Wireless Systems - Lecture
Network coding techniques
Network coding techniques
Elena Fasolo
Elena Fasolo
PhD Student - SIGNET Group
fasoloel@dei.unipd.it
March, 7th 2004
Definition of network coding (NC)
Network coding techniques
Elena Fasolo
DEFINITION
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
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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.
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Improves the performance in data broadcasting
Most suitable setting: all to all communications
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Communication networks
TERMINOLOGY
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Network coding techniques
Elena Fasolo
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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)
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WX has capacity equal to 2
The canonical example (I)
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Without network coding
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Network coding techniques
Elena Fasolo
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Simple store and forward
Multicast rate of 1.5 bits per time unit
The canonical example (II)
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With network coding
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Network coding techniques
Elena Fasolo
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X-OR  is one of the
simplest form of data
coding
Multicast rate of 2 bits
per time unit
Disadvantages
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Coding/decoding scheme
has to be agreed upon
beforehand
NC and wireless communications
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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
Network coding techniques
Elena Fasolo
(b)
b1
A
b2
A
C
B
(a)
C
r
b1
A
B
(c)
b2
C
B
Linear network coding
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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
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Network coding techniques
Elena Fasolo
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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. 2003.
Theoretical model for linear NC
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Network coding techniques
Elena Fasolo
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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
Linear coding phase
Network coding techniques
Elena Fasolo
Transmitted symbol
Local encoding vector
Global encoding vector
Network coding techniques
Elena Fasolo
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
Inverting Gt
Network coding techniques
Elena Fasolo
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Gt will be invertible with high probability if local
encoding vectors are random and the field size
is sufficiently large [3]
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P = 1 - |F| (where |F| is the cardinality of the finite field
of coefficients)
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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
Theory vs. Practice
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Theory:
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Network coding techniques
Elena Fasolo
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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:
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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
Practical Random NC
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Main idea [4]:
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Select the linear coefficients in a finite field of
opportune size in a random way
Send the encoding vector within the same packet
Network coding techniques
Elena Fasolo
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Packetization: Header removes need for centralized
knowledge of graph topology and encoding/decoding
functions
Nodes stores within their buffers the received packets
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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. 2003.
Network coding techniques
Elena Fasolo
Practical Algorithm
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Each nodes sends out packets obtained as a random
linear combination of packets stored in its buffer
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Each node receives packets which are a linear
combinations of source packets and it stores them
into a matrix
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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
Innovative packets or not
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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
Network coding techniques
Elena Fasolo
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Generations
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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
256 - is sufficient)
 Generations are useful to take into account the
differences in data types, generation instants,
priorities, etc.
Network coding techniques
Elena Fasolo
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Packet Format
Network coding techniques
Elena Fasolo
At source nodes
At the intermediated
nodes
Summarizing
Transmission
opportunity:
generate packet
Random
Combination
Network coding techniques
Elena Fasolo
edge
Arriving packets
(jitter, loss,
variable rate)
edge
Buffer
NODE
Asynchronous
transmission
Observations about the decoding phase
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Block decoding:
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Early decoding (recommended):
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Network coding techniques
Elena Fasolo
Collect h or more packets, hope to invert Gt
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
aij
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0
It can be decoded
Much shorter decoding delay than block decoding
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Approximately constant, independent of block length h
Costs and benefits
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Cost:
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Overhead of transmitting h extra symbols per packet
Example:
h = 50 and field size = 28  overhead ≈ 50/1400 ≈ 3%
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Network coding techniques
Elena Fasolo
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Benefits:
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Receivers can decode even if
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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
Energy efficient broadcasting with NC [5]
RING NETWORK
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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
Network coding techniques
Elena Fasolo
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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. 2004.
Energy efficient broadcasting with NC
GRID NETWORK
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Consider grid network
(toroidal)
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Network coding techniques
Elena Fasolo
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Lemma: Tnc/Tw ≥ ¾
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n = m2 nodes
Without NC = Tw ≥ n2 / 3
With NC = Tnc ≥ n2 / 4
Achievable by physical
piggybacking
Broadcasting in random networks [6]
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At each node v in the graph is associated a forwarding
factor, dv.
Source node v transmits its source symbols (or packets)
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Network coding techniques
Elena Fasolo
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When a node receives an innovative symbol (packet),
it broadcasts a linear combination over the span of the
received coding vectors
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max{ 1, | dv | } times.
An additional time with probability p = dv - max{ 1, | dv | } if p > 0.
int(dv) times
And TX a further copy with probability p = dv – int(dv) if p > 0
Two heuristics:
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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.
Simulation results
Network coding techniques
Elena Fasolo
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
Network coding techniques
Elena Fasolo
NC in multicast communications
Summary
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Network Coding can be used in practice
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Network Coding is being applied to
Network coding techniques
Elena Fasolo
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Packetization
Buffering
Generation
Internet, Live broadcast, storage, messaging,
peer2peer file sharing (“eMULE of the future”), …
Wireless ad hoc, mobile, and sensor networks
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Many open issues
Network coding techniques
Elena Fasolo
Wireless Systems - Lecture
Thank you!