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intro interference networks bounds conclusion
Graphs and Algebra in Modern Communication.
Felice ManganielloSchool of Math and Stat Sciences @ Clemson University
Cybersecurity Research Lab @ Ryerson University
May 26, 2020
Joint work withKschischang (UofT), Ravagnani (TU/e), Savary (Clemson)
Kai (UMichigan), Pedro (UMaryland), Paige (U of Mary Washington),
Kimberly (Bowdoin College) and Nathan (Haverford College)
Partially funded by NSF Grant No. DMS:1547399.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion
1 Introduction
2 Interference NetworksDefinitionMultiple Unicast networkLinear Achievability and complexity
3 Achievability bounds for Multiple Unicast NetworksLower BoundUpper Bounds
4 Conclusions
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion
Noisy-Channel Coding Theorem (1948)
Theorem (Noisy-Channel Coding Theorem - Shannon - 1948)
“Any channel, however affected by noise, possesses a specificchannel capacity - a rate of conveying information that can neverbe exceeded without error, but that can always be attained with anarbitrarily small probability of error.”
Solved: Turbo codes (LTE networks), Polar & spatially-coupledLDPC codes (5G networks)
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Noisy-Channel Coding Theorem (1948)
Theorem (Noisy-Channel Coding Theorem - Shannon - 1948)
“Any channel, however affected by noise, possesses a specificchannel capacity - a rate of conveying information that can neverbe exceeded without error, but that can always be attained with anarbitrarily small probability of error.”
Solved: Turbo codes (LTE networks), Polar & spatially-coupledLDPC codes (5G networks)
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion
Noisy-Channel Coding Theorem (1948)
Theorem (Noisy-Channel Coding Theorem - Shannon - 1948)
“Any channel, however affected by noise, possesses a specificchannel capacity - a rate of conveying information that can neverbe exceeded without error, but that can always be attained with anarbitrarily small probability of error.”
Solved: Turbo codes (LTE networks), Polar & spatially-coupledLDPC codes (5G networks)
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion
Networks - a graph perspective
A network is a directed acyclic graph N = (V, E ,S,R,Fq).
• Sources: nodes with no incoming edges, S ( V.
• Sinks: nodes with no outgoing edges, R ( V.
• Edges represent perfect unit capacity channels.
• each sink R ∈ R demands messages from DR ⊆ S.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion
Networks - a graph perspective
A network is a directed acyclic graph N = (V, E ,S,R,Fq).
• Unicast Problem: |S| = |R| = 1 and DR = S.
• Multicast Problem: |R| ≥ 1 and DR = S for all R ∈ R.
• Multiple Unicast problem: S = {S1, . . . ,Sn},R = {R1, . . . ,Rn}, and DRi
= {Si}.Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Unicast Network
• Communication rate: ρ ≤ mincut(S ,R).
• Menger’s Theorem: mincut(S ,R) = maximum number ofpairwise edge–disjoint paths.
• Routing maximizes R.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion
Unicast Network
• Communication rate: ρ ≤ mincut(S ,R).
• Menger’s Theorem: mincut(S ,R) = maximum number ofpairwise edge–disjoint paths.
• Routing maximizes R.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion
Unicast Network
• Communication rate: ρ ≤ mincut(S ,R).
• Menger’s Theorem: mincut(S ,R) = maximum number ofpairwise edge–disjoint paths.
