Thrust 2Network Coding and

Forwarding

Team

ν Muriel Medardν Michelle Effrosν Andrea Goldsmithν Ralf Koetterν Asu Ozdaglarν Atilla Eryilmazν Ivana Maric

ν Network coding has proven to be an effective way to increasenetwork capacity over traditional routing schemes for multicast.

ν The advent of network coding enables the formulation of multicastas a flow problem.

ν Resource allocation problem with network coding can be cast as asimple optimization problem, which admits distributed solutionmethods.

ν Current work in network coding:θ Integration of physical layer considerations with network coding

and optimizationθ Integration of source coding, network coding, channel codingθ Network coding in noisy situations where the algebraic structure is

combined with estimation approaches in order to create “soft”network coding

θ Network coding for error correction in the presence of adversarialusers

Vision

This Talk

ν Network Coding and Optimizationθ Zhao, Medard, Ozdaglar

ν Network Coding and Relayingθ Katti, Maric, Goldsmith, Katabi, Medard

ν Network Coding, Optimization, and MultipleAccessθ Parandehgheibi, Eryilmaz, Medard, Ozdaglar

ν Posters

Network Coding in Wireless Networks

ν Network coding isparticularlyadvantageous inwireless networksdue to thebroadcast nature ofthe medium.

s

t1t2

Multicast: s ! { t1,t2 }

s1

s2

t1

t2

Unicasts: s1 ! t1 & s2 ! t2

a b

a + b

a b

bb

aa

a + b

Network Coding in Wireless Networks

s3

s2

s1

t1

t2

t3

• Three Unicasts:

si ! ti, i = 1,2,3.a

a

a

bb

b

c

c

c

… ♦

Time slots

♦ ♦ ♦

a+b+c

a+b+

c

a+b+c

• Network Coding achievesa rate of 1/4 bits/slot perflow.

• Routing can only achieve arate of 1/6 bits/slot per flow.

Decentralized Subgraph Optimization

ν The problem of subgraph optimization formulticast connections with network codingcan be solved using the subgradient methodin a distributed manner.

ν We examine the rate of convergence,complexity and implementation issues of thismethod.

ν We focus on the problem of min-energymulticast in both static and dynamic multi-hopwireless networks.

j

Problem formulation

ν Our Approach:θ Formulate the minimum subgraph problem as a linear

programming problem;θ Solve it by using subgradient methods on its dual;θ Recovering the primal solution from the dual solutions.

i

ai,j : Link cost

xi,j(t) : Virtual flow over (i,j) for terminal t

zi,j : Actual flow over (i,j)

F(t) : Feasible set of x(t)

Theoretical Results

ν Result 1: Incremental subgradient method on the dualproblem yields linear convergence rate with appropriatelychosen step-sizes.

ν Result 2: The primal solution recovered from the dualsolutions also converges linearly to the optimal solution,given that the dual converges linearly.

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Primal Dual

Simulation Results (1)

The subgradientmethod providessignificant reductionsin multicast energyas compared tocentralized routingalgorithms (forexample, the MIPalgorithm) after just afew iterations.

Simulation Results (2)

The algorithm isrobust to changes inthe network and canconverge to newoptimal solutionsquickly, as long asthe rate of change inthe network is slowas compared to thespeed of computationand transmission.

Joint Relaying and Network Coding inWireless Networks

source1

source2

relay

destination1

destination2

messageW1

messageW2

W2

W1

a

b

a+b

a+b

ν Relaying: Network layer time-shares the relay between different flowsθ No cooperation between multiple flows

ν Joint Relaying and Network Coding: “Analog network coding”, networkcoding done over physical signals in the wireless channelθ Significantly increases wireless throughput

ν Analyzed channel models:

Multicast and Unicast in Wireless Settings

ν Allowed broadcasting and interference

⇒ channel outputs:Y3=X1+X2+Z3

Y4=X1+X2+X3+Z4

Y5=X1+X2+X3+Z5

source 1

source 2

relay

node 5

node 4

X1

X2

Y3

Y4

Y5

X3

messageW1

messageW2

Multicast relay channel Two-way relay channel

messageW1

s1 s2

messageW2

relay

X1

X2

Y3

Our Approach1) Consider routing ⇒ signal at the relay:

X3=f(X1) for t=[0,τ] relaying for source 1

X3=f(X2) for t=[τ,T] relaying for source 2

2) Consider a simple network coding scheme: X3=αY3

• This combines two data streams at the relay as:X3 = αY3 = α(X1+X2+Z3)

• Evaluate the achievable rates

compare

Goals:

ν Demonstrate that combining data streams helps Motivates (simple!) coding for wireless networksν Generalize to large networks

• Determine outer bounds

Multicast with Network Coding inWireless Networks

ν Goal:θ To maximize multicasting throughput

considering wireless channel capacityconstraints for Gaussian MultipleAccess and Broadcast transmissions

θ Provide a distributed implementation

ν Previous works:θ [LRMKKHAZ06] No Interferenceθ [XY06] CDMA-based limited

interferenceθ Collision-based limited interference

s

t1t2

MAC

MACMAC

BCBC

BC

CDMA based Collision based

Layered Network Modelν Given a network with directed acyclic graph, add virtual

nodes to create level sets, where levels are scheduled totransmit in order.

L1 L2 L3 L4 ..RXIdleIdleRxIdleIdleL4

..TXRXIdleTXRXIdleL3

..IdleTXRXIdleTXRXL2

..IdleIdleTXIdleIdleTXL1

…T6T5T4T3T2T1

s

t1

t2

Problem Formulationν Under broadcasting limited to common information.

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DistributedImplementationby SubgradientMethod

Extension to Degraded Broadcastν Allowing for independent information to be sent to different

receivers.

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ZBC, ZMAC are not independent of rates ziJ and depend ondecoding scheme.

Future work:

• Considering Undirected graphmodel

• Optimal Transmission Schedule

• Power Control

Hypergraph: H = (N, A)

{ }}{\,|),( iNJNiJiA !"=

Hyperedges:

Posters

ν "Joint relaying and network coding in wirelessnetworks,"I. Maric, M. Medard, A. Goldsmith

ν “Network coding on noisy links,”R. Koetter, F. Kschischang, S. Yang

ν "On joint source and network coding,”A. Lee, M. Medard, K. Z. Haigh, S. Gowanand P. Rubel.