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 PBNA

Feasibility 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 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

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 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

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”

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 PBNA

Feasibility of PBNA

o Conclusion

11

Outline

12

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"

13

NA in the Middle: Pros & Cons

Pros:

Achieve ½ in exactly 2 time slots

Cons:

Finding code is NOT easy

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“

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

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

o Motivation:

Network ≈ Wireless Interference Channel

o Approaches:

NA in the middle, Precoding-Based NA

o PBNA

Feasibility 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 & Q3

If is not constant, is asymptotically achievable via PBNA if and only if

39

Key Idea Behind Q-1Graph-related properties

𝜎 1 𝜏1𝑒1

𝑒4

𝑒2

𝑒3

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

e

e’

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 PBNA

Feasibility 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 ?