network alignment: treating networks as wireless interference channel
DESCRIPTION
Network Alignment: Treating Networks as Wireless Interference Channel. Chun Meng Univ. of California, Irvine. Outline. Motivation: Network ≈ Wireless Interference Channel Approaches: N A in the middle, Precoding -Based NA PBNA Feasibility of PBNA Conclusion. State of the Art - I. - PowerPoint PPT PresentationTRANSCRIPT
Network Alignment: Treating Networks as Wireless Interference Channel
Chun MengUniv. of California, Irvine
o Motivation:Network ≈ Wireless Interference Channel
o Approaches:NA in the middle, Precoding-Based NA
o PBNAFeasibility of PBNA
o Conclusion
2
Outline
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”
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”
5
Restrictive Framework𝑋 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
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
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
8
Interference AlignmentCommon 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
9
Brief Intro of IAo 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”
10
Network Is NOT Wireless Channel
o symbols from finite field
o : polynomial of coding variables
o real & complex numbers
o : structureless
o Motivation:Network ≈ Wireless Interference Channel
o Approaches:NA in the middle, Precoding-Based NA
o PBNAFeasibility of PBNA
o Conclusion
11
Outline
12
NA in the Middle𝑋 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"
13
NA in the Middle: Pros & ConsPros:
Achieve ½ in exactly 2 time slots
Cons:Finding code is NOT easy
14
Precoding-Based NA - IS1
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“
15
Precoding-Based NA - II
M11V1x1
M12V2x2
M13V3x3
M22V2x2
M21V1x1
M23V3x3
M33V3x3
M32V2x2
M31V1x1
Align interferences
16
Precoding-Based NA - III
Alignment conditions
Rank conditions
17
Precoding-Based NA - Advantages
• Achievable rate ½ min-cut[1]
• Code design is simpleEncoding & decoding are predetermined regardless of topology
18
Get a Better Understanding
V1 can NOT be chosen freely!
19
Reformulated Feasibility Cond.
Condensed alignment cond.
Reformulated rank cond.
20
Algebraic Formulation - I
is not constant. V1 can NOT be arbitrary matrix
21
Algebraic Formulation - II
22
Algebraic Formulation - III
is full rank
Linearly independent
23
Algebraic Formulation - IV is achievable via PBNA if
If is not constant, is asymptotically achievable via PBNA if
24
Algebraic Formulation - V is constant. Setting AB=C, V1 can be arbitrary matrix
25
Algebraic Formulation - VI
If is constant, is asymptotically achievable via PBNA if
pi(x) is not constant
26
Summarizationo 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
o Motivation:Network ≈ Wireless Interference Channel
o Approaches:NA in the middle, Precoding-Based NA
o PBNAFeasibility of PBNA
o Conclusion
27
Outline
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
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
30
Coupling Relations
network for which the relation holds, it is realizable
31
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
32
Networks vs. Wireless Channel
Have structures
Coupling relations
Feasibility conditions are violated
Structureless
Can change independently
IA is always feasible
33
NOT All Coupling Relations are Realizable
Max degree of xee’ ≤ 2 Max degree of xee’ ≥ 3
Q1: Which coupling relations are realizable?
34
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
35
How About Other Precoding Matrices?
Q3: If can not be used, how about others?
The ONLY one ?
36
Answer to Q1Q1: Which coupling relations are realizable?
Answer:
37
Answer to Q3
Answer:
Q3: If can not be used, how about others?
NO !
38
Combining the Answers to Q1 & Q3If is not constant, is asymptotically achievable via PBNA if and only if
39
Key Idea Behind Q-1Graph-related properties
𝜎 1 𝜏1𝑒1
𝑒4
𝑒2𝑒3
𝑒5
𝑒6
40
Graph-Related Properties - IHow to check pi(x) is not constant?
1 2
1 3
1 2
1 3
1 2
1 3
41
Graph-Related Properties - IILinearization Property
Assign values to x
Max degree = 1
42
Graph-Related Properties - IIIIntuition behind Linearization Property
1
1
3
2
ee’
43
Graph-Related Properties - IVSquare-Term Property
Implication:
Assign values to x
44
Graph-Related Properties - VIntuition behind Square-Term Property
1 2
1 3
e
e’
1 3
1 2
e
e’
45
Finding Realizable Coupling Relations - I
Objective:
Step I
Assign values to x
Max degree of f(z) and g(z) = 1
46
Finding Realizable Coupling Relations - II
Step II
Define
No square term in the numerator
47
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
48
How to Answer Q3 ?Q3: If can not be used, how about others?
How to construct V1 ?
49
Example: Construct V1
50
All Precoding Matrices Are Equivalent
can not be used to coupling relation Any V1 cannot be used
51
Topology of Coupling Relations - IQ2: What is the network topology for ?
1
1
3
2
52
Topology of Coupling Relations - II
1
1
2
3
53
Topology of Coupling Relations - III
𝑋 1
𝑋 2
𝑋 3
𝑍 3
𝑍1
𝑍 2
54
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
55
Trivial Case - Example
1
2
3
2
1
3𝑒1
𝑒2
o Motivation:Network ≈ Wireless Interference Channel
o Approaches:NA in the middle, Precoding-Based NA
o PBNAFeasibility of PBNA
o Conclusion
56
Outline
57
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?
58
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 ?
59
http://odysseas.calit2.uci.edu/doku.php/public:publication
Thank you ! Questions ?