• Routing maximizes R.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion
Multicast network [Li et al., 2003, Koetter and Medard, 2003]
Theorem (Linear Network Multicasting Theorem)
Let N = (V, E ,S,R,Fq). A multicast rate of
minR∈R
mincut(S,R)
is achievable, for sufficiently large q, with linear network coding.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion
Multicast network [Li et al., 2003, Koetter and Medard, 2003]
Theorem (Linear Network Multicasting Theorem)
Let N = (V, E ,S,R,Fq). A multicast rate of
minR∈R
mincut(S,R)
is achievable, for sufficiently large q, with linear network coding.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion
Multicast network [Li et al., 2003, Koetter and Medard, 2003]
Theorem (Linear Network Multicasting Theorem)
Let N = (V, E ,S,R,Fq). A multicast rate of
minR∈R
mincut(S,R)
is achievable, for sufficiently large q, with linear network coding.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion
Multicast network [Li et al., 2003, Koetter and Medard, 2003]
Theorem (Linear Network Multicasting Theorem)
Let N = (V, E ,S,R,Fq). A multicast rate of
minR∈R
mincut(S,R)
is achievable, for sufficiently large q, with linear network coding.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion
Multicast network [Li et al., 2003, Koetter and Medard, 2003]
Theorem (Linear Network Multicasting Theorem)
Let N = (V, E ,S,R,Fq). A multicast rate of
minR∈R
mincut(S,R)
is achievable, for sufficiently large q, with linear network coding.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion
Multicast network [Li et al., 2003, Koetter and Medard, 2003]
Theorem (Linear Network Multicasting Theorem)
Let N = (V, E ,S,R,Fq). A multicast rate of
minR∈R
mincut(S,R)
is achievable, for sufficiently large q, with linear network coding.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Insufficiency of LNC [Dougherty et al., 2005]
Theorem
There exists an solvable network that has no linear solution overany finite field and any vector dimension.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
1 Introduction
2 Interference NetworksDefinitionMultiple Unicast networkLinear Achievability and complexity
3 Achievability bounds for Multiple Unicast NetworksLower BoundUpper Bounds
4 Conclusions
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Networks and Interference
“Interference is a major impairment to the reliable communicationin multi-user wireless networks, due to the broadcast andsuperposition nature of wireless medium.” [Zhao et al., 2016]
Let N be a network with S sources setand R receivers set. A network hasinterference if
• DR 6= S for some R ∈ R• for some R ∈ R there is a paths
between S /∈ DR and R.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Networks and Interference
“Interference is a major impairment to the reliable communicationin multi-user wireless networks, due to the broadcast andsuperposition nature of wireless medium.” [Zhao et al., 2016]
Let N be a network with S sources setand R receivers set. A network hasinterference if
• DR 6= S for some R ∈ R• for some R ∈ R there is a paths
between S /∈ DR and R.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Networks and Interference
“Interference is a major impairment to the reliable communicationin multi-user wireless networks, due to the broadcast andsuperposition nature of wireless medium.” [Zhao et al., 2016]
Let N be a network with S sources setand R receivers set. A network hasinterference if
• DR 6= S for some R ∈ R• for some R ∈ R there is a paths
between S /∈ DR and R.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Multiple Unicast network
Definition
A multiple unicast network is a network N such that |S| = |R| andDRi
= {Si} for i = 1, . . . , |S|.
Communication strategy:
• 1 round → no interference-freecommunication possible.
• multiple rounds → time sharing.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Multiple Unicast network
Definition
A multiple unicast network is a network N such that |S| = |R| andDRi
= {Si} for i = 1, . . . , |S|.
Communication strategy:
• 1 round → no interference-freecommunication possible.
• multiple rounds → time sharing.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Multiple Unicast network
Definition
A multiple unicast network is a network N such that |S| = |R| andDRi
= {Si} for i = 1, . . . , |S|.
Communication strategy:
• 1 round
→ no interference-freecommunication possible.
• multiple rounds → time sharing.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Multiple Unicast network
Definition
A multiple unicast network is a network N such that |S| = |R| andDRi
= {Si} for i = 1, . . . , |S|.
Communication strategy:
• 1 round → no interference-freecommunication possible.
• multiple rounds → time sharing.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Multiple Unicast network
Definition
A multiple unicast network is a network N such that |S| = |R| andDRi
= {Si} for i = 1, . . . , |S|.
Communication strategy:
• 1 round → no interference-freecommunication possible.
• multiple rounds → time sharing.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Multiple Unicast network (cont’d)
Communication strategy:
• multiple rounds → time sharing.
• 1 round → interference alignment,meaning use of subspaces tocommunicate without interference
• original paper[Cadambe and Jafar, 2008].
