1. Network coding techniques
Wireless Systems - Lecture
Network coding techniques
Elena Fasolo
PhD Student - SIGNET Group
March, 7th 2004

2. Definition of network coding (NC)
Network coding is a particular in-network data processing technique that exploits the characteristics of the wireless medium
(in particular, the broadcast communication channel)
in order to increase the capacity or the throughput of the network
Pioneering work: [1] R. Ahlswede, N. Cai, S.-Y. R. Li, and R.W. Yeung, “Network information flow,” IEEE Trans. on Information Theory, vol. 46, no. 4, July 2000.
Improves the performance in data broadcasting
Most suitable setting: all to all communications

3. Communication networks
TERMINOLOGY
Communication network = finite directed graph
Acyclic communication network = network without any direct cyclic
Source node = node without any incoming edges (square)
Channel = noiseless communication link for the transmission of a data unit per unit time (edge)
WX has capacity equal to 2

4. The canonical example (I)
Without network coding
Simple store and forward
Multicast rate of 1.5 bits per time unit

5. The canonical example (II)
With network coding
X-OR  is one of the simplest form of data coding
Multicast rate of 2 bits per time unit
Disadvantages
Coding/decoding scheme has to be agreed upon beforehand

6. NC and wireless communications
Problem: send b1 from A to B and b2 from B to A using node C as a relay
A and B are not in communication range (r)
Without network coding, 4 transmissions are required.
With network coding, only 3 transmissions are needed b1 A B C r
(b)
(c) b2 b1 b2 C C B A A B

7. Linear network coding
When we refer to linear network coding [2], we intend that:
The output flow at a given node is obtained as a linear combination of its input flows. The coefficients of the combination are, by definition, selected from a finite field
Coding can be implemented at low computational cost
Moreover, the information traversing a non source node has the following property:
The content of any information flowing out of a set of non source nodes can be derived from the accumulated information that has flown into the set of nodes
[2] S.-Y. R. Li, R. W. Yeung, and N. Cai, “Linear network Coding”, IEEE Trans. on Information Theory, vol. 49, no. 2, Feb

8. Theoretical model for linear NC
Graph (V,E) having unit capacity edges
Sender s in V, set of receivers T={t,…} in V
Source node of
h symbols
Intermediate node
Destination node

9. Global encoding vector
Linear coding phase
Transmitted symbol
Local encoding vector
Global encoding vector

10. Decoding phase
Node t can recover the source symbols x1, , xh as long as the matrix Gt, formed by the global encoding vectors, has (full) rank h. -1

11. Inverting Gt
Gt will be invertible with high probability if local encoding vectors are random and the field size is sufficiently large [3]
P = 1 - |F| (where |F| is the cardinality of the finite field of coefficients)
Example:
If field size = 216 and |E| = 28 then Gt will be invertible with probability ≥ 1−2−8 = 0.996
[3] R. Koetter,M.Medard, “An algebraic approach to network coding”, IEEE/ACM Trans. on Networking, Nov.2003

12. Theory vs. Practice Theory: Practice:
Symbols flow synchronously throughout network
Edges have unit (or known integer) capacities
Centralized and full knowledge of topology, which is used to compute encoding and decoding functions
Practice:
Information travels asynchronously in packets
Packets subject to random delays and losses
Edge capacities often unknown, time-varying
Difficult to obtain centralized knowledge, or to arrange reliable broadcast of functions
Need for simple solutions, applicable in practice

13. Practical Random NC Main idea [4]:
Select the linear coefficients in a finite field of opportune size in a random way
Send the encoding vector within the same packet
Packetization: Header removes need for centralized knowledge of graph topology and encoding/decoding functions
Nodes stores within their buffers the received packets
Buffering: Allows asynchronous packets arrivals & departures with arbitrarily varying rates, delay, loss
[4] P. A. Chou, T.Wu, and K. Jain, “Practical network coding”, in 51st Allerton Conf. Communication, Control and Computing, Oct

