribution of packets chosen is uniform. Coding technique used is non-random.
Fig. 9. For 5 clients and with fixed cardinality of Overheard set, random cardinality of required set and distribution of packets chosen is uniform. Coding technique used is random. Fig. 6. For 10 clients and with random cardinality of required and Overheard sets and distribution of packets chosen is uniform. Coding technique used is random. • •
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Fig. 7. For 20 clients and with random cardinality of required and Overheard sets and distribution of packets chosen is uniform. Coding technique used is non-random. •
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is random Observation:Coding gain can go as high as 1.25 for 2% of cases studied and on average gain is about 1.08. Fig 9 and Fig 10 shows the results for 5 clients and with fixed cardinality of Overheard set, random cardinality of required set and distribution of packets chosen is uniform, Coding technique used is random and nonrandom respectively. Observation: Average coding gain increases with increase in number of packets in overheard set. As compared to coding gain with random technique, non-random gives much higher coding gain with sharp rise. Fig 11 and Fig 12 shows the results for 10 clients and with fixed cardinality of Overheard set, random cardinality of required set and distribution of packets chosen is uniform. Coding technique used is random and nonrandom respectively. Observation: Average coding gain increases with increase in number of packets in overheard set. As compared to coding gain with random technique, non-random gives much higher coding gain with sharp rise. Fig 13 and Fig 14 shows the results for 1 up to 10
Fig. 13. For 1 up to 10 clients and with fixed cardinality of Overheard set and required set is random and distribution of packets chosen is Gaussian. Coding technique used is non-random.
Fig. 10. For 5 clients and with fixed cardinality of Overheard set, random cardinality of required set and distribution of packets chosen is uniform. Coding technique used is non-random.
Fig. 14. For 1 up to 10 clients and with fixed cardinality of Overheard set and required set is random and distribution of packets chosen is Gaussian. Coding technique used is random. Fig. 11. For 10 clients and with fixed cardinality of Overheard set, random cardinality of required set and distribution of packets chosen is uniform. Coding technique used is random.
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Fig. 12. For 10 clients and with fixed cardinality of Overheard set, random cardinality of required set and distribution of packets chosen is uniform. Coding technique used is non-random.
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clients and with fixed cardinality of Overheard set and required set is random and distribution of packets chosen is Gaussian. Coding technique used is non-random and random respectively. Observation:Average coding gain increases with increase in number of packets in overheard set shows more network coding gain. As compared to coding gain with random technique, non-random gives much higher coding gain. Fig 15 and Fig 16 shows the results for 3 up to 10 clients and with fixed cardinality of Overheard set, random cardinality of required set and distribution of packets chosen is uniform. Coding technique used is non random. Fig 15 shows coding gain whereas Fig 16 shows time taken. Observation:Average coding gain decreases with increase in number of clients. More packets in overheard set shows more network coding gain. Fig 17 and Fig 18 shows coding gain and time taken respectively for 3 up to 10 clients and with fixed cardinality of Overheard set, random cardinality of required set and distribution of packets chosen is Gaussian. Coding technique used is non random.
Fig. 18. Time taken for 3 up to 10 clients and with fixed cardinality of Overheard set, random cardinality of required set and distribution of packets chosen is Gaussian. Coding technique used is non random
Fig. 15. Coding Gain for 3 up to 10 clients and with fixed cardinality of Overheard set, random cardinality of required set and distribution of packets chosen is uniform. Coding technique used is non random.
Fig. 19. Coding Gain for 3 up to 10 clients and with fixed cardinality of Required set, random cardinality of Overheard set and distribution of packets chosen is uniform. Coding technique used is non random. Fig. 16. Time taken for 3 up to 10 clients and with fixed cardinality of Overheard set, random cardinality of required set and distribution of packets chosen is uniform. Coding technique used is non random. •
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Fig. 17. Coding Gain for 3 up to 10 clients and with fixed cardinality of Overheard set, random cardinality of required set and distribution of packets chosen is Gaussian. Coding technique used is non random
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Observation: Average coding gain decreases with increase in number of clients. More packets in overheard set shows more network coding gain. Fig 19 and Fig 20 shows coding gain and time taken respectively for 3 up to 10 clients and with fixed cardinality of Required set, random cardinality of Overheard set and distribution of packets chosen is uniform. Coding technique used is non random. Observation: Average coding gain decreases with increase in number of clients whereas server takes more time to code as number of clients increases. More packets in Required set shows more network coding gain and more time. Fig 19 and Fig 20 shows coding gain and time taken respectively for 3 up to 10 clients and with fixed cardinality of Required set, random cardinality of Overheard set and distribution of packets chosen is Gaussian. Coding technique used is non random. Observation:Average coding gain decreases with increase in number of clients. More packets in Required set shows more network coding gain.
VI. C ONCLUSION We studied random and non-random network coding for wireless network scenario. As solving the wireless network coding problem optimally is shown to be NP-hard, we concentrate on the random coding solutions for simulation section. We consider several practical scenarios and parameters that might affect the overall gain in terms of saving the number of transmissions. The results shows how the distribution of packets in Required-set and Overheard-set, the cardinality of packets in the Overheard-set of a client, and the number of clients affect the gain of using network coding schemes over traditional schemes. The experiments shows that in all the cases network coding shows positive gains which is most prominent when the number of clients and the cardinality of packets in Overheard-set is higher. The experiment results strongly supports the usefulness of the wireless network coding. In future we would like to extend the results to the scenario when the channel condition is not ideal (i.e. lossless). We would like to explore the affect and benefits of the wireless network coding schemes under the faulty channels like bit flip channels as well as completely Byzantine channels.
Fig. 20. Time taken for 3 up to 10 clients and with fixed cardinality of Required set, random cardinality of Overheard set and distribution of packets chosen is uniform. Coding technique used is non random.
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Fig. 21. Coding Gain for 3 up to 10 clients and with fixed cardinality of Required set, random cardinality of Overheard set and distribution of packets chosen is Gaussian. Coding technique used is non random.
Fig. 22. Time taken for 3 up to 10 clients and with fixed cardinality of Required set, random cardinality of Overheard set and distribution of packets chosen is Gaussian. Coding technique used is non random.
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