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Multiple Unicast network (cont’d)
Communication strategy:
• multiple rounds → time sharing.
• 1 round → interference alignment,meaning use of subspaces tocommunicate without interference
• original paper[Cadambe and Jafar, 2008].
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Multiple Unicast network (cont’d)
Communication strategy:
• multiple rounds → time sharing.
• 1 round →
interference alignment,meaning use of subspaces tocommunicate without interference
• original paper[Cadambe and Jafar, 2008].
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Multiple Unicast network (cont’d)
Communication strategy:
• multiple rounds → time sharing.
• 1 round → interference alignment,meaning use of subspaces tocommunicate without interference
• original paper[Cadambe and Jafar, 2008].
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Multiple Unicast network (cont’d)
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Multiple Unicast network (cont’d)
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Multiple Unicast network (cont’d)
Communication strategy:
• multiple rounds → time sharing.
• 1 round → interference alignment,meaning use of subspaces tocommunicate without interference
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Multiple Unicast network - Notation
• si number of antennas available tosource Si and s =
∑Ni=1 si .
• ti number of antennas available tosource Ri and t =
∑Ni=1 ti .
• The adjacency matrix H ∈ {0, 1}t×s ,
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Multiple Unicast network - Notation
• si number of antennas available tosource Si and s =
∑Ni=1 si .
• ti number of antennas available tosource Ri and t =
∑Ni=1 ti .
• The adjacency matrix H ∈ {0, 1}t×s ,
H =
H11
H22
HNN
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Multiple Unicast network - Notation
• si number of antennas available tosource Si and s =
∑Ni=1 si .
• ti number of antennas available tosource Ri and t =
∑Ni=1 ti .
• The adjacency matrix H ∈ {0, 1}t×s ,
H =
H11 H12 . . . H1N
H21 H22 . . . H2N...
......
HN1 HN2 . . . HNN
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Example
H =
1 0 0 0 0 00 1 0 0 1 01 0 1 0 0 00 0 0 1 0 00 0 0 1 1 00 1 0 0 0 1
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Linear Encoders and Decoders
E =
E1 0 . . . 00 E2 . . . 0...
...0 0 . . . EN
D =
D1 0 . . . 00 D2 . . . 0...
...0 0 . . . DN
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Linear Encoders and Decoders
E =
E1 0 . . . 00 E2 . . . 0...
...0 0 . . . EN
D =
D1 0 . . . 00 D2 . . . 0...
...0 0 . . . DN
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Linear Network Communication
mi ,1...
mi ,`i
j 6=i
mj ,1...
mj ,`j
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Linear Network Communication
Ei
mi ,1...
mi ,`i
j 6=i Ej
mj ,1...
mj ,`j
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Linear Network Communication
HiiEi
mi ,1...
mi ,`i
+∑
j 6=i HijEj
mj ,1...
mj ,`j
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Linear Network Communication
DiHiiEi
mi ,1...
mi ,`i
+∑
j 6=i DiHijEj
mj ,1...
mj ,`j
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Linear Network Communication
m̂i ,1...
m̂i ,`i
= DiHiiEi
mi ,1...
mi ,`i
+∑
j 6=i DiHijEj
mj ,1...
mj ,`j
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Linear Network Communication
m̂i ,1...
m̂i ,`i
= DiHiiEi
mi ,1...
mi ,`i
+∑
j 6=i DiHijEj
mj ,1...
mj ,`j
=
mi ,1...
mi ,`i
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Linear Achievability of Multiple Unicast networks
Definition
A network N is linearly achievable for ρ = (ρ1, . . . , ρN) ∈ ZN , orsimply ρ-linearly achievable, if there exist two matrices D,E (withentries in Fq) such that
DiHijEj = 0 and rank(DiHiiEi ) = ρi .
• DiHiiEi is a `i × `i matrix.• wlog Di can be chosen to be in RCEF and messages are sent
at pivot positions.
DiHiiEi
m1
0m3
...mρi
=
1 0 . . . 0 0 . . . 0? 0 . . . 0 0 . . . 00 1 . . . 0 0 . . . 0...
......
