The High, the Low and the Ugly

Muriel Médard

Collaborators

• Nadia Fawaz, Andrea Goldsmith, Minji Kim, Ivana Maric

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Regimes of SNR

• Recent work has considered different SNR regimes

• High SNR:– Deterministic models– Analog coding models

• Low SNR:– Hypergraph models

• Other SNRs: the ugly

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Wireless Network

• Open problem: capacity & code construction for wireless relay networks– Channel noise– Interference

• [Avestimehr et al. ‘07]“Deterministic model” (ADT model)– Interference – Does not take into account channel noise– In essence, high SNR regime

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

Model as error free

links

R1

R2

Y(e1)

Y(e2)e1

e2

e3Y(e3)

Y(e3) = β1Y(e1) + β2Y(e2)

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. [Avestimehr et al. ‘07]

• Matroidal [Goemans et al. ‘09]

• Code construction algorithms [Amaudruz et al. ‘09][Erez et al. ‘10]

Our Contributions

• Connection to Algebraic Network Coding [Koetter and Médard. ‘03]:– Use of higher field size– Model broadcast constraint with hyper-edges– Capture ADT network problem with a single system matrix M

• Prove that min-cut of ADT networks = rank(M)• Prove Min-cut Max-flow for unicast/multicast holds• Extend optimality of linear operations to non-multicast sessions• Incorporate failures and erasures• Incorporate cycles

– Show that random linear network coding achieves capacity– Do not prove/disprove 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

ADT Network Model

• Original ADT model:– 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:

Algebraic Framework

• X(S, i): source process i

• Y(e): process at port e

• Z(T, i): destination process i

• Linear operations – at the source S: α(i, ej)

– at the nodes V: β(ej, ej’)

– at the destination T: ε(ej, (T, i))

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

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e6

e7

e8

e9

e10

e11

e12

a

b

c

df

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

e2

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a

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df

F =

Broadcast constraint (hyperedge)

MAC constraint(addition)

Internal operations(network code)

System Matrix M = A(I – F )-1BT

Z = X(S) M

e1

e2

e3

e4

e5

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• Input-output relationship of the network

Captures rate

Captures network code, topology(Field size as well)

Theorem: Min-cut of ADT Networks

• From the original paper by Avestimehr et al. – 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 to optimizing over all cuts– System matrix captures the structure of the network– Constructive: the assignment of variables gives a network code

Min-cut Max-flow Theorem

• For a unicast/multicast connection from source S to destination T, the following are equivalent:

– A unicast/multicast connection of rate R is feasible.– mincut(S,Ti) ≥ R for all destinations Ti.– There exists an assignment of variables such that M is invertible.

1. Proof idea:1. & 2. equivalent by previous work. 3.→1. If M is invertible, then connection has been established.1.→3. If connection established, M = I. Therefore, M is invertible.

2. Corollary: Random linear network coding achieves capacity for a unicast/multicast connection.

Extensions to Non-multicast Connections

• [Multiple multicast] Multiple sources S1 S2 … Sk wants to transmit to all destinations T1 T2… TN

– Connection feasible if and only if mincut({S1 S2 … Sk}, Tj) ≥ sum of rate from sources S1 S2 … Sk

• Proof idea: Introduce a super-source S, and apply the multicast min-cut max-flow theorem.

Extensions to Non-multicast Connections

• [Disjoint Multicast] The connection is feasible if and only if:

• Proof idea:Introduce a super-destination T, and apply the multicast min-cut max-flow theorem.

Extensions to Non-multicast Connections

• [Two-level Multicast] A set of destinations, Tm, participate in a multicast connection; rest of the destinations, Td, in a disjoint multicast. The connection is feasible if and only if: – Tm: Satisfy single multicast connection requirement.– Td: Satisfy disjoint multicast connection requirement.– Random linear network coding at intermediate nodes; but a

carefully chosen encoding matrix at source achieves capacity

Disjoint multicast

Single multicast

Incorporating Erasures in ADT network

• Wireless networks: stochastic in nature– Random erasures occur

• ADT network– Models wireless deterministically with parallel bit-pipes– Min-cut as well as previous code construction algorithm

needs to be recomputed every time the network changes

• Algebraic framework:– Robust against some set of link failures (network code

will remain successful regardless of these failures)– The time average of rank(M) gives the true min-cut of

the network– Applies to the connections described previously

Incorporating cycles in ADT network

• Wireless networks intrinsically have bi-directional links; therefore, cycles exists

• ADT network mode– A directed network without cycles: links from the

source to the destinations

• Algebraic framework– To incorporate cycles, need a notion of time (causal);

therefore, introduce delay on links D– Express network processes in power series in D– The same theorems as for delay-less network apply

Network Coding and ADT

• ADT network can be expressed with Algebraic Network Coding Formulation [Koetter and Médard ‘03].– 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 = rank(M)• Prove Min-cut Max-flow for unicast/multicast holds

• Extend optimality of linear operations to non-multicast sessions– Disjoint multicast, Two-level multicast, multiple source multicast,

generalized min-cut max-flow theorem

• Show that random linear network coding achieves capacity

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

• BUT IS IT THE RIGHT MODEL?

Different Types of SNR

• Diamond network [Schein]

• As a increases: the gap between analog network coding and cut set increases [Avestimehr, Diggavi & Tse]

• In networks, increasing the gain and the transmit power are not equivalent, unlike in point-to-point links

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Let SNR Increase with Input Power

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Analog Network Coding is Optimal at High SNR

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What about Low SNR?

• Consider again hyperedges

• At high SNR, interference was the main issue and analog network coding turned it into a code

• At low SNR, it is noise

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What about Low SNR?

• Consider again hyperedges

• At high SNR, interference was the main issue and analog network coding turned it into a code

• At low SNR, it is noise

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Peaky Binning Signal

• Non-coherence is not bothersome, unlike the high-SNR regime

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What Min-cut?

• Open question: Can the gap to the cut-set upper-bound be closed?

• An ∞ capacity on the link R-D would be sufficient to achieve the cut like in SIMO

• Because of power limit at relay, it cannot make its observation fully available to destination.

• Conjecture: cannot reach the cut-set upper-bound

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What About Other Regimes?

• The use of hyperedges is important to take into account dependencies

• In general, it is difficult to determine how to proceed (see the difficulties with the relay channel) – the ugly

• Equivalence leads to certain bounds for multiple access and broadcast channels, but these bounds may be loose

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