Linear network coding

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Linear network coding is a mathematical technique which may be used to improve a network's throughput, efficiency and scalability, as well as resilience to attacks and eavesdropping. Instead of simply relaying the packets of information they receive, the nodes of a network take several packets and combine them together for transmission. This may be used to attain the maximum possible information flow in a network.

It has been proven in theory that linear coding is enough to achieve the upper bound in multicast problems with one source.[1] However linear coding is not sufficient in general (e.g. multisource, multisink with arbitrary demands), even for more general versions of linearity such as convolutional coding and filter-bank coding.[2] Finding optimal coding solutions for general network problems with arbitrary demands remains an open problem.

Encoding and decoding[edit]

In a linear network coding problem, a group of nodes are involved in moving the data from source nodes to sink nodes. Each node generates new packets which are linear combinations of earlier received packets, multiplying them by coefficients chosen from a finite field, typically of size .

Each node, with indegree, , generates a message from the linear combination of received messages by the relation:

where the values are the coefficients selected from . Note that, since operations are computed in a finite field, the generated message is of the same length as the original messages. Each node forwards the computed value along with the coefficients, , used in the level, .

Sink nodes receive these network coded messages, and collect them in a matrix. The original messages can be recovered by performing Gaussian elimination on the matrix.[3] In reduced row echelon form, decoded packets correspond to the rows of the form .

A brief history[edit]

A network is represented by a directed graph . is the set of nodes or vertices, is the set of directed links (or edges), and gives the capacity of each link of . Let be the maximum possible throughput from node to node . By the max-flow min-cut theorem, is upper bounded by the minimum capacity of all cuts, which is the sum of the capacities of the edges on a cut, between these two nodes.

Karl Menger proved that there is always a set of edge-disjoint paths achieving the upper bound in a unicast scenario, known as the max-flow min-cut theorem. Later, the Ford–Fulkerson algorithm was proposed to find such paths in polynomial time. Then, Edmonds proved in the paper "Edge-Disjoint Branchings" the upper bound in the broadcast scenario is also achievable, and proposed a polynomial time algorithm.

However, the situation in the multicast scenario is more complicated, and in fact, such an upper bound can't be reached using traditional routing ideas. Ahlswede, et al. proved that it can be achieved if additional computing tasks (incoming packets are combined into one or several outgoing packets) can be done in the intermediate nodes.[4]

The butterfly network example[edit]

Butterfly Network.

The butterfly network [4] is often used to illustrate how linear network coding can outperform routing. Two source nodes (at the top of the picture) have information A and B that must be transmitted to the two destination nodes (at the bottom), which each want to know both A and B. Each edge can carry only a single value (we can think of an edge transmitting a bit in each time slot).

If only routing were allowed, then the central link would be only able to carry A or B, but not both. Suppose we send A through the center; then the left destination would receive A twice and not know B at all. Sending B poses a similar problem for the right destination. We say that routing is insufficient because no routing scheme can transmit both A and B simultaneously to both destinations.

Using a simple code, as shown, A and B can be transmitted to both destinations simultaneously by sending the sum of the symbols through the center – in other words, we encode A and B using the formula "A+B". The left destination receives A and A + B, and can calculate B by subtracting the two values. Similarly, the right destination will receive B and A + B, and will also be able to determine both A and B.

Random Linear Network Coding[edit]

Random linear network coding [5] is a simple yet powerful encoding scheme, which in broadcast transmission schemes allows close to optimal throughput using a decentralized algorithm. Nodes transmit random linear combinations of the packets they receive, with coefficients chosen from a Galois field. If the field size is sufficiently large, the probability that the receiver(s) will obtain linearly independent combinations (and therefore obtain innovative information) approaches 1. It should however be noted that, although random linear network coding has excellent throughput performance, if a receiver obtains an insufficient number of packets, it is extremely unlikely that they can recover any of the original packets. This can be addressed by sending additional random linear combinations until the receiver obtains the appropriate number of packets.

Open issues[edit]

Linear network coding is still a relatively new subject. Based on previous studies, there are three important open issues in RLNC:

  1. High decoding computational complexity due to using the Gauss-Jordan elimination method
  2. High transmission overhead due to attaching large coefficients vectors to encoded blocks
  3. Linear dependency among coefficients vectors which can reduce the number of innovative encoded blocks

Wireless Network Coding[edit]

The broadcast nature of wireless (coupled with network topology) determines the nature of interference. Simultaneous transmissions in a wireless network typically result in all of the packets being lost (i.e., collision, see Multiple Access with Collision Avoidance for Wireless). A wireless network therefore requires a scheduler (as part of the MAC functionality) to minimize such interference. Hence any gains from network coding are strongly impacted by the underlying scheduler and will deviate from the gains seen in wired networks. Further, wireless links are typically half-duplex due to hardware constraints; i.e., a node can not simultaneously transmit and receive due to the lack of sufficient isolation between the two paths.