0 0 . . . 1 0 . . . 0
m1
0m3
...mρi
=
m1
?m3
...mρi
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Linear Achievability Region
Definition
The linear achievability region of network N , denoted Lin(N ), isthe subset of RN for which N is ρ-linearly achievable.
H =
1 0 1 00 1 0 01 0 1 00 0 0 1
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Algebraic model
We represent a network N by matrices with
DHE =
D1H11E1 . . . D1H1NEN
D2H21E1 . . . D2H2NEN...
. . ....
DNHN1E1 . . . DNHNNEN
where E ∈ Fq[e, d ]{s×`} and D ∈ Fq[e, d ]{`×t}.
Optimization problem. Find matrices E ,D such that
• DiHijEj = 0 −→ homogeneous bilinear system.
• rankDiHiiEi is maximal.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Algebraic model
We represent a network N by matrices with
DHE =
D1H11E1 . . . D1H1NEN
D2H21E1 . . . D2H2NEN...
. . ....
DNHN1E1 . . . DNHNNEN
where E ∈ Fq[e, d ]{s×`} and D ∈ Fq[e, d ]{`×t}.
Optimization problem. Find matrices E ,D such that
• DiHijEj = 0 −→ homogeneous bilinear system.
• rankDiHiiEi is maximal.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Algebraic model
We represent a network N by matrices with
DHE =
D1H11E1 . . . D1H1NEN
D2H21E1 . . . D2H2NEN...
. . ....
DNHN1E1 . . . DNHNNEN
where E ∈ Fq[e, d ]{s×`} and D ∈ Fq[e, d ]{`×t}.
Optimization problem. Find matrices E ,D such that
• DiHijEj = 0
−→ homogeneous bilinear system.
• rankDiHiiEi is maximal.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Algebraic model
We represent a network N by matrices with
DHE =
D1H11E1 . . . D1H1NEN
D2H21E1 . . . D2H2NEN...
. . ....
DNHN1E1 . . . DNHNNEN
where E ∈ Fq[e, d ]{s×`} and D ∈ Fq[e, d ]{`×t}.
Optimization problem. Find matrices E ,D such that
• DiHijEj = 0 −→ homogeneous bilinear system.
• rankDiHiiEi is maximal.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion definition multiple unicast achievability
Algebraic model
We represent a network N by matrices with
DHE =
D1H11E1 . . . D1H1NEN
D2H21E1 . . . D2H2NEN...
. . ....
DNHN1E1 . . . DNHNNEN
where E ∈ Fq[e, d ]{s×`} and D ∈ Fq[e, d ]{`×t}.
Optimization problem. Find matrices E ,D such that
• DiHijEj = 0 −→ homogeneous bilinear system.
• rankDiHiiEi is maximal.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion lower upper
1 Introduction
2 Interference NetworksDefinitionMultiple Unicast networkLinear Achievability and complexity
3 Achievability bounds for Multiple Unicast NetworksLower BoundUpper Bounds
4 Conclusions
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion lower upper
Some preliminary results
Lemma
If N is ρ-linearly achievable thenρi ≤ rank Hii = ri .
Lemma
Let N be such that ti = si = ri = `i for all 1 ≤ i ≤ N. Then N is`-linearly achievable for ` = (`1, . . . , `N) only if D, Hand E arefullrank.
Theorem
Let N be such that ti = si = ri = `i for all 1 ≤ i ≤ N. If N hasinterference then N is not linearly achievable for ` (for all finitefields).
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Some preliminary results
Lemma
If N is ρ-linearly achievable thenρi ≤ rank Hii = ri .
Lemma
Let N be such that ti = si = ri = `i for all 1 ≤ i ≤ N. Then N is`-linearly achievable for ` = (`1, . . . , `N) only if D, Hand E arefullrank.
Theorem
Let N be such that ti = si = ri = `i for all 1 ≤ i ≤ N. If N hasinterference then N is not linearly achievable for ` (for all finitefields).
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Some preliminary results
Lemma
If N is ρ-linearly achievable thenρi ≤ rank Hii = ri .