14. Practical Algorithm
Each nodes sends out packets obtained as a random linear combination of packets stored in its buffer
Each node receives packets which are a linear combinations of source packets and it stores them into a matrix
If the matrix of a node has full rank (h) or a submatrix with full rank (r < h) exists, the node can decode h (or r) packets at the same time

15. Innovative packets or not
When a node receives a packet, it decides whether to store the packet or discard it
Innovative packet: it increases the current rank of the matrix
Non innovative packet: it does not increase the rank of the matrix. It means that the packet contains redundant information and it is not needed to decode the source packets
Hence, non innovative packets are dropped

16. Generations Need to synchronize
All packets related to same source vectors x1,…, xh are said to be in the same generation; h is the generation size
All packets in same generation are tagged with same generation number (one byte - mod is sufficient)
Generations are useful to take into account the differences in data types, generation instants, priorities, etc.

17. Packet Format
At source nodes
At the intermediated
nodes

18. Arriving packets (jitter, loss, variable rate)
Summarizing
Random
Combination
Arriving packets (jitter, loss, variable rate)
edge
Buffer
NODE
Transmission opportunity:
generate packet
Asynchronous transmission

19. Observations about the decoding phase
Block decoding:
Collect h or more packets, hope to invert Gt
Early decoding (recommended):
Perform Gaussian elimination after each RX packet
At every node, detect & discard non-innovative packets
Gt tends to be lower triangular, so it is typically possible to decode x1,…,xk with fewer more than k packets
Much shorter decoding delay than block decoding
Approximately constant, independent of block length h
aij
It can be decoded

20. Costs and benefits Cost: Benefits:
Overhead of transmitting h extra symbols per packet
Example:
h = 50 and field size = 28  overhead ≈ 50/1400 ≈ 3%
Benefits:
Receivers can decode even if
Network topology & encoding functions are unknown
Nodes & edges added & removed in ad hoc manner
Packet loss, node & link failures with unknown locations
Local encoding vectors are time-varying & random

21. Energy efficient broadcasting with NC [5]
RING NETWORK
All nodes are senders; all nodes are receivers
Tnc = # transmissions needed to broadcast with network coding
Tw = # transmissions without network coding
Lemma: Tnc/Tw ≥ ½
Without NC = 6 transmissions (Tw ≥ n - 2 )
With NC = Tnc ≥ (n – 1)/ 2
Achievable by physical piggybacking
[5] J. Widmer, C. Fragouli, and J.-Y. L. Boudec, “Low–complexity energy–efficient broadcasting in wireless ad–hoc networks usign network coding”, in Proc.IEEE Information Theory Workshop, Oct

22. Energy efficient broadcasting with NC
GRID NETWORK
Consider grid network (toroidal)
n = m2 nodes
Lemma: Tnc/Tw ≥ ¾
Without NC = Tw ≥ n2 / 3
With NC = Tnc ≥ n2 / 4
Achievable by physical piggybacking

23. Broadcasting in random networks [6]
At each node v in the graph is associated a forwarding factor, dv.
Source node v transmits its source symbols (or packets)
max{ 1, | dv | } times.
An additional time with probability p = dv - max{ 1, | dv | } if p > 0.
When a node receives an innovative symbol (packet), it broadcasts a linear combination over the span of the received coding vectors
int(dv) times
And TX a further copy with probability p = dv – int(dv) if p > 0
Two heuristics:
dv = k / |N(v)|
dv = k / min |N2(v)| where N2(v) are the number of 2-hops neighbors
[6] C. Fragouli, J. Widmer, and J.-Y. L. Boudec, “A network coding approach to energy efficient broadcasting”, Proceedings of INFOCOM06, April 2006.

24. All to all communication scenario
Simulation results
All to all communication scenario
Energy consumption: number of transmissions and receptions needed to gather all the required packets
Delay: number of time units needed to decode all the required packets

25. NC in multicast communications

26. Summary Network Coding can be used in practice
Packetization
Buffering
Generation
Network Coding is being applied to
Internet, Live broadcast, storage, messaging, peer2peer file sharing (“eMULE of the future”), …
Wireless ad hoc, mobile, and sensor networks
Many open issues

27. Wireless Systems - Lecture
Thank you!