Network Coding for Three Unicast Sessions:Network Coding for Three Unicast Sessions:Interference Alignment Approaches

Abinesh Ramakrishnan*, Abhik Das†, Hamed Maleki*,Athina Markopoulou*, Syed Jafar*, Sriram Vishwanath†

(*) UC Irvine and (†) UT Austin

OutlineOutline

o Motivationo Network Alignment (NA: NC+IA)

o Network Alignment Approaches o at the edgeo in the middle

Wh N o When is NA necessary? o Can other schemes achieve >=½ rate or more?

C l io Conclusion

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AlignmentAlignment

• Interference Alignment (IA) – IA originally introduced for wireless interference channels

• as a systematic way to guarantee half the rate per user

– allows to solve fewer equations for some unknowns h • needed when messages are mixed

• Network Alignment (NA=NC+IA)– IA techniques can also be applied in networks– Applied to repair problem in dist. storage [WD’09, SR’10, CJM’10]– Applied to network coding for multiple unicasts [DVJM’10]

3

The Network as a ChannelA lAnalogy

S1 D1

Multiple Unicast Network Interference Channel

S2 D2

S3 D3

• Both represented by a Linear Transfer Function [Mij]. • +: [Mij] no longer determined by nature but defined by us• -: Spatial dependencies introduced by the graph. Feasibility?

4

Network Alignment ApproachesNetwork Alignment ApproachesCoding in the Middle Coding at the Edge

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Alignment by Coding at the EdgeAlignment by Coding at the Edge• Asymptotic scheme: Symbol Extension [CJ’08]

– recently applied to network coding for multiple unicastsrecently applied to network coding for multiple unicasts[Das, Vishwanath, Jafar, Markopoulou, ISIT’10]

S1 D1(2n+1) – symbol extension( ) yz1

(n+1) x 1

S2 D2z2

n x 1

S3 D3z3

6

z3n x 1

Alignment by Coding at the EdgeAlignment by Coding at the Edge• Asymptotic scheme: Symbol Extension [CJ’08]

– recently applied to network coding for multiple unicasts [DVJM’10]recently applied to network coding for multiple unicasts [DVJM 10]

S1 D1(2n+1) – symbol extension( ) yz1

(n+1) x 1V1

(2n+1) x (n+1)

S2 D2z2

n x 1V2

(2n+1) x n

S3 D3z3 V

(2n+1) x n

7

z3n x 1 (2n+1) x n

V3

Alignment by Coding at the EdgeAlignment by Coding at the Edge• Asymptotic scheme: Symbol Extension [CJ’08]

– recently applied to network coding for multiple unicasts [DVJM’10]recently applied to network coding for multiple unicasts [DVJM 10]

S1 D1(2n+1) – symbol extension( ) yz1

(n+1) x 1 x1=V1z1

S2 D2z2

n x 1 x2=V2z2

S3 D3z3

8

z3n x 1 x3=V3z3

Alignment by Coding at the EdgeAlignment by Coding at the Edge• Asymptotic scheme: Symbol Extension [CJ’08]

– recently applied to network coding for multiple unicasts [DVJM’10]recently applied to network coding for multiple unicasts [DVJM 10]

S1 D1(2n+1) – symbol extension

M11V1z1

( ) yz1

(n+1) x 1M12V2z2 + M13V3z3

x1=V1z1

S2 D2z2

n x 1 x2=V2z2

M22V2z2

S3 D3z3

M21V1z1 + M23V3z3

M33V3z3

9

z3n x 1 x3=V3z3

M33V3z3

M31V1z1 + M32V2z2

Alignment by Coding at the EdgeAlignment by Coding at the Edge• Asymptotic scheme: Symbol Extension [CJ’08]

– recently applied to network coding for multiple unicasts [DVJM’10]recently applied to network coding for multiple unicasts [DVJM 10]

S1 D1(2n+1) – symbol extension

z1

( ) yz1

(n+1) x 1 x1=V1z1

S2 D2z2

n x 1 x2=V2z2

z2

S3 D3z3

z3

10

z3n x 1 x3=V3z3

Feasibility is a challenge (unlike wireless) due to dependencies in [Mij]

