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Network coding techniques

Elena Fasolo

PhD Student - SIGNET Group

fasoloel@dei.unipd.it

Wireless Systems - Lecture

March, 7th 2004

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Definition of network coding (NC)

DEFINITIONNetwork 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

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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

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The canonical example (I)

Without network coding Simple store and forward Multicast rate of 1.5 bits per time unit

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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

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NC and wireless communications

b1 b2b2

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

A A BB CC

b1

A BC

r

(a)

(b) (c)

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Linear network coding

[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.

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

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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

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Linear coding phase

Global encoding vector

Local encoding vectorTransmitted symbol

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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

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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

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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 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

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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. 2003.

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Practical Algorithm

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

Each nodes sends out packets obtained as a random linear combination of packets stored in its buffer

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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

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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 256 - is sufficient)

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

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Packet Format

At source nodes

At the intermediated nodes

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Summarizing

Random Combination

Arriving packets (jitter, loss,

variable rate)

edge

edge

edge

edge

Buffer

NODE

Transmission opportunity: generate packet

Asynchronous transmission

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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

0aij

It can be decoded

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Costs and benefits

Cost: 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

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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

RING NETWORK

[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.

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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

GRID NETWORK

Energy efficient broadcasting with NC

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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.

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Simulation results

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

All to all communication scenario

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NC in multicast communications

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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

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Thank you!

Wireless Systems - Lecture