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![Page 1: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/1.jpg)
Network Alignment: Treating Networks as Wireless Interference Channel
Chun MengUniv. of California, Irvine
![Page 2: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/2.jpg)
o Motivation:
Network ≈ Wireless Interference Channel
o Approaches:
NA in the middle, Precoding-Based NA
o PBNA
Feasibility of PBNA
o Conclusion
2
Outline
![Page 3: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/3.jpg)
Intra-Session NC
Achievable rate = min-cut[1,2]
LP-formulation[3]
Code design: RNC[4], deterministic[5]
3
State of the Art - I
[1] R. Ahlswede, et al, “Network information flow”[2] R. Koetter and M. M edard, “An algebraic approach to network coding”′[3] Z. Li, et al, “On Achieving Maximum Multicast Throughput in Undirected Networks”[4] T. Ho, et al, “A random linear network coding approach to multicast”[5] S. Jaggi, et al, “Polynomial Time Algorithms for Multicast Network Code Construction”
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Inter-Session NC
Only approximation of bounds [1]
Exponential number of variables
Code design: NP-hard[5]
LP, evolutionary approach
4
State of the Art - II
[1] N. Harvey, et al, “On the Capacity of Information Networks”[2] A. R. Lehman and E. Lehman, “Complexity classification of network information flow problems”[3] D. Traskov, et al, “Network coding for multiple unicasts: An approach based on linear optimization”[4] M. Kim, et al, “An evolutionary approach to inter-session network coding”
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5
Restrictive Framework
𝑋 1
𝑋 2
𝑋 3
𝑍 2
𝑍1
𝑍 3
R. Koetter and M. M edard, “An algebraic approach to network coding”′
Interference must be canceled out
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6
Network vs. Wireless Channel - I
Network with multiple unicasts SISO
Channel gain: introduced by nature
𝑋 1
𝑋 2
𝑋 3
𝑍 2
𝑍1
𝑍 3
𝑥1
𝑥2
𝑥3
𝑦 1
𝑦 2
𝑦 3
Transfer function: introduced by network
Min-cut = 1
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7
Networks vs. Wireless Channel - II
Network with multiple unicasts MIMO
𝐗1
𝐗2
𝐗3
𝐙2
𝐙1
𝐙3
𝐱1
𝐱 2
𝐱 3
𝐲 1
𝐲 2
𝐲 3
Min-cut > 1
Transfer matrix Channel matrix
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8
Interference Alignment
Common problem:
Too MANY unknowns!
Solution:
Align interferences to reduce the number of
unknowns
V. Cadambe and S. Jafar, “Interference Alignment and Degrees of Freedom of the K-User Interference Channel”
Benefit:
Everyone gets one half of the cake
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9
Brief Intro of IA
o Originally introduced by Cadambe & Jafar
o Approaches:• Asymptotic alignment, • Ergodic alignment, • Lattice alignment, • Blind alignment
o Applications• K-user wireless interference channel, • K-user MIMO interference channel, • Cellular networks, • Multi-hop interference networks, • Exact repair in distributed storage
Syed A. Jafar, “Interference Alignment — A New Look at Signal Dimensions in a Communication Network”
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10
Network Is NOT Wireless Channel
o symbols from finite field
o : polynomial of coding variables
o real & complex numbers
o : structureless
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o Motivation:
Network ≈ Wireless Interference Channel
o Approaches:
NA in the middle, Precoding-Based NA
o PBNA
Feasibility of PBNA
o Conclusion
11
Outline
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NA in the Middle
𝑋 1
𝑋 2
𝑋 3 𝑍 31
𝑍 21
𝑍11
t=1 𝑋 1
𝑋 2
𝑋 3 𝑍 32
𝑍 22
𝑍12t=
2
≠ = =NA in the middle:
B. Nazer, et al, "Ergodic Interference Alignment"
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13
NA in the Middle: Pros & Cons
Pros:
Achieve ½ in exactly 2 time slots
Cons:
Finding code is NOT easy
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14
Precoding-Based NA - I
S1
S2
S3
D1
D2
D3
2n+1 uses of networkor 2n+1 symbol extensionx1
x2
x3
n+1
n
n
y1=V1x1
y2=V2x2
y3=V3x3
2n+1
2n+1
2n+1
V. R. Cadambe and S. A. Jafar, "Interference Alignment and Degrees of Freedom of the K-User Interference Channel“
![Page 15: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/15.jpg)
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Precoding-Based NA - II
M11V1x1
M12V2x2
M13V3x3
M22V2x2
M21V1x1
M23V3x3
M33V3x3
M32V2x2
M31V1x1
Align interferences
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16
Precoding-Based NA - III
Alignment
conditions
Rank
conditions
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17
Precoding-Based NA - Advantages
• Achievable rate ½ min-cut[1]
• Code design is simpleEncoding & decoding are predetermined regardless of topology
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Get a Better Understanding
V1 can NOT be chosen freely!
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Reformulated Feasibility Cond.
Condensed alignment cond.
Reformulated rank cond.