Lemma
Let N be such that ti = si = ri = `i for all 1 ≤ i ≤ N. Then N is`-linearly achievable for ` = (`1, . . . , `N) only if D, Hand E arefullrank.
Theorem
Let N be such that ti = si = ri = `i for all 1 ≤ i ≤ N. If N hasinterference then N is not linearly achievable for ` (for all finitefields).
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Lower achievability bound
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Lower achievability bound
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intro interference networks bounds conclusion lower upper
Lower achievability bound
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Sufficient condition for solvability
H =
H11 H12 . . . H1N
H21 H22 . . . H2N...
...HN1 HN2 . . . HNN
Definition
We call rank of interference the valueoi := rank(Hij | j 6= i).
Theorem
A network N is(r1 − o1, . . . , rN − oN)-linearly achievableand matrices E and D can be computedusing Gaussian elimination 2N times.
H =
1 0 0 0 0 00 1 0 0 1 01 0 1 0 0 00 0 0 1 0 00 0 0 1 1 00 1 0 0 0 1
o1 = 1, o2 = 1, o3 = 2
N is (1, 1, 0)-linearlyachievable.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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Sufficient condition for solvability
H =
H11 H12 . . . H1N
H21 H22 . . . H2N...
...HN1 HN2 . . . HNN
Definition
We call rank of interference the valueoi := rank(Hij | j 6= i).
Theorem
A network N is(r1 − o1, . . . , rN − oN)-linearly achievableand matrices E and D can be computedusing Gaussian elimination 2N times.
H =
1 0 0 0 0 00 1 0 0 1 01 0 1 0 0 00 0 0 1 0 00 0 0 1 1 00 1 0 0 0 1
o1 = 1, o2 = 1, o3 = 2
N is (1, 1, 0)-linearlyachievable.
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Proof of the lower bound
Denote by `ker(Hij | j 6= i) be the left kernel of (Hij | j 6= i).
• DiHij = 0 if and only if the rows of Di are contained in`ker (Hij | j 6= i).
• (Hij | j 6= i) is a matrix with ti rows and rank oi .
• dim `ker (Hij | j 6= i) is ti − oi .
• Choose the rows of Di to span the `ker (Hij | j 6= i).
• Without loss of generality Hii is a partial identity of rank ri .
• rankDiHii ≥ (ti − oi )− (ti − ri ) = ri − oi .
• Let Ei be any invertible matrix, then rankDiHiiEi ≥ ri − oi .
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Proof of the lower bound
Denote by `ker(Hij | j 6= i) be the left kernel of (Hij | j 6= i).
• DiHij = 0 if and only if the rows of Di are contained in`ker (Hij | j 6= i).
• (Hij | j 6= i) is a matrix with ti rows and rank oi .
• dim `ker (Hij | j 6= i) is ti − oi .
• Choose the rows of Di to span the `ker (Hij | j 6= i).
• Without loss of generality Hii is a partial identity of rank ri .
• rankDiHii ≥ (ti − oi )− (ti − ri ) = ri − oi .
• Let Ei be any invertible matrix, then rankDiHiiEi ≥ ri − oi .
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion lower upper
Proof of the lower bound
Denote by `ker(Hij | j 6= i) be the left kernel of (Hij | j 6= i).
• DiHij = 0 if and only if the rows of Di are contained in`ker (Hij | j 6= i).
• (Hij | j 6= i) is a matrix with ti rows and rank oi .
• dim `ker (Hij | j 6= i) is ti − oi .
• Choose the rows of Di to span the `ker (Hij | j 6= i).
• Without loss of generality Hii is a partial identity of rank ri .
• rankDiHii ≥ (ti − oi )− (ti − ri ) = ri − oi .
• Let Ei be any invertible matrix, then rankDiHiiEi ≥ ri − oi .
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 65
intro interference networks bounds conclusion lower upper
Proof of the lower bound
Denote by `ker(Hij | j 6= i) be the left kernel of (Hij | j 6= i).
• DiHij = 0 if and only if the rows of Di are contained in`ker (Hij | j 6= i).
