Multi-Cast Channels with Hierarchical Flow Jonathan Ponniah San - - PowerPoint PPT Presentation

Multi-Cast Channels with Hierarchical Flow

Jonathan Ponniah San Jose State University Liang-Liang Xie University of Waterloo ISIT 2020

Outline

• Previous Work
• The One-Relay Channel
• Flow Decomposition
• Main Result
• Overview of Proof
• Conclusion

2

Timeline

3

One-Source Multi-Relay Channel Yassaee and Aref, 2008 One-Source Multi-Relay Channel Xie and Kumar, 2005 Noisy Network Coding Kim and El Gamal., 2011 Multi-Source Multi-Relay Multi-Cast Channels Xie and Kumar, 2007 Multiple-Access Relay Channel Sankar and Kramer, 2007 On Optimal Compressions Wu and Xie., 2013 Short-Length Noisy Network Coding Hou and Kramer, 2016 On Compress-Forward Schemes Ponniah, 2019 Flow Decomposition Ponniah, 2019 Capacity Theorems for the Relay Chanel El Gamal and Cover, 1973 Compress-Forward Decode-Forward

Timeline

4

One-Source Multi-Relay Channel Yassaee and Aref, 2008 One-Source Multi-Relay Channel Xie and Kumar, 2005 Noisy Network Coding Kim and El Gamal., 2011 Multi-Source Multi-Relay Multi-Cast Channels Xie and Kumar, 2007 Multiple-Access Relay Channel Sankar and Kramer, 2007 On Optimal Compressions Wu and Xie., 2013 Short-Length Noisy Network Coding Hou and Kramer, 2016 On Compress-Forward Schemes Ponniah, 2019 Flow Decomposition Ponniah and Xie, 2019 Capacity Theorems for the Relay Chanel El Gamal and Cover, 1973 Compress-Forward Decode-Forward Backward Decoding

Timeline

5

One-Source Multi-Relay Channel Yassaee and Aref, 2008 One-Source Multi-Relay Channel Xie and Kumar, 2005 Noisy Network Coding Kim and El Gamal., 2011 Multi-Source Multi-Relay Multi-Cast Channels Xie and Kumar, 2007 Multiple-Access Relay Channel Sankar and Kramer, 2007 On Optimal Compressions Wu and Xie., 2013 Short-Length Noisy Network Coding Hou and Kramer, 2016 On Compress-Forward Schemes Ponniah, 2019 Flow Decomposition Ponniah and Xie, 2019 Capacity Theorems for the Relay Chanel El Gamal and Cover, 1973 Compress-Forward Decode-Forward Regular Decoding

The One-Relay Channel

Theorem: For some the following rate is achievable: p(x1)p(x2)

Source Relay Destination 1 2 3 R < min{I(X1; Y2|X2), I(X1X2; Y3)}.

Probability of Error:

Decode-Forward

Encoding:

• In block b the source encodes:
• In block b the relay encodes:

m(b) ∈ {1, . . . , 2nR}

m(b − 1)

Codebook Generation: p(x1)p(x2)

Channel Usage: Regular Decoding:

• In block b the relay decodes:
• In block b the destination decodes:

m(b)

m(b − 1)

divided into B blocks of n channel uses

• Goes to zero at the relay if:
• Goes to zero at the destination:

R < I(X1; Y2|X2)

R < I(X2; Y3) + I(X1; Y3|X2) = I(X2X3; Y3)

Probability of Error:

Decode-Forward

Encoding:

• In block b the source encodes:
• In block b the relay encodes:

m(b) ∈ {1, . . . , 2nR}

m(b − 1)

Codebook Generation: p(x1)p(x2)

Channel Usage: Regular Decoding:

• In block b the relay decodes:
• In block b the destination decodes:

m(b)

m(b − 1)

divided into B blocks of n channel uses

• Goes to zero at the relay if:
• Goes to zero at the destination:

R < I(X1; Y2|X2)

R < I(X2; Y3) + I(X1; Y3|X2) = I(X2X3; Y3)

encoding delay

Probability of Error:

Decode-Forward

Encoding:

• In block b the source encodes:
• In block b the relay encodes:

m(b) ∈ {1, . . . , 2nR}

m(b − 1)

