the high, the low and the ugly muriel médard. collaborators nadia fawaz, andrea goldsmith, minji...
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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
e5
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
e3
e4
e5
e6
e7
e8
e9
e10
e11
e12
a
b
c
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
e6
e7
e8
e9
e10
e11
e12
a
b
c
df
• 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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