• (Hij | j 6= i) is a matrix with ti rows and rank oi .
• dim `ker (Hij | j 6= i) is ti − oi .
• Choose the rows of Di to span the `ker (Hij | j 6= i).
• Without loss of generality Hii is a partial identity of rank ri .
• rankDiHii ≥ (ti − oi )− (ti − ri ) = ri − oi .
• Let Ei be any invertible matrix, then rankDiHiiEi ≥ ri − oi .
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 66
intro interference networks bounds conclusion lower upper
Proof of the lower bound
Denote by `ker(Hij | j 6= i) be the left kernel of (Hij | j 6= i).
• DiHij = 0 if and only if the rows of Di are contained in`ker (Hij | j 6= i).
• (Hij | j 6= i) is a matrix with ti rows and rank oi .
• dim `ker (Hij | j 6= i) is ti − oi .
• Choose the rows of Di to span the `ker (Hij | j 6= i).
• Without loss of generality Hii is a partial identity of rank ri .
• rankDiHii ≥ (ti − oi )− (ti − ri ) = ri − oi .
• Let Ei be any invertible matrix, then rankDiHiiEi ≥ ri − oi .
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 67
intro interference networks bounds conclusion lower upper
Proof of the lower bound
Denote by `ker(Hij | j 6= i) be the left kernel of (Hij | j 6= i).
• DiHij = 0 if and only if the rows of Di are contained in`ker (Hij | j 6= i).
• (Hij | j 6= i) is a matrix with ti rows and rank oi .
• dim `ker (Hij | j 6= i) is ti − oi .
• Choose the rows of Di to span the `ker (Hij | j 6= i).
• Without loss of generality Hii is a partial identity of rank ri .
• rankDiHii ≥ (ti − oi )− (ti − ri ) = ri − oi .
• Let Ei be any invertible matrix, then rankDiHiiEi ≥ ri − oi .
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 68
intro interference networks bounds conclusion lower upper
Proof of the lower bound
Denote by `ker(Hij | j 6= i) be the left kernel of (Hij | j 6= i).
• DiHij = 0 if and only if the rows of Di are contained in`ker (Hij | j 6= i).
• (Hij | j 6= i) is a matrix with ti rows and rank oi .
• dim `ker (Hij | j 6= i) is ti − oi .
• Choose the rows of Di to span the `ker (Hij | j 6= i).
• Without loss of generality Hii is a partial identity of rank ri .
• rankDiHii ≥ (ti − oi )− (ti − ri ) = ri − oi .
• Let Ei be any invertible matrix, then rankDiHiiEi ≥ ri − oi .
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 69
intro interference networks bounds conclusion lower upper
Encoding-Dependent Linear Achievability
Theorem
A network N is ρ = (ρ1, . . . , ρn)-linearly achievable if and only ifthere exist E1, . . . ,En such that for all i ∈ [N] if holds that
ρi ≤ dim `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j).
Moreover the bound is always tight.
• Let Vi = `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j)
• If Di is such that DiHijEj = 0, then the rows of Di belong to`ker(HijEj | j 6= i).
• Let vi ,1, . . . , vi ,`i be the rows of Di , then
rankDiHiiEi = dim〈vi ,1HiiEi , . . . , vi ,`iHiiEi 〉≤ dim `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j)
• rankDiHiiEi = dimVi iff {[vi ,1], . . . , [vi ,`i ]} spans Vi .
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 70
intro interference networks bounds conclusion lower upper
Encoding-Dependent Linear Achievability
Theorem
A network N is ρ = (ρ1, . . . , ρn)-linearly achievable if and only ifthere exist E1, . . . ,En such that for all i ∈ [N] if holds that
ρi ≤ dim `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j).
Moreover the bound is always tight.
• Let Vi = `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j)
• If Di is such that DiHijEj = 0, then the rows of Di belong to`ker(HijEj | j 6= i).
• Let vi ,1, . . . , vi ,`i be the rows of Di , then
rankDiHiiEi = dim〈vi ,1HiiEi , . . . , vi ,`iHiiEi 〉≤ dim `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j)
• rankDiHiiEi = dimVi iff {[vi ,1], . . . , [vi ,`i ]} spans Vi .
