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Network coding techniques


Network coding is a particular in-network data processing technique that ... [2] S.-Y. R. Li, R. W. Yeung, and N. Cai, 'Linear network Coding', IEEE Trans. ... – PowerPoint PPT presentation

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Title: Network coding techniques

Network coding techniques
Wireless Systems - Lecture
  • Elena Fasolo
  • PhD Student - SIGNET Group

March, 7th 2004
Definition of network coding (NC)
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
  • 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

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

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

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

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
  • With network coding, only 3 transmissions are

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
  • 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
  • Graph (V,E) having unit capacity edges
  • Sender s in V, set of receivers Tt, in V

Source node of h symbols
Intermediate node
Destination node
Linear coding phase
Local encoding vector
Transmitted symbol
Global encoding vector
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.
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
Theory vs. Practice
  • Theory
  • 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
  • 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

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
  • Buffering Allows asynchronous packets arrivals
    departures with arbitrarily varying rates, delay,

4 P. A. Chou, T.Wu, and K. Jain, Practical
network coding, in 51st Allerton Conf.
Communication, Control and Computing, Oct. 2003.
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 lt h) exists, the node
    can decode h (or r) packets at the same time

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

  • 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
  • Generations are useful to take into account the
    differences in data types, generation instants,
    priorities, etc.

Packet Format
At source nodes
At the intermediated nodes
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
  • 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

It can be decoded
Costs and benefits
  • Cost
  • Overhead of transmitting h extra symbols per
  • Example
  • h 50 and field size 28 ? overhead 50/1400
  • 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
  • Local encoding vectors are time-varying random

Energy efficient broadcasting with NC 5
  • 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,
Lowcomplexity energyefficient broadcasting in
wireless adhoc networks usign network coding,
in Proc.IEEE Information Theory Workshop, Oct.
Energy efficient broadcasting with NC
  • 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

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
  • max 1, dv times.
  • An additional time with probability p dv - max
    1, dv if p gt 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 gt 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
Simulation results
All to all communication scenario
Energy consumption number of transmissions and
receptions needed to gather all the required
Delay number of time units needed to decode all
the required packets
NC in multicast communications
  • 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

Thank you!
Wireless Systems - Lecture