Broadcasting and Multicasting Nested Message Sets · special cases: [Cov72, Ber73, Gal74],[KM77]...

Linear deterministic models Why is this model worthwhile to investigate? Simplification: a combination network model Linear deterministic BC Final remarks

Broadcasting and MulticastingNested Message Sets

Shirin Saeedi Bidokhti (TUM)joint work with Vinod Prabhakaran (TIFR) and Suhas Diggavi (UCLA)

May 12, 2013

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Problem setup

Transmitter

Receiver Receiver Receiver Receiver Receiver

Communication Media

2 / 36

Problem setup: broadcasting

Transmitter

Broadcast Channel

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Problem setup: multicasting

Transmitter

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Problem setup: nested message sets

Transmitter

W1,W2,W3

Receiver

W1, W2

Receiver

W1, W2, W3

Receiver

W1, W2, W3

Communication Media

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Problem setup

Transmitter

,W2,W3

Receiver

, W2, W3

Receiver

, W2, W3

Communication Media

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Problem setup

Transmitter

Receiver

W1, W2

Receiver

W1, W2

Receiver

W1, W2

Communication Media

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Problem setup

Transmitter

W1,W2,W3

Receiver

W1, W2

Receiver

W1, W2, W3

Receiver

W1, W2, W3

Communication Media

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Practical motivation

video streaming forms more than 50% of the traffic over mobilenetworks

receiver devices have different QoS demands and one can gain bycoding for these demands

we consider nested message multicast setting, motivated byapplications in Scalable Video Coding (SVC): a base layer and severalrefinement layers

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Previous work

p(y1, y2|x)Encoder

Decoder

-XW0,W1,W2

W0, W2

W0, W1

single-hop broadcast channelCover (1972)special cases: [Cov72, Ber73, Gal74], [KM77]multilevel broadcast channels with nested message sets [BZT07, NE09]3-receiver linear deterministic BC with nested message sets [PDT07]

multi-hop networkclassical (wireline) networks [ACLY00],[EF03,NY04, RW09]

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Previous work

p(y1, y2|x)Encoder

Decoder

-XW0,��W1,W2

W0, W2

W0,��W1

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Previous work

p(y1, y2|x)Encoder

Decoder

-XW0,W1,W2

W0, W2

W0, W1

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Previous work

p(y1, y2|x)Encoder

Decoder

-XW0,W1,W2

W0, W2

W0, W1

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Previous work

p(y1, y2|x)Encoder

Decoder

-XW0,W1,W2

W0, W2

W0, W1

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In this talk...

We look at the broadcast channel through deterministic models

linear deterministic modelour goal in this work is not to find approximate solutions, but toinvestigate new techniques that may be adapted to more general channels

The type of question that we are interested in is the finding of ultimaterates of communication in the nested message set broadcast andmulticast scenarios

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In this talk...

We look at the broadcast channel through deterministic models

linear deterministic modelour goal in this work is not to find approximate solutions, but toinvestigate new techniques that may be adapted to more general channels

The type of question that we are interested in is the finding of ultimaterates of communication in the nested message set broadcast andmulticast scenarios

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Outline

1 Linear deterministic models

2 Why is this model worthwhile to investigate?

3 Simplification: a combination network modelThe challengeLinear encoding schemesOptimality results

4 Linear deterministic BClinear deterministic channel: more generallyCapacity region

5 Final remarks

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Outline

5 Final remarks

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Linear deterministic broadcast channels

Encoder

Decoder

- -��3QQsJJJ

XW1,W2

W1, W2

W1 ∈ FR1 W2 ∈ FR2 , X ∈ Fm, Hi ∈ Fni×m, Yi = HiX

[1 0 0 00 0 1 1

]︸︷︷︸

x1x2x3x4

x3 + x4

[1 0 0 00 1 0 0

]︸︷︷︸

x1x2x3x4

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Encoder

Decoder

- -��3QQsJJJ

XW1,W2

W1, W2

[1 0 0 00 0 1 1

]︸︷︷︸

x1x2x3x4

x3 + x4

[1 0 0 00 1 0 0

]︸︷︷︸

x1x2x3x4

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Encoder

Decoder

- -��3QQsJJJ

XW1,W2

W1, W2

I1 = {1, 2}: Public receivers

[1 0 0 00 0 1 1

]︸︷︷︸

x1x2x3x4

x3 + x4

[1 0 0 00 1 0 0

]︸︷︷︸

x1x2x3x4

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Encoder

Decoder

- -��3QQsJJJ

XW1,W2

W1, W2

I1 = {1, 2}: Public receiversI2 = {3, 4}: Private receivers

[1 0 0 00 0 1 1

]︸︷︷︸

x1x2x3x4

x3 + x4

[1 0 0 00 1 0 0

]︸︷︷︸

x1x2x3x4

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Linear deterministic broadcast channels: objective

Y1 = H1

x1x2x3x4

Y2 = H2

x1x2x3x4

[x1 + 2x2

Y3 = H3

x1x2x3x4

x1x2 + x3

Y4 = H4

x1x2x3x4

x1x1 + x2

x2 + 3x3 + 2x4

Can the source convey the message

w1w2w3

common

private

What we control:The source operations.

X = AW =

a1 a2 a3

a4 a5 a6

a6 a7 a8

a9 a10 a11

This is a matrix completion problem...

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Y1 = H1

x1x2x3x4

Y2 = H2

x1x2x3x4

[x1 + 2x2

Y3 = H3

x1x2x3x4

x1x2 + x3

Y4 = H4

x1x2x3x4

x1x1 + x2

x2 + 3x3 + 2x4

w1w2w3

common

private

X = AW =

a1 a2 a3

a4 a5 a6

a6 a7 a8

a9 a10 a11

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Y1 = H1

x1x2x3x4

Y2 = H2

x1x2x3x4

[x1 + 2x2

Y3 = H3

x1x2x3x4

x1x2 + x3

Y4 = H4

x1x2x3x4

x1x1 + x2

x2 + 3x3 + 2x4

w1w2w3

common

private

X = AW =

a1 a2 a3

a4 a5 a6

a6 a7 a8

a9 a10 a11

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Outline

5 Final remarks

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Motivation I: MIMO broadcast channel in the high SNR

Transmitter

Receiver

Y1=H1X+Z1

Y2=H2X+Z2

Y3=H3X+Z3

the high SNR regimestudy of the interaction of the signals (rather than the background noise)

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Motivation II: wireline networks

source

receiver 1

receiver 2

receiver 3

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source

receiver 1

receiver 2

receiver 3

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source

receiver 1

receiver 2

receiver 3

Y1 = H1

X1X2X3X4

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source

receiver 1

receiver 2

receiver 3

Y1 = H1X

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source

receiver 1

receiver 2

receiver 3

Y1 = H1X

[X1 + 2X2

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source

receiver 1

receiver 2

receiver 3

Y1 = H1X

Y2 = H2X

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source

receiver 1

receiver 2

receiver 3

Y1 = H1X

Y2 = H2X

X1 + 2X2X1 + X4

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source

receiver 1

receiver 2

receiver 3

Y1 = H1X

Y2 = H2X

Y3 = H3X

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Outline

5 Final remarks

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A class of linear deterministic channels

1 0 0 0]

x1x2x3x4

[0 1 0 00 0 1 0

]x1x2x3x4

1 0 0 00 1 0 00 0 0 1

x1x2x3x4

x1x2x4

1 0 0 00 0 1 00 0 0 1

x1x2x3x4

x1x3x4

D1 D2 D3D3 D4D4

x1 x2 x3 x4

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Notation

D1 D2 D3D3 D4D4

E{1} E{2} Eφ

ES, S ⊆ I1: all resource connected to every (public) receiver i ∈ S andnot connected to any public receiver j /∈ S

EpS , S ⊆ I1, p ∈ I2: resource in ES and connected to private receiver p

XS (resp. XpS): vector of symbols carried over ES (resp. Ep

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Notation

D1 D2 D3D3 D4D4

E{1} E{2} Eφ

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Notation

D1 D2 D3D3 D4D4

E{1} E{2} Eφ

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The challenge

W1 = [w1,1]

W2 = [w2,1,w2,2]

D1 D2 D3D3 D4D4

Mixing of the common and private messages is necessary; but in a controlledmanner

One has to reveal (partial) information about the private message to publicreceivers!

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The challenge

W1 = [w1,1]

W2 = [w2,1,w2,2]

D1 D2 D3D3 D4D4

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The challenge

W1 = [w1,1]

W2 = [w2,1,w2,2]

D1 D2 D3D3 D4D4

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The challenge

W1 = [w1,1]

W2 = [w2,1,w2,2]

D1 D2 D3D3 D4D4

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The challenge

W1 = [w1,1]

W2 = [w2,1,w2,2]

D1 D2 D3D3 D4D4

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The challenge

W1 = [w1,1]

W2 = [w2,1,w2,2]

D1 D2 D3D3 D4D4

w1,1 w1,1 + w2,1 w2,1 w2,2

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Rate splitting and linear encoding scheme

let W = [w1,1 . . .w1,R1 w2,1 . . .w2,R2 ]T

let X = A ·Wreveal information about the private messages to public receiversthrough a zero-structured encoding matrix

R1←→α{1,2}↔

α{1}↔α{2}↔ αφ↔

0 0 00 0

|E{1,2}|

|E{1}|

|E{2}|

R2 = α{1,2} + α{2} + α{1} + αφ

choose appropriate parameters, and complete the matrix

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let W = [w1,1 . . .w1,R1 w2,1 . . .w2,R2 ]T

R1←→α{1,2}↔

α{1}↔α{2}↔ αφ↔

0 0 00 0

|E{1,2}|

|E{1}|

|E{2}|

R2 = α{1,2} + α{2} + α{1} + αφ

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let W = [w1,1 . . .w1,R1 w2,1 . . .w2,R2 ]T

R1←→α{1,2}↔

α{1}↔α{2}↔ αφ↔

0 0 00 0

|E{1,2}|

|E{1}|

|E{2}|

R2 = α{1,2} + α{2} + α{1} + αφ

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let W = [w1,1 . . .w1,R1 w2,1 . . .w2,R2 ]T

R1←→α{1,2}↔

α{1}↔α{2}↔ αφ↔

0 0 00 0 0

0 00 0

|Ep{1,2}|

|Ep{1}|

|Ep{2}|

|Epφ|

R2 = α{1,2} + α{2} + α{1} + αφ

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let W = [w1,1 . . .w1,R1 w2,1 . . .w2,R2 ]T

R1←→α{1,2}↔

α{1}↔α{2}↔ αφ↔

0 0 00 0

|E{1,2}|

|E{1}|

|E{2}|

R2 = α{1,2} + α{2} + α{1} + αφ

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At first glance...

A rate pair (R1,R2) is achievable if there exist variables αS, S ⊆ I1, s.t.

Structural constraints:

αS ≥ 0 ∀S ⊆ I1

R2 =∑

Decoding constraints at public receiver i ∈ I1:

R1 +∑S3i

αS ≤∑S3i

Decoding constraints at private receiver p ∈ I2:

R2 ≤∑S∈T

αS +∑

S∈T c

|EpS | ∀T ⊆ 2I1 superset saturated

R1 + R2 ≤∑S⊆I1

|EpS |

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At first glance...

(R1,R2) is achievable if

R1 ≤ minreceivers i

mincuti

R1 + R2 ≤ minprivate receivers p

mincutp

2R1 + R2 ≤ minprivate receivers p

{mincut1 + mincut2 + remaining resources of receiver p

and this turns out to be optimal for two public and any number of privatereceivers, characterizing the capacity region.

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At first glance...

(R1,R2) is achievable if

R1 ≤ minreceivers i

mincuti

R1 + R2 ≤ minprivate receivers p

mincutp

2R1 + R2 ≤ minprivate receivers p

{mincut1 + mincut2 + remaining resources of receiver p

and this turns out to be optimal for two public and any number of privatereceivers, characterizing the capacity region.

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When there are more than two public receivers...

(0, 2) is not achievable using the previous scheme!

W1 = []

W2 = [w2,1,w2,2]

D1 D2 D3 D4D4 D5D5 D6D6

w2,1 w2,1 + w2,2 w2,2

The private information revealed to different subsets of public receivers neednot be independent

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When there are more than two public receivers...

(0, 2) is not achievable using the previous scheme!

W1 = []

W2 = [w2,1,w2,2]

D1 D2 D3 D4D4 D5D5 D6D6

w2,1 w2,1 + w2,2 w2,2

The private information revealed to different subsets of public receivers neednot be independent

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Appropriate pre-encoding

W1 = []

W2 = [w2,1,w2,2]

D1 D2 D3 D4D4 D5D5 D6D6

X{1} X{2} X{3}

pre-encode W2 = [w2,1,w2,2]T into W ′2 = [w′2,1,w

′2,2,w

′2,3]

now use an structured encoding matrix X{1}X{2}X{3}

1 0 00 1 00 0 1

w′2,1w′2,2w′2,3

.20 / 36

Achievable rate-region

A rate pair (R1,R2) is achievable if there exist variables αS, S ⊆ I1, s.t.

αS ≥ 0 ∀φ 6=S ⊆ I1

R2 =∑

Decoding constraints at public receiver i ∈ I1:

R1 +∑S3i

αS ≤∑S3i

Decoding constraints at private receiver p ∈ I2:

R2 ≤∑S∈T

αS +∑

S∈T c

|EpS | ∀T ⊆ 2I1 superset saturated

R1 + R2 ≤∑S⊆I1

|EpS |

The converse holds for three (or fewer) public and any number of privatereceivers, characterizing the capacity region.

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Optimality results

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Example

Is rate pair (R1 = 1,R2 = 2) achievable?

D1 D2 D3 D4D4 D5D5

Is (R1 = 1,R2 = 2) in the innerbound?NO

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Example

D1 D2 D3 D4D4 D5D5

Is (R1 = 1,R2 = 2) in the innerbound?

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Example

D1 D2 D3 D4D4 D5D5

Is (R1 = 1,R2 = 2) in the innerbound?NO

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Rate pair (R1 = 1,R2 = 2) is not in the innerbound

4R1 + 2R2 ≤ 7

α{1,2,3} . . . , α{1} ≥ 0

R2 = α{1,2,3} + . . .+ αφDecoding constraints at public receivers:

R1 + α{1,2,3} + α{1,3} + α{1,2} + α{1} ≤∑

S31 |ES|

×2←

R1 + α{1,2,3} + α{2,3} + α{1,2} + α{2} ≤∑

S32 |ES|

×1←

R1 + α{1,2,3} + α{2,3} + α{1,3} + α{3} ≤∑

S33 |ES|

×1←

Decoding constraints at private receivers p ∈ I2:

R2 ≤ α{1} + α{2} + α{3} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Epφ|

×1, p=5←

R2 ≤ α{1} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Ep{2}|+ |E

p{3}|+ |E

×1, p=4←

...R1 + R2 ≤

∑|Ep

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4R1 + 2R2 ≤ 7

α{1,2,3} . . . , α{1} ≥ 0 ←R2 = α{1,2,3} + . . .+ αφDecoding constraints at public receivers:

R1 + α{1,2,3} + α{1,3} + α{1,2} + α{1} ≤∑

S31 |ES|

×2←

R1 + α{1,2,3} + α{2,3} + α{1,2} + α{2} ≤∑

S32 |ES|

×1←

R1 + α{1,2,3} + α{2,3} + α{1,3} + α{3} ≤∑

S33 |ES|

×1←

R2 ≤ α{1} + α{2} + α{3} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Epφ|

×1, p=5←

R2 ≤ α{1} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Ep{2}|+ |E

p{3}|+ |E

×1, p=4←

...R1 + R2 ≤

∑|Ep

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4R1 + 2R2 ≤ 7

R1 + α{1,2,3} + α{1,3} + α{1,2} + α{1} ≤∑

S31 |ES|×2←

R1 + α{1,2,3} + α{2,3} + α{1,2} + α{2} ≤∑

S32 |ES|

×1←

R1 + α{1,2,3} + α{2,3} + α{1,3} + α{3} ≤∑

S33 |ES|

×1←

R2 ≤ α{1} + α{2} + α{3} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Epφ|

×1, p=5←

R2 ≤ α{1} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Ep{2}|+ |E

p{3}|+ |E

×1, p=4←

...R1 + R2 ≤

∑|Ep

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4R1 + 2R2 ≤ 7

R1 + α{1,2,3} + α{1,3} + α{1,2} + α{1} ≤∑

S31 |ES|×2←

R1 + α{1,2,3} + α{2,3} + α{1,2} + α{2} ≤∑

S32 |ES|×1←

R1 + α{1,2,3} + α{2,3} + α{1,3} + α{3} ≤∑

S33 |ES|

×1←

R2 ≤ α{1} + α{2} + α{3} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Epφ|

×1, p=5←

R2 ≤ α{1} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Ep{2}|+ |E

p{3}|+ |E

×1, p=4←

...R1 + R2 ≤

∑|Ep

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4R1 + 2R2 ≤ 7

R1 + α{1,2,3} + α{1,3} + α{1,2} + α{1} ≤∑

S31 |ES|×2←

R1 + α{1,2,3} + α{2,3} + α{1,2} + α{2} ≤∑

S32 |ES|×1←

R1 + α{1,2,3} + α{2,3} + α{1,3} + α{3} ≤∑

S33 |ES|×1←

R2 ≤ α{1} + α{2} + α{3} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Epφ|

×1, p=5←

R2 ≤ α{1} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Ep{2}|+ |E

p{3}|+ |E

×1, p=4←

...R1 + R2 ≤

∑|Ep

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4R1 + 2R2 ≤ 7

R1 + α{1,2,3} + α{1,3} + α{1,2} + α{1} ≤∑

S31 |ES|×2←

R1 + α{1,2,3} + α{2,3} + α{1,2} + α{2} ≤∑

S32 |ES|×1←

R1 + α{1,2,3} + α{2,3} + α{1,3} + α{3} ≤∑

S33 |ES|×1←

R2 ≤ α{1} + α{2} + α{3} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Epφ|

×1, p=5←

R2 ≤ α{1} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Ep{2}|+ |E

p{3}|+ |E

×1, p=4←

...R1 + R2 ≤

∑|Ep

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4R1 + 2R2 ≤ 7

R1 + α{1,2,3} + α{1,3} + α{1,2} + α{1} ≤∑

S31 |ES|×2←

R1 + α{1,2,3} + α{2,3} + α{1,2} + α{2} ≤∑

S32 |ES|×1←

R1 + α{1,2,3} + α{2,3} + α{1,3} + α{3} ≤∑

S33 |ES|×1←

R2 ≤ α{1} + α{2} + α{3} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Epφ|

×1, p=5←

R2 ≤ α{1} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Ep{2}|+ |E

p{3}|+ |E

×1, p=4←...R1 + R2 ≤

∑|Ep

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Structural constraints: 4R1 + 2R2 ≤ 7α{1,2,3} . . . , α{1} ≥ 0 ←R2 = α{1,2,3} + . . .+ αφDecoding constraints at public receivers:

R1 + α{1,2,3} + α{1,3} + α{1,2} + α{1} ≤∑

S31 |ES|×2←

R1 + α{1,2,3} + α{2,3} + α{1,2} + α{2} ≤∑

S32 |ES|×1←

R1 + α{1,2,3} + α{2,3} + α{1,3} + α{3} ≤∑

S33 |ES|×1←

R2 ≤ α{1} + α{2} + α{3} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Epφ|

×1, p=5←

R2 ≤ α{1} + α{1,2} + α{1,3} + α{2,3} + α{1,2,3} + |Ep{2}|+ |E

p{3}|+ |E

×1, p=4←...R1 + R2 ≤

∑|Ep

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The idea: an outerbound similar to the innerbound

Innerbound

α{1,2,3} ≥ 0

R1 ≤ 1−∑S31

R1 ≤ 2−∑S32

R1 ≤ 2−∑S33

R2 ≤∑

S∈{{1},{1,2},{1,3},{2,3},{1,2,3}

}αS +∑

S∈{{2},{3}φ}c

|EpS |

R2 ≤∑

S∈{{1},{{2},{{3},{1,2},{1,3},{{2,3},{1,2,3}

}αS +∑

S∈{φ}c

|EpS |

Outerbound

H(X{1,2,3}|W1) ≥ 0

R1 ≤ 1− 1n

H(X{1},X{1,2},X{1,3},X{1,2,3}|W1)

R1 ≤ 2− 1n

H(X{2}, X{2,3}|W1)

R1 ≤ 2− 1n

H(X{2,3}, X{3}|W1)

R2 ≤1n

H(X{1}, X{2,3}|W1) + 1

R2 ≤1n

H(X{1}, X{3}, X{2,3}, X{2}|W1)

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The idea: submodularity of the entropy function

4R1 + 2R2

≤ 7− 1n

(2H(X{1}|W1) + H(X{2}, X{2,3}|W1) + H(X{2,3}, X{3}|W1)−H(X{1}, X{2,3}|W1)− H(X{1}, X{3}, X{2,3}, X{2}|W1)

)≤ 7 by sub-modularity

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Summary of the converse proof

1 Write upper-bounds on R1 and R2 using standard techniques and findbounds similar to inner-bounds

2 Take the appropriate weighted sum that cancels out all α parameters

3 Use sub-modularity of entropyUse the fact that α parameters all cancel outSome technicalityDoes not extend to I1 = {1, 2, 3, 4}

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Up to now...

general channels

linear deterministic model

combinationnetworkmodel

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Outline

5 Final remarks

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linear deterministic channels: more generally

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linear deterministic channels: more generally

Y1 = H1

x1x2x3x4

Y2 = H2

x1x2x3x4

[x1 + 2x2

Y3 = H3

x1x2x3x4

x1x2 + x3

Y4 = H4

x1x2x3x4

x1x1 + x2

x2 + 3x3 + 2x4

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Towards more general linear deterministic models

Nullspace of H1Nullspace of H2

Nullspace of H12

Vφ | V{1} | V{2} | V{1,2}

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Nullspace of H12

Vφ | V{1} | V{2} | V{1,2}

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Nullspace of H12

Vφ | V{1} | V{2} | V{1,2}

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Nullspace of H12

Vφ | V{1} | V{2} | V{1,2}

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Nullspace of H12

Vφ | V{1} | V{2} | V{1,2}

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Nullspace of H12

V =[Vφ |

V{1} | V{2} | V{1,2}

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Nullspace of H12

V =[Vφ | V{1} |

V{2} | V{1,2}

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Nullspace of H12

V =[Vφ | V{1} | V{2} |

V{1,2}

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Nullspace of H12

V =[Vφ | V{1} | V{2} | V{1,2}

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Sketch of the achievability proof

change of basis to create simpler equivalent channels: the source sendsX = VAW.

techniques of rate splitting and superposition coding at the sourcethrough a zero-structured code A with parameters to be designed.

decodability conditions at each user and their translations to rankconditions of the channel matrices.

one choice of the structured code that satisfies all the rank conditions.

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Capacity region

The capacity region of a linear deterministic broadcast channel with twopublic receivers (in I1 = {1, 2}) and any number of private receivers (inI2 = {3, · · · ,K}) is given by

R1 ≤ mini∈I

R1 + R2 ≤ minp: private

2R1 + R2 ≤ minp: private

{r{1} + r{2} + r{1,2,i} − r{1,2}},

The rates given above are expressed in log|F|(·).

r{i} , rank(Hi) r{i1,··· ,i|S|} , rank

Hi1...

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Outline

5 Final remarks

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Summary

took a deterministic approach to the problem of broadcasting nestedmessages

characterized a general achievable rate-region over thecombination-network BC: the capacity region when there are three (orfewer) public and any number of private receivers

discussed techniques to use sub modularity in a more systematic way toshow optimality results.

characterized the capacity region for linear deterministic BC for twopublic and any number of private receivers

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Interesting open questions

How to extend the results to MIMO Gaussian broadcast channels?

How may the schemes (over combination network model) be insightfulto linear deterministic broadcast channels with more than two publicreceivers?

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Broadcasting and Multicasting Nested Message Sets · special cases: [Cov72, Ber73, Gal74],[KM77]...

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