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Algebraic Formulation - I
is not constant. V1 can NOT be arbitrary matrix
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Algebraic Formulation - II
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Algebraic Formulation - III
is full rank
Linearly independent
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Algebraic Formulation - IV
is achievable via PBNA if
If is not constant, is asymptotically achievable via PBNA if
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Algebraic Formulation - V
is constant. Setting AB=C, V1 can be arbitrary matrix
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Algebraic Formulation - VI
If is constant, is asymptotically achievable via PBNA if
pi(x) is not constant
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26
Summarization
o If is not constant, is asymptotically achievable via PBNA if
o If is constant, is asymptotically achievable via PBNA if
pi(x) is not constant
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o Motivation:
Network ≈ Wireless Interference Channel
o Approaches:
NA in the middle, Precoding-Based NA
o PBNA
Feasibility of PBNA
o Conclusion
27
Outline
![Page 28: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/28.jpg)
28
Unfriendly Networks - I If is constant, is asymptotically achievable via PBNA if
pi(x) is not constant
𝑋 1
𝑋 2
𝑋 3
𝑍 2
𝑍1
𝑍 3
𝑒
![Page 29: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/29.jpg)
29
Unfriendly Networks - IIIf is not constant, is asymptotically achievable via PBNA if
𝑋 1
𝑋 2
𝑋 3
𝑍 3
𝑍1
𝑍 2
𝑒1
𝑒2
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Coupling Relations
network for which the relation holds, it is realizable
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Coupling Relations are Mostly Bad
Bad guys
Good guy
𝑋 1
𝑋 2
𝑋 3
𝑍 2
𝑍1
𝑍 3𝑒1
𝑒2
Arbitrary precoding matrix V1 is OK
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Networks vs. Wireless Channel
Have structures
Coupling relations
Feasibility conditions are violated
Structureless
Can change independently
IA is always feasible
![Page 33: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/33.jpg)
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NOT All Coupling Relations are Realizable
Max degree of xee’ ≤ 2 Max degree of xee’ ≥ 3
Q1: Which coupling relations are realizable?
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Topology and Coupling Relations
Q2: What is the network topology for ?
𝑋 1
𝑋 2
𝑋 3
𝑍 2
𝑍1
𝑍 3
𝑋 1
𝑋 2
𝑋 3
𝑍 3
𝑍1
𝑍 2
![Page 35: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/35.jpg)
35
How About Other Precoding Matrices?
Q3: If can not be used, how about others?
The ONLY one ?
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36
Answer to Q1Q1: Which coupling relations are realizable?
Answer:
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Answer to Q3
Answer:
Q3: If can not be used, how about others?
NO !
![Page 38: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/38.jpg)
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Combining the Answers to Q1 & Q3
If is not constant, is asymptotically achievable via PBNA if and only if
![Page 39: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/39.jpg)
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Key Idea Behind Q-1Graph-related properties
𝜎 1 𝜏1𝑒1
𝑒4
𝑒2
𝑒3
𝑒5
𝑒6
![Page 40: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/40.jpg)
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Graph-Related Properties - IHow to check pi(x) is not constant?
1 2
1 3
1 2
1 3
1 2
1 3
![Page 41: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/41.jpg)
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Graph-Related Properties - IILinearization Property
Assign values to x
Max degree = 1
![Page 42: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/42.jpg)
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Graph-Related Properties - IIIIntuition behind Linearization Property
1
1
3
2
e
e’
![Page 43: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/43.jpg)
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Graph-Related Properties - IVSquare-Term Property
Implication:
Assign values to x
![Page 44: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/44.jpg)
44
Graph-Related Properties - VIntuition behind Square-Term Property
1 2
1 3
e
e’
1 3
1 2
e
e’
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45
Finding Realizable Coupling Relations - I
Objective:
Step I
Assign values to x
Max degree of f(z) and g(z) = 1
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Finding Realizable Coupling Relations - II
Step II
Define
No square term in the numerator
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Finding Realizable Coupling Relations - III
Step III
[1] J. Han, et al, “Analysis of precoding-based intersession network coding and the corresponding 3-unicast interference alignment scheme”
Unrealizable
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How to Answer Q3 ?
Q3: If can not be used, how about others?
How to construct V1 ?
![Page 49: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/49.jpg)
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Example: Construct V1
![Page 50: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/50.jpg)
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All Precoding Matrices Are Equivalent
can not be used to coupling relation Any V1 cannot be used
![Page 51: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/51.jpg)
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Topology of Coupling Relations - IQ2: What is the network topology for ?
1
1
3
2
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Topology of Coupling Relations - II
1
1
2
3
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Topology of Coupling Relations - III
𝑋 1
𝑋 2
𝑋 3
𝑍 3
𝑍1
𝑍 2
![Page 54: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/54.jpg)
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Trivial Case is constant and T is identity matrix
Perfectly aligned
If is constant, can be achieved via PBNA in exactly two time slots if and only if
pi(x) is not constant
![Page 55: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/55.jpg)
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Trivial Case - Example
1
2
3
2
1
3𝑒1
𝑒2
![Page 56: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/56.jpg)
o Motivation:
Network ≈ Wireless Interference Channel
o Approaches:
NA in the middle, Precoding-Based NA
o PBNA
Feasibility of PBNA
o Conclusion
56
Outline
![Page 57: Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine.](https://reader030.fdocuments.us/reader030/viewer/2022032801/56649dd25503460f94ac936b/html5/thumbnails/57.jpg)
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Conclusion
o How to apply interference alignment to networks?
o Q1: Which coupling relations are realizable?
o Q2: What is the network topology for ?
o Q3: If can not be used, how about others?
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Open Questions
o Is it possible to achieve in limited number of time
slots ?
o How about other IA schemes ?
o In what condition does IA behave better than routing ?
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http://odysseas.calit2.uci.edu/doku.php/public:publication
Thank you ! Questions ?