The High, the Low and the Ugly

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The High, the Low and the Ugly

• Muriel Médard

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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. 07Deterministic 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

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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.
  09Erez et al. 10

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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 models ability
  to approximate the wireless networks but show
  that ADT network problems can be captured by the
  algebraic network coding framework

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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
interference

• Algebraic model

broadcast
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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 a(i, ej)
• at the nodes V ß(ej, ej)
• at the destination T e(ej, (T, i))

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System Matrix M A(I F )-1BT

• Linear operations
• Encoding at the source S a(i, ej)
• Decoding at the destination T e(ej, (T, i))

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System Matrix M A(I F )-1BT

• Linear operations
• Coding at the nodes V ß(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)

Broadcast constraint (hyperedge)
F
MAC constraint(addition)
Internal operations(network code)
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System Matrix M A(I F )-1BT

• Input-output relationship of the network

Z X(S) M
Captures rate
Captures network code, topology(Field size as
well)
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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

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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.
• Proof idea1. 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.
• Corollary Random linear network coding achieves
  capacity for a unicast/multicast connection.

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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.

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Extensions to Non-multicast Connections

• Disjoint Multicast The connection is feasible
  if and only if
• Proof ideaIntroduce a super-destination T, and
  apply the multicast min-cut max-flow theorem.

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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

Single multicast
Disjoint multicast
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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

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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

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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?

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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 8 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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