Here the communication network is represented by a directed acyclic graph ( , )
G= V E with unit capacity edges and that the value of the min cut between the source and each of the sinks ish. The source node S is required to transmit simultaneously, h, unit-rate independent information streams { , ... }s s1 2 sh , and a set
of N sinks { , ... }t t1 2 tN is required to receive the multicast data from the source S.
The source needs to apply the multicast coding technique in this multicast transmission and it requires identification of the minimal configurations between the
source itself and the set of N sinks. Moreover these minimal configurations have abilities to minimise the network coding resources during the multicast transmission.
Fundamentally, the coding nodes are enriched in terms of buffer memory, computational capability and operating power, and these additional abilities are defined as the coding resources. These resources are rapidly consumed and ultimately exhausted by computational complexity, packet delay, congestion, packet misrouting and so forth. The packet delay, congestion and packet misrouting contribute to cause synchronising errors at the coding nodes and decoding errors at the sinks.
The network coding resources for multicasting are comprehensively discussed by Fragouli and Soljanin [1] who describe the major complexity components as Set-up
complexity and Operational complexity. The former denotes the complexity of
designing the network coding scheme, which includes selecting the paths through the information flows and determining the operations (coding, forwarding etc.) that the nodes of the network perform. The latter encompasses the running cost of using network coding, that is, the amount of computational and network resources required per information unit successfully delivered. Moreover, this complexity is strongly correlated with the network coding scheme employed. For example, Figure 4-3 shows the coding scheme which can be used to deliver the multicast traffic with optimum network and coding resources usages.
The operational complexity is further discussed using assumptions that the source
S simultaneously emits multicast packets{σ σ1, 2...σh}which are elements of some
consisting of hNpaths. In linear network coding, these elements are linearly combined and forwarded by some intermediated nodes of G’, and these combined packets are elements of q. The linear combination of hinformation streams requires Ο(h2)finite field operations. The complexity is further affected by the size
of the finite field over which operations take place as the cost of finite field arithmetic
grows with the field size. For example, typical algorithms for multiplication or inversion over a field of size q=2nrequire (n2)binary operations. Moreover the field size affects the required storage capabilities at intermediate network nodes. The computational complexity is further affected by the number of coding points inG'. Coding points are, in general, more expensive due to need to equip them with encoding capabilities. In addition, coding points incur delay and increase the overall complexity of the network [3]. The computational complexity at each coding point of
G’ is considerably increased by a number of in-links per coding point and which exhausts the coding resources via increasing operational network complexity [3].
To recover the source packets{σ σ1, 2...σh}, which have been linearly combined
over qby the coding nodes, each sink needs to solve a system of h h× linear equations, which requires Ο(h3)operations over qif Gaussian elimination is used.
S D A G C t1 t3 a a b b a a F E t2 B b a + b a a a + b a + b a + b d d a + b d a + b d b a d Time Input buffer at Node - C b a Time Input buffer at Node - E a a + b d Time Input buffer at Node – t1 a + b d Input buffer at Node – t2 a + b a Input buffer at Node – t3 a + b Time Time a – misrouted packet
Figure 5-1: Congestion, packets delay and packet misrouting exhaust network coding resources and cause decoding errors
Figure 5-1 is used to explain the issues mentioned in section 5.1 and their effects on the network coding resources. Packet ‘a’ is congested in link AC by a packet, in fact it is delayed by time d. Node C is receiving packet ‘b’ on link BC and it has to store this in a Node C input buffer during the time d until ‘a’ arrives. This is defined as a ‘synchronous error to coding operation’ and consumes Node C’s power to maintain its buffer memory. As a result, coded packet 'a⊕b' is routed throughout the network with time delay d. Therefore Sinks t1 and t2 face a synchronous error
like Node C and this is defined as a ‘synchronous error to decoding operation’. Moreover Node G misroutes packet ‘a’ through link GE and an unwanted coding operation proceeds at Node E. The coded packet 'a⊕b' is routed throughout the network and t2 is able to receive identical packets 'a⊕b'and ‘a⊕b' meaning it is
impossible for this sink to obtain the original packets ‘a’ and ‘b’ by solving linear equations. This issue is defined as a ‘decoding error’ and t2 re-requests the multicast
data from S, which then attempts to redeliver the multicast data ‘a’ and ‘b’ not only to t2 but also to t1 and t3. Therefore network and coding resources are allocated to
retransmit the same set of the multicast data ‘a’ and ‘b’. The decoding error causes fatal damage to the network and coding resources.