Linear network coding

Encoding and decoding

In a linear network coding problem, a group of nodes P are involved in moving the data from S source nodes to K 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 G F ( 2 s ) .

Each node, p k with indegree, I n D e g ( p k ) = S , generates a message X k from the linear combination of received messages { M i } i = 1 S by the relation:

where the values g k i are the coefficients selected from G F ( 2 s ) . 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 X k along with the coefficients, g k i , used in the k th level, g k i .

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. In reduced row echelon form, decoded packets correspond to the rows of the form e i = [ 0...010...0 ] .

A brief history

A network is represented by a directed graph G = ( V , E , C ) . V is the set of nodes or vertices, E is the set of directed links (or edges), and C gives the capacity of each link of E . Let T ( s , t ) be the maximum possible throughput from node s to node t . By the max-flow min-cut theorem, T ( s , t ) 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.

The butterfly network example

The butterfly network 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

Random linear network coding 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

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

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. It has been shown that in both cases, network coding can increase the end-to-end throughput.

Applications

Network coding is seen to be useful in the following areas: