Network Coding and Reliable Communications Group

Algebraic Network Coding Approach to Deterministic Wireless Relay Networks

MinJi Kim, Muriel Médard

Network Coding and Reliable Communications Group

Wireless Network• Open problem: capacity & code construction for wireless relay networks

– Channel noise– Interference

• High SNR– Noise → 0 & large gain– Large transmit power

• High SNR rate region:– TDM, Noise free additive channel [Ray et al. ‘03]– Note: This holds for higher field size (not just binary) [Ray et al. ‘03]

• [Avestimehr et al. ‘07]“Deterministic model” (ADT model)– Interference – Model noise deterministically– Use binary field– In essence, high SNR regime

Model as error free

links

R1

R2

Y(e1)

Y(e2)e1

e2

e3Y(e3)

Y(e3) = Y(e1) + Y(e2)

R1

R2

log(1+P2/N)

log(1+P1/N)

log(1+P2/(P1+N))

log(1+P1/(P2+N))

R1

R2

High SNR

Network Coding and Reliable Communications Group

ADT Network Background• Min-cut: minimal rank of an incidence matrix of a certain cut

between the source and destination [Avestimehr et al. ‘07]– Requires optimization over a large set of matrices– Min-cut Max-flow Theorem holds for unicast/multicast sessions

• Matroidal [Goemans et al. ’09]– Algebraic Network Coding is also Matroidal [Dougherty et al. ’07]

• Unicast code construction algorithms [Goemans et al. ‘09][Yazdi & Savari ‘09][Amaudruz & Fragouli ‘09]

• Multicast code construction algorithms [Erez et al. ‘10][Ebrahimi & Fragouli ‘10]

Network Coding and Reliable Communications Group

Our Contributions• Connection to Algebraic Network Coding [Koetter & Médard ‘03]:

– Use of higher field size [Ray et al. ’03]1.

2. Can’t achieve capacity for multicast with justbinary field [Feder et al. ’03][Rasala-Lehman & Lehman ’04][Fragouli et al. ‘04]

3. [Jaggi et al. ‘06] “permute-and-add”:Show that network codes in higher field size Fq can be converted to binary-vector code in (F2)n without loss in performance

R1

R2

log(1+P2/N)

log(1+P1/N)

log(1+P2/(P1+N))

log(1+P1/(P2+N))

R1

R2

High SNR

Network Coding and Reliable Communications Group

Our Contributions• Connection to Algebraic Network Coding [Koetter & Médard ‘03]:

– Use of higher field size [Ray et al. ’03]– Model broadcast constraint with hyper-edges– Capture ADT network problem with a single system matrix M

• Prove that min-cut of ADT networks = max rank(M)• Prove Min-cut Max-flow for unicast/multicast holds• Extend optimality of linear operations to certain non-multicast sessions• Incorporate failures and erasures [Lun et al. ‘04]• Incorporate cycles

– Show that random linear network coding achieves capacity [Ho et al. ‘03]

– Do not address ADT network model’s ability to approximate the wireless networks; but show that ADT network problems can be captured by the algebraic network coding framework

Network Coding and Reliable Communications Group

ADT Network Model• Original ADT model (Binary field)

– Broadcast: multiple edges (bit pipes) from the same node– Interference: additive MAC over binary field

Higher SNR: S-V1

Higher SNR: S-V2

broadcast

interference• Algebraic model:

Network Coding and Reliable Communications Group

Algebraic Framework

• Assume higher field size• X(S, i): source process i• Y(e): process at port e• Z(T, i): destination process i• Assume linear operations

– at the source S: α(i, ej)

– at the nodes V: β(ej, ej’)

– at the destination T: ε(ej, (T, i))

Network Coding and Reliable Communications Group

System Matrix M= A(I – F )-1BT

• Linear operations

– Encoding at the source S: α(i, ej)

– Decoding at the destination T: ε(ej, (T, i))

e1

e2

e3

e4

e5

e6

e7

e8

e9

e10

e11

e12

a

b

c

df

Network Coding and Reliable Communications Group

System Matrix M= A(I – F )-1BT

• Linear operations – Coding at the nodes V: β(ej, ej’)

– F represents physical structure of the ADT network– Fk: non-zero entry = path of length k between nodes exists– (I-F)-1 = I + F + F2 + F3 + … : connectivity of the network

(impulse response of the network)

e1

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a

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F =

Broadcast constraint (hyperedge)

MAC constraint(addition)

Internal operations(network code)

Linear code with some coefficients fixed by the network!

Network Coding and Reliable Communications Group

System Matrix M = A(I – F )-1BT

Z = X(S) M

e1

e2

e3

e4

e5

e6

e7

e8

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e10

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a

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df

• Input-output relationship of the network

Captures rate

Captures network code, topology(Field size as well)

Network Coding and Reliable Communications Group

Theorem: Min-cut of ADT Networks

• From [Avestimehr et al. ‘07]– Requires optimizing over ALL

cuts between S and T– Not constructive: assumes infinite

block length, internal node operations not considered

• Show that the rank of M is equivalent– System matrix captures the structure of the network– Constructive: the assignment of variables gives a network code

e1

e2

e3

e4

e5

e6

e7

e8

e9

e10

e11

e12

a

b

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df

Network Coding and Reliable Communications Group

Min-cut Max-flow Theorem• For a unicast/multicast connection from source S to destination T, the

following are equivalent:1. A unicast/multicast connection of rate R is feasible2. mincut(S,Ti) ≥ R for all destinations Ti

3. There exists an assignment of variables such that M is invertible• Proof idea:

1. & 2. equivalent by previous work3.→1. If M is invertible, then connection has been established1.→3. If connection established, M = I. Therefore, M is invertible

• Alternate proof of sufficiency of linear operations for multicast in ADT networks [Avestimehr et al. ‘07]

Network Coding and Reliable Communications Group

Corollaries• Extend Min-cut Max-flow theorem to other connections:

– [Multiple multicast]: Sources S1 S2 … Sk wants to transmit to all destinations T1 T2… TN

– [Disjoint multicast]:– [Two-level multicast]: Two sets of destinations, a set Tm for multicast connection, another set Td for

disjoint multicast connection.

S2

T1

TN

Network

S1

Sk

S

T1

T3

Network

a, b, c, d

T2

a

b, c

d

Destination wants

S

T1

T3

a, b, c, d

a, b, c, d

a, b, c, d

Destination wants

T4

T6

T5

a

b, c

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Network

Network Coding and Reliable Communications Group

Corollaries• Extend Min-cut Max-flow theorem to other connections:

– [Multiple multicast]: Sources S1 S2 … Sk wants to transmit to all destinations T1 T2… TN

– [Disjoint multicast]:– [Two-level multicast]: Two sets of destinations, a set Tm for multicast connection, another set Td for disjoint multicast connection.

• Random linear network coding achieves capacity for unicast, multicast, and above connections.• Extend results to ADT networks with…

– Delay– Cycles– Erasures/Failures

Network Coding and Reliable Communications Group

Conclusions• ADT network can be expressed with Algebraic Network Coding Formulation

– Use of higher field size– Model broadcast constraint with hyper-edge– Capture ADT network problem with a single system matrix M

• Prove an algebraic definition of min-cut = max rank(M)• Prove Min-cut Max-flow for unicast/multicast holds • Show that random linear network coding achieves capacity• Extend optimality of linear operations to non-multicast sessions

– Disjoint multicast, Two-level multicast, multiple source multicast, generalized min-cut max-flow theorem

– Random linear network coding achieves capacity

• Incorporate delay and failures (allows cycles within the network)