Although, originally network coding was proposed to be used at Network layer (see OSI model), in wireless networks, network coding has been widely used in either MAC layer or PHY layer.[6] It has been shown that network coding when used in wireless mesh networks need attentive design and thoughts to exploit the advantages of packet mixing, else advantages cannot be realized. There are also a variety of factors influencing throughput performance, such as media access layer protocol, congestion control algorithms, etc. It is not evident how network coding can co-exist and not jeopardize what existing congestion and flow control algorithms are doing for our Internet.[7]

Applications[edit]

Since linear network coding is a relatively new subject, its adoption in industries is still pending. Unlike other coding, linear network coding is not entirely applicable in a system due to its narrow specific usage scenario. Theorists are trying to connect to real world applications.[8] In fact, it was found that BitTorrent approach is far superior than network coding.

It is envisaged that network coding is useful in the following areas:

Maturity & Issues[edit]

Since this area is relatively new and the mathematical treatment of this subject is currently limited to a handful of people, network coding has yet found its way to commercialization in products and services. It is unclear at this stage if this subject will prevail, or cease as a good mathematical exercise.[23]

Researchers have clearly pointed out that special care is needed to explore how network coding can co-exist with existing routing, media access, congestion, flow control algorithms and TCP protocol. If not, network coding may not offer much advantages and can increase computation complexity and memory requirements.[24]

See also[edit]

References[edit]

  1. ^ S. Li, R. Yeung, and N. Cai, "Linear Network Coding"(PDF), in IEEE Transactions on Information Theory, Vol 49, No. 2, pp. 371–381, 2003
  2. ^ R. Dougherty, C. Freiling, and K. Zeger, "Insufficiency of Linear Coding in Network Information Flow" (PDF), in IEEE Transactions on Information Theory, Vol. 51, No. 8, pp. 2745-2759, August 2005 ( erratum)
  3. ^ Chou, Philip A.; Wu, Yunnan; Jain, Kamal (October 2003), "Practical network coding", Allerton Conference on Communication, Control, and Computing, Any receiver can then recover the source vectors using Gaussian elimination on the vectors in its h (or more) received packets .
  4. ^ a b Ahlswede, Rudolf; N. Cai; Shuo-Yen Robert Li; Raymond Wai-Ho Yeung (2000). "Network Information Flow". IEEE Transactions on Information Theory, IT-46. 46 (4): 1204–1216. doi:10.1109/18.850663. 
  5. ^ T. Ho, R. Koetter, M. Medard, D. R. Karger and M. Effros, "The Benefits of Coding over Routing in a Randomized Setting" in 2003 IEEE International Symposium on Information Theory. doi:10.1109/ISIT.2003.1228459
  6. ^ M.H.Firooz, Z. Chen, S. Roy and H. Liu, (Wireless Network Coding via Modified 802.11 MAC/PHY: Design and Implementation on SDR) in IEEE Journal on Selected Areas in Communications, 2013.
  7. ^ XORs in The Air: Practical Wireless Network Coding
  8. ^ "How Practical is Network Coding? by Mea Wang, Baochun Li". 
  9. ^ http://arxiv.org/abs/1212.2291
  10. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2007-11-08. Retrieved 2007-06-16. 
  11. ^ http://securenetworkcoding.wikidot.com/
  12. ^ http://home.eng.iastate.edu/~yuzhen/publications/ZhenYu_INFOCOM_2008.pdf
  13. ^ http://netcod.org/papers/11AcedanskiDMK-final.pdf
  14. ^ http://arxiv.org/pdf/cs/0702015.pdf
  15. ^ http://people.csail.mit.edu/rahul/papers/cope-ton2008.pdf
  16. ^ "CORE: COPE with MORE in Wireless Meshed Networks". 2013 IEEE 77th Vehicular Technology Conference (VTC Spring). doi:10.1109/VTCSpring.2013.6692495. 
  17. ^ http://arena.cse.sc.edu/papers/rocx.secon06.pdf
  18. ^ http://www.cs.wisc.edu/~shravan/infocom-07-2.pdf
  19. ^ "NetworkCoding - batman-adv - Open Mesh". www.open-mesh.org. Retrieved 2015-10-28. 
  20. ^ Welcome to IEEE Xplore 2.0: Looking at Large Networks: Coding vs. Queueing
  21. ^ http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4549741
  22. ^ Data dissemination in wireless networks with network coding
  23. ^ "How Practical is Network Coding?". 
  24. ^ "XORs in The Air" (PDF). 
  • Fragouli, C.; Le Boudec, J. & Widmer, J. "Network coding: An instant primer" in Computer Communication Review, 2006.

Ali Farzamnia, Sharifah K. Syed-Yusof, Norsheila Fisa "Multicasting Multiple Description Coding Using p-Cycle Network Coding", KSII Transactions on Internet and Information Systems, Vol 7, No 12, 2013.

External links[edit]