Codebook Generation: p(x1)p(x2)

Channel Usage: Regular Decoding:

• In block b the relay decodes:
• In block b the destination decodes:

m(b)

m(b − 1)

divided into B blocks of n channel uses

• Goes to zero at the relay if:
• Goes to zero at the destination:

R < I(X1; Y2|X2)

R < I(X2; Y3) + I(X1; Y3|X2) = I(X2X3; Y3)

Probability of Error:

Decode-Forward

Encoding:

• In block b the source encodes:
• In block b the relay encodes:

m(b) ∈ {1, . . . , 2nR}

m(b − 1)

Codebook Generation: p(x1)p(x2)

Channel Usage: Regular Decoding:

• In block b the relay decodes:
• In block b the destination decodes:

m(b)

m(b − 1)

divided into B blocks of n channel uses

• Goes to zero at the relay if:
• Goes to zero at the destination:

R < I(X1; Y2|X2)

R < I(X2; Y3) + I(X1; Y3|X2) = I(X2X3; Y3)

Decode-Forward

Encoding:

• In block b the source encodes:
• In block b the relay encodes:

m(b) ∈ {1, . . . , 2nR}

m(b − 1)

Regular Decoding:

• In block b the relay decodes:
• In block b the destination decodes:

m(b)

m(b − 1)

Source Relay Destination 1 2 3 “Decoding Phase”

}

Decode-Forward

Encoding:

• In block b the source encodes:
• In block b the relay encodes:

m(b) ∈ {1, . . . , 2nR}

m(b − 1)

Regular Decoding:

• In block b the relay decodes:
• In block b the destination decodes:

m(b)

m(b − 1)

Source Relay Destination 1 2 3 “Encoding Phase”

}

Flow Decomposition

Encoding:

• In block b the source encodes:
• In block b the relay encodes:

m(b) ∈ {1, . . . , 2nR}

m(b − 1)

Regular Decoding:

• In block b the relay decodes:
• In block b the destination decodes:

m(b)

m(b − 1)

1 2 3

1

“Flow”

Flow Decomposition

Encoding:

• In block b the source encodes:
• In block b the relay encodes:

m(b) ∈ {1, . . . , 2nR}

m(b − 1)

Regular Decoding:

• In block b the relay decodes:
• In block b the destination decodes:

m(b)

m(b − 1)

1 2 3 “Layered Partition”

1

1 “Flow”

Flow

1 2 3 4 5 6 7 8

15

2 2 15 15 44 44 44

f(1) = 1

2

− → {2, 3}

15

− → {4}

44

− → {5, 6, 7}

A Flow Set

16

1 2 3 4 5 6 7 8

2 2 15 15 44 44 44

F = {f(1), f(2), f(3), . . . , f(N)}

A Cut-Set Interpretation of the DF Region

1 2 3 4 5 6 7

RS < I(XF (S); Y |X ˜

F (S))

∀S ⊆ S

F({1}) = {1, 2, 3, 4, 5, 6, 7}

“super-source”

17

1 2 3 4 5 6 7 9 10 11 12 13 14

F({5, 12}) = {5, 6, 7, 10, 12, 13, 14}

“super-source”

18

A Cut-Set Interpretation of the DF Region

RS < I(XF (S); Y |X ˜

F (S))

∀S ⊆ S

Layered Partitions

Layer 0 Layer 1 Layer 5 Layer 8

19

Layered Partitions

Layer 0 Layer 2 Layer 2097 Layer 768902305486 Layer 19 Layer 3

20

Hierarchical Flow

21

F = {f(1), f(2), f(3), . . . , f(N)}

Main Result

For any hierarchical flow set and rate vector in the DF region:

F R

22

Theorem 1 (i) There exists a set of causal flow decompositions (ii) There exists an algorithm that finds in

{(F, Li) : i ∈ N}

{(F, Li) : i ∈ N}

that achieve ;

R

time.

P(N)EXP(S)

Main Result

R1

R2

The MARC DF Region

23

R1

R2

Layered Partition #1

24

Main Result

R1

R2

Layered Partition #2

25

Main Result

Main Result

R1

R2

Layered Partition #3

26

Overview of Proof

Encoding:

• In block b node i generates the message:
• In block b node i encodes the index:

mi(b) ∈ {1, . . . , 2nRi}

where is the index assigned to the vector: w(b) ∈ {1, . . . , 2n P

s:i∈f(s) Ri}

w(b)

w(b) = (w1, w2, . . . , wN)

and ws = ( ms(b − ks,i) i ∈ f(s) “1” i / ∈ f(s)

Overview of Proof

Encoding:

• In block b node i generates the message:
• In block b node i encodes the index:

mi(b) ∈ {1, . . . , 2nRi}

where is the index assigned to the vector: w(b) ∈ {1, . . . , 2n P

s:i∈f(s) Ri}

w(b)

w(b) = (w1, w2, . . . , wN)

and ws = ( ms(b − ks,i) i ∈ f(s) “1” i / ∈ f(s)

encoding delay index “Index Coding with Encoding Delays”

Overview of Proof

Regular Decoding:

• Given the destination derives the “virtual” flows:
• For any and , let:

L = {L1, . . . , L|L|}

{v(s) : s ∈ S}

S ⊆ S

0 ≤ l ≤ |L| − 1

Al(S) := {i ∈ v(s) : s ∈ S} ∩ Ll ˜ Al(S) := {∪l

q=0Lq} \ Al(S)

• The destination finds the message vector that satisfies the

following typicality checks for : ˆ m(b) := ( ˆ m1, . . . , ˆ mN)

0 ≤ l ≤ |L| − 1

({xi( ˆ w(b − l)) : i ∈ Al(S)}, {Xi(b − l) : i ∈ ˜ Al(S)}, Yd(b − l)) ∈ T (n)

✏

(XAl(S)∪ ˜

Al(S), Yd)

Overview of Proof

Regular Decoding:

• Given the destination derives the “virtual” flows:
• For any and , let:

L = {L1, . . . , L|L|}

{v(s) : s ∈ S}

S ⊆ S

0 ≤ l ≤ |L| − 1

Al(S) := {i ∈ v(s) : s ∈ S} ∩ Ll ˜ Al(S) := {∪l

q=0Lq} \ Al(S)

• The destination finds the message vector that satisfies the

following typicality checks for : ˆ m(b) := ( ˆ m1, . . . , ˆ mN)

0 ≤ l ≤ |L| − 1

({xi( ˆ w(b − l)) : i ∈ Al(S)}, {Xi(b − l) : i ∈ ˜ Al(S)}, Yd(b − l)) ∈ T (n)

✏

(XAl(S)∪ ˜

Al(S), Yd)

“typicality checks generated from layered partitions”

Overview of Proof

{1}

{2, 3}

{3}

{1, 2}

{1, 3}

{2}

{1}

{1, 2}

{1, 3}

{3}

{2}

{2, 3}

L1

L2

L3

L4

L11

L5

L7

L8

L6

L12

L13

L10

L9

Overview of Proof

{1}

{2, 3}

{3}

{1, 2}

{1, 3}

{2}

{1}

{1, 2}

{1, 3}

{3}

{2}

{2, 3}

L1

L2

L3

L4

L11

L5

L7

L8

L6

L12

L13

L10

L9

regular decoding scheme

Overview of Proof

{1}

{2, 3}

{3}

{1, 2}

{1, 3}

{2}

{1}

{1, 2}

{1, 3}

{3}

{2}

{2, 3}

L1

L2

L3

L4

L11

L5

L7

L8

L6

L12

L13

L10

L9

regular decoding scheme The DF Region (3-user MARC)

Overview of Proof

{1}

{2, 3}

{3}

{1, 2}

{1, 3}

{2}

{1}

{1, 2}

{1, 3}

{3}

{2}

{2, 3}

L1

L2

L3

L4

L11

L5

L7

L8

L6

L12

L13

L10

L9

R

regular decoding scheme L1 = shift(L12, {1, 3})

The DF Region (3-user MARC)

Conclusion

• Regular decoding schemes collectively achieve the DF region for multi-cast

channels with hierarchical flow

• Future work: computability
• Under what conditions is a rate-vector achievable by decode-forward

schemes?

• All-cast channels with arbitrary flow

35

Thanks!

36

jonathan.ponniah@sjsu.edu