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 71
intro interference networks bounds conclusion lower upper
Encoding-Dependent Linear Achievability
Theorem
A network N is ρ = (ρ1, . . . , ρn)-linearly achievable if and only ifthere exist E1, . . . ,En such that for all i ∈ [N] if holds that
ρi ≤ dim `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j).
Moreover the bound is always tight.
• Let Vi = `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j)
• If Di is such that DiHijEj = 0, then the rows of Di belong to`ker(HijEj | j 6= i).
• Let vi ,1, . . . , vi ,`i be the rows of Di , then
rankDiHiiEi = dim〈vi ,1HiiEi , . . . , vi ,`iHiiEi 〉≤ dim `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j)
• rankDiHiiEi = dimVi iff {[vi ,1], . . . , [vi ,`i ]} spans Vi .
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 72
intro interference networks bounds conclusion lower upper
Encoding-Dependent Linear Achievability
Theorem
A network N is ρ = (ρ1, . . . , ρn)-linearly achievable if and only ifthere exist E1, . . . ,En such that for all i ∈ [N] if holds that
ρi ≤ dim `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j).
Moreover the bound is always tight.
• Let Vi = `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j)
• If Di is such that DiHijEj = 0, then the rows of Di belong to`ker(HijEj | j 6= i).
• Let vi ,1, . . . , vi ,`i be the rows of Di , then
rankDiHiiEi = dim〈vi ,1HiiEi , . . . , vi ,`iHiiEi 〉≤ dim `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j)
• rankDiHiiEi = dimVi iff {[vi ,1], . . . , [vi ,`i ]} spans Vi .
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 73
intro interference networks bounds conclusion lower upper
Encoding-Dependent Linear Achievability
Theorem
A network N is ρ = (ρ1, . . . , ρn)-linearly achievable if and only ifthere exist E1, . . . ,En such that for all i ∈ [N] if holds that
ρi ≤ dim `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j).
Moreover the bound is always tight.
• Let Vi = `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j)
• If Di is such that DiHijEj = 0, then the rows of Di belong to`ker(HijEj | j 6= i).
• Let vi ,1, . . . , vi ,`i be the rows of Di , then
rankDiHiiEi = dim〈vi ,1HiiEi , . . . , vi ,`iHiiEi 〉≤ dim `ker(HijEj | j 6= i)/̀ ker(HijEj | ∀ j)
• rankDiHiiEi = dimVi iff {[vi ,1], . . . , [vi ,`i ]} spans Vi .
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion lower upper
Example
H =
1 0 0 0 0 00 1 0 0 1 01 0 1 0 0 00 0 0 1 0 00 0 0 1 1 00 1 0 0 0 1
,
E1 =
(1 00 0
),E2 =
(1 00 1
),E3 =
(0 00 1
)
H31E1 = 0 H32E2 =
(0 10 0
)H33E3 =
(0 00 1
)
`ker H31E1 = F2q `ker H32E2 = 〈(0, 1)〉 `ker H33E2 = 〈(1, 0)〉
ρ3 = dim `ker(H3jEj | j 6= 3)/̀ ker(H3jEj | ∀ j) = dim 〈(0, 1)〉/{0} = 1
D3 =
(0 10 0
)
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
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intro interference networks bounds conclusion lower upper
Example
H =
1 0 0 0 0 00 1 0 0 1 01 0 1 0 0 00 0 0 1 0 00 0 0 1 1 00 1 0 0 0 1
, E1 =
(1 00 0
),E2 =
(1 00 1
),E3 =
(0 00 1
)
H31E1 = 0 H32E2 =
(0 10 0
)H33E3 =
(0 00 1
)
`ker H31E1 = F2q `ker H32E2 = 〈(0, 1)〉 `ker H33E2 = 〈(1, 0)〉