Feasibility conditionsy[DVJM’10]

By construction:• mij(ξ), i,j =1,2,3, non-trivial polynomialsj j p y• mii(ξ) ≠ c mij(ξ) for any c in Fp\{0}

Notation:

Sufficient conditions for asymptotic alignment [DVJM’10] : • for all n, and pi, qi (i=0,1,2,..n) it should be:

[M11V1 M12V2] is full rank

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QuestionsQuestions

When can we align:• Intuition behind conditions? Relation to Network Structure?• Infinitely many and complicated conditions to check. Simplify?• How mild are these conditions? (e.g. hold almost always in wireless)

Why should we align? • How much benefit/loss compared to alternatives?• Is alignment necessary? Is alignment necessary?

How to align?• Is asymptotic alignment the only way? Algorithms?

Focus mostly on: 3 unicasts, min-cut =1 per session.

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Understanding the conditionsUnderstanding the conditions

• Let’s look at a subset of all conditions– (p0=1, qo=1) and (p0=1, q1=1)

• Interestingly, these are also necessary for any scheme to achieve rate >=1/2 in the wireless interference channel [CJ, ToIT’09]f [ J, ]

13

Understanding the conditionsUnderstanding the conditions

1

2

1

2

3

2

3

2

14

3 3

Conditions and network structureConditions and network structure

1 11

2

11

2

1

2 2

33 3

2

1 11 1 1

2

11

2

1

15

2

3

2

3

Simplifying the conditions?Simplifying the conditions?

• Conjecture:jThe “small” conditions are sufficient for asymptotic alignment.

?

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Alignment Approaches g ppto Code at the Edge or in the Middle?

Coding in the Middle Coding at the Edge

Symbol Extension Method:17

Symbol Extension Method:• intelligence at the edge, middle is simple• widely applicable• large number of symbols and finite field

Example of Coding in the MiddleExample of Coding in the Middle• “Ergodic’’ Alignment

– inspired by [Nazer Gastpar Jafar Vishwanath “Ergodic IA” ISIT’09]inspired by [Nazer, Gastpar, Jafar, Vishwanath, Ergodic IA , ISIT 09]– choose coding coefficients over two time slots so that:

⎥⎥⎤

⎢⎢⎡

= )1(1312

)1(11

)1( mmmmmm

M ⎥⎥⎤

⎢⎢⎡

= )2(1312

)2(11

)2( mmmmmm

M

– then subtract and obtain 1 symbol in 2 time slots

l d

⎥⎥

⎦⎢⎢

⎣

=)1(

333231

232221

mmmmmmM

⎥⎥

⎦⎢⎢

⎣

=)2(

333231

232221

mmmmmmM

• Feasibility condition: – each mii is not a function of the mij‘s– more restrictive than the condition for asymptotic alignment

St n ths:• Strengths:– 2 time slots are enough (no need to wait for channel opportunities)– smaller field size and number of symbols than asymptotic scheme

• Limitation:Limitation:– Intelligence resides in the network. May be complex for large networks.

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How mild are the conditions? Random Graphs

19

How mild are the conditions? Real Topologies

20

How mild are the conditions? Real Topologies

21

OutlineOutline

o Motivationo NC+IA=NA

o Network Alignment Approaches o At the edgeo In the middle

o When is NA necessary? o Can other schemes achieve ½ the min-cut or more?

C l io Conclusion

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Example: Extended ButterflyExample Extended Butterfly

• K unicast sessions going through the same bottleneck• Routing achieves rate = 1/k Routing achieves rate 1/k • Network coding (with all side links) achieves rate = 1 • Alignment (with sufficiently rich side links) achieves rate =1/2

1 1

ax1+bx2+cx3

x1

1 1

x2

2 2

23x33 3

Example 1 3 unicast sessions, min-cut=1

• No side links ⎥⎤

⎢⎡

bcba

M• Rate 1/3 by any scheme⎥⎥⎥

⎦⎢⎢⎢

⎣

=cbacbaM

1 1

ax1+bx2+cx3

x1

1 1

x2

2 2

24

x33 3

Example 13 unicast sessions, min-cut=1

• One side link ⎥⎤

⎢⎡ +

bccba

M13

• Still rate 1/3 by any scheme⎥⎥⎥

⎦⎢⎢⎢

⎣

=cbacbaM

1 1

ax1+bx2+cx3

x1

1 1

x2

2 2

25

x33 3

Example 1 3 unicast sessions, min-cut=1

• Two receivers have side links (which ones matter) ⎥⎤

⎢⎡ +

bccba

M13

• Here: still 1/3 rate, NA not possible⎥⎥⎥

⎦⎢⎢⎢

⎣ +=

ccbacbaM

32

c

c

c

c

1 1

ax1+bx2+cx3

x1

1 1

x2

2 2

26

x33 3

Example 1 3 unicast sessions, min-cut=1

• Two receivers have side links ⎥⎤

⎢⎡ +

bccba

M13c c

• 1/2 min-cut achievable in 2 slots• by alignment or • or by sharing between session 2 and butterfly 1-3

⎥⎥⎥

⎦⎢⎢⎢

⎣ +=

cbcacbaM

31c c

1 1

ax1+bx2+cx3

x1

1 1

x2

2 2

27

x33 3

Example 1 3 unicast sessions, min-cut=1

• Three receivers have one side link.The optimal rate is ½ the min cut • The optimal rate is ½ the min-cut,

• NA achieves it; routing does not; there are no butterflies• Alignment needed if intelligence is allowed only at the edge• Coding in the middle (RNC + deterministic at 1’,2’,3’) can also achieve ½ the min-cut

1 1

1’

ax1+bx2+cx3

x1

1 1

2’

x3

x2

2 2

3’

x1

28

x33 3

3

x2

When is NA necessary?summary

• Arbitrary network, 3 unicast sessions with min-cut=1:y– Theorem: Whenever NA is possible, another approach (e.g., routing,

butterflies, NC in the middle w/o alignment) can also achieve half the min-cut.P f li – Proof outline:

• Sparsity bound S=1/3: the extended butterfly examples extend to any network• Sparsity bound >=1/2: consider networks where routing rate <= 1/2, and construct a

deterministic NC scheme (in the middle, w/o alignment) that achieves ½ the min-cut

• NA necessary (depending on the topology) to achieve ½ for:– K>3 unicastsK– or min-cut>1

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Example 2:4 unicast sessions, min-cut=1

• Alignment needed: • receiver 3 has 2 equations, 4 unknowns in 2 slots

d 3’ h 3 4 k 2 l• even node 3’ has 3 equations, 4 unknowns in 2 slots• alignment achieves ½ min-cut, no other scheme does

αx1+βx2+γx3+δx4

x1

αx1 βx2 γx3 δ 4

1 1

x2

2 2

x33 3

3’

30x4

4 4

Example 2:5 unicast sessions, min-cut=1

• Alignment needed - effect amplified• all receivers have 2 (nodes in the middle have 2) equation,s with 5 unknowns in 2 slots

li hi ½ i h h d• alignment achieves ½ min-cut, no other scheme does

1 1x1

1

2

1

2x2

2

3 3

2

x33

x4

4 4

31x5

5 5

Example 3: 3 unicasts sessions, min-cut = 2

• Alignment needed:½ ½• It achieves ½min-cut, which is optimal; no other scheme achieves ½ rate.

a1x+b1y+c1zPick c3 s.t.:ba2x+b2y+c2z

32

1

2

1

ccc

bb

+=

1 1c3z

3

2 2

32

3

OutlineOutlineo Motivation

o NC+IA=NA, analogy and dependencies

o Network Alignment Approaches o At the edgeo In the middle

o When is NA possible/necessary? o Can other schemes achieve ½ the min-cut or more?

o Conclusion

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ConclusionConclusion• Network Alignment (NA=NC+IA)

– a systematic approach for network coding across multiple unicasts– guarantees half the min-cut per session

• Future Directions:

– Characterize Feasibility and Performance of NA and their relation to Network Structure

– Develop practical Alignment Algorithms.

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