ρ3 = dim `ker(H3jEj | j 6= 3)/̀ ker(H3jEj | ∀ j) = dim 〈(0, 1)〉/{0} = 1
D3 =
(0 10 0
)
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 76
intro interference networks bounds conclusion lower upper
Example
H =
1 0 0 0 0 00 1 0 0 1 01 0 1 0 0 00 0 0 1 0 00 0 0 1 1 00 1 0 0 0 1
, E1 =
(1 00 0
),E2 =
(1 00 1
),E3 =
(0 00 1
)
H31E1 = 0 H32E2 =
(0 10 0
)H33E3 =
(0 00 1
)
`ker H31E1 = F2q `ker H32E2 = 〈(0, 1)〉 `ker H33E2 = 〈(1, 0)〉
ρ3 = dim `ker(H3jEj | j 6= 3)/̀ ker(H3jEj | ∀ j) = dim 〈(0, 1)〉/{0} = 1
D3 =
(0 10 0
)
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 77
intro interference networks bounds conclusion lower upper
Example
H =
1 0 0 0 0 00 1 0 0 1 01 0 1 0 0 00 0 0 1 0 00 0 0 1 1 00 1 0 0 0 1
, E1 =
(1 00 0
),E2 =
(1 00 1
),E3 =
(0 00 1
)
H31E1 = 0 H32E2 =
(0 10 0
)H33E3 =
(0 00 1
)
`ker H31E1 = F2q `ker H32E2 = 〈(0, 1)〉 `ker H33E2 = 〈(1, 0)〉
ρ3 = dim `ker(H3jEj | j 6= 3)/̀ ker(H3jEj | ∀ j) = dim 〈(0, 1)〉/{0} = 1
D3 =
(0 10 0
)
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 78
intro interference networks bounds conclusion lower upper
Example
H =
1 0 0 0 0 00 1 0 0 1 01 0 1 0 0 00 0 0 1 0 00 0 0 1 1 00 1 0 0 0 1
, E1 =
(1 00 0
),E2 =
(1 00 1
),E3 =
(0 00 1
)
H31E1 = 0 H32E2 =
(0 10 0
)H33E3 =
(0 00 1
)
`ker H31E1 = F2q `ker H32E2 = 〈(0, 1)〉 `ker H33E2 = 〈(1, 0)〉
ρ3 = dim `ker(H3jEj | j 6= 3)/̀ ker(H3jEj | ∀ j) = dim 〈(0, 1)〉/{0} = 1
D3 =
(0 10 0
)
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 79
intro interference networks bounds conclusion
Conclusions
Future projects
• Find Lin(N ) for all multiple unicast networks.
• Find good algorithmic methods to solve the optimizationproblem.
• Prove that linearity is actually optimal for the explainedmultiple unicast networks.
• Generalize these results to other interference networks
Thank you.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 80
intro interference networks bounds conclusion
Conclusions
Future projects
• Find Lin(N ) for all multiple unicast networks.
• Find good algorithmic methods to solve the optimizationproblem.
• Prove that linearity is actually optimal for the explainedmultiple unicast networks.
• Generalize these results to other interference networks
Thank you.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication
Page 81
intro interference networks bounds conclusion
References
Cadambe, V. R. and Jafar, S. A. (2008).Interference alignment and degrees of freedom of the k-user interference channel.
IEEE Transactions on Information Theory, 54(8):3425–3441.
Dougherty, R., Freiling, C., and Zeger, K. (2005).Insufficiency of linear coding in network information flow.IEEE Transactions on Information Theory, 51(8):2745–2759.
Koetter, R. and Medard, M. (2003).An algebraic approach to network coding.IEEE/ACM Transactions on Networking, 11(5):782–795.
Li, S. . R., Yeung, R. W., and Ning Cai (2003).Linear network coding.IEEE Transactions on Information Theory, 49(2):371–381.
Zhao, N., Yu, F. R., Jin, M., Yan, Q., and Leung, V. C. M. (2016).Interference alignment and its applications: A survey, research issues, andchallenges.IEEE Communications Surveys Tutorials, 18(3):1779–1803.
Manganiello, SMSS@Clemson & CRL@Ryerson Graphs and Algebra in Modern Communication