eighth asian-european workshop on information theory: fundamental concepts in information theory

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Ahlswede et. al. 2000

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Conclusion

Network Coding and PolyMatroid/Co-PolyMatroid:A Short Survey

Joe Suzuki

Osaka University

May 17-19, 2013Eighth Asian-European Workshop on Information Theory

Kamakura, Kanagawa

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1 / 16

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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

Road Map

From Multiterminal Information Theory to Network Coding

Why Polymatroid/Co-Polymatroid?

Comparing three papers

Future Problems

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1 T. S. Han ”Slepian-Wolf-Cover theorem for a network of channels”,Inform. Control, vol. 47, no. 1, pp.67 -83 1980

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2 R. Ahlswede , N. Cai , S. Y. R. Li and R. W. Yeung ”Networkinformation flow”, IEEE Trans. Inf. Theory, vol. IT-46, pp.1204-1216 2000

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3 Han Te Sun “Multicasting Multiple Correlated Sources to MyltipleSinks over a Noisy Channel Network”, IEEE Trans. on Inform.Theory, Jan. 2011

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2 / 16

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

Network N = (V ,E ,C )

G = (V ,E ): DAGV : finite set (nodes)E ⊂ {(i , j)|i = j , i , j ∈ V } (edge)Φ,Ψ ⊂ V , Φ ∩Ψ = ϕ (source and sink nodes)

Source X ns = (X

(1)s , · · · ,X (n)

s ) (s ∈ Φ): stationary ergodicXΦ = (Xs)s∈Φ, XT = (Xs)s∈T (T ⊂ Ψ)

Channel C = (ci ,j), ci ,j := limn→∞

1

nmaxX ni

I (X ni ,X

nj ) (capacity)

statistically independent for each (i , j) ∈ Estrong converse property

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3 / 16

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

Existing Results assuming DAGs

SinksSources single multiple

single Ahlswede et. al. 2000

multiple Han 1980 Han 2011

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4 / 16

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

Capacity Function ρN (S), S ⊂ Φ

(M, M): pair (cut) of M ⊂ V and M := V \M　EM := {(i , j) ∈ E |i ∈ M, j ∈ M} (cut set)

c(M, M) :=∑

(i ,j)∈E ,i∈M,j∈M

cij

ρt(S) := minM:S⊂M,t∈M

c(M, M)

for each ϕ = S ⊂ Φ, t ∈ Ψ

ρN (S) := mint∈Ψ

ρt(S)

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5 / 16

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

Example 1

Φ = {s1, s2}, Ψ = {t1, t2}, cij = 1, (i , j) ∈ E

@@R �� @@R ��

�� @@R �� @@R? ? ?

s1 s2

t1 t2

ρt1({s2}) = ρt2({s1}) = 1 , ρt1({s1}) = ρt2({s2}) = 2

ρt1({s1, s2}) = ρt2({s1, s2}) = 2

ρN ({s1}) = min(ρt1({s1}), ρt2({s2})) = 1

ρN ({s2}) = min(ρt1({s2}), ρt2({s2})) = 1

ρN ({s1, s2}) = min(ρt1({s1, s2}), ρt2({s1, s2})) = 2

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6 / 16

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

Example 2

Φ = {s1, s2}, Ψ = {t1, t2}, 0 < p < 1, cij is replaced byh(p) := −p log2 p − (1− p) log2(1− p) for −→

@@R �� @@R ��

�� @@R �� @@R? ? ?

s1 s2

t1 t2

ρt1({s2}) = ρt2({s1}) = h(p) , ρt1({s1}) = ρt2({s2}) = 1 + h(p)

ρt1({s1, s2}) = ρt2({s1, s2}) = min{1 + 2h(p), 2}

ρN ({s1}) = min(ρt1({s1}), ρt1({s2})) = h(p)

ρN ({s2}) = min(ρt1({s2}), ρt2({s2})) = h(p)

ρN ({s1, s2}) = min(ρt1({s1, s2}), ρt2({s1, s2})) = min{1+2h(p), 2}

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7 / 16

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

(n, (Rij)(i ,j)∈E , δ, ϵ)-code

Xs : possible values Xs can takefsj : X n

s → [1, 2n(Rsj−δ)] for each s ∈ Φ, (s, j) ∈ E

hsj = ψsj ◦ wsj ◦ φsj ◦ fsj : X ns → [1, 2n(Rsj−δ)]

fij :∏

k:(k,j)∈E

[1, 2n(Rki−δ)] → [1, 2n(Rij−δ)] for each i ∈ Φ, (i , j) ∈ E

hij = ψij ◦ wij ◦ φij ◦ fij :∏

k:(k,j)∈E

[1, 2n(Rki−δ)] → [1, 2n(Rij−δ)]

λn,t := Pr{XΦ,t = X nΦ} ≤ ϵ

gt :∏

k:(k,t)∈E

[1, 2n(Rkt−δ)] → X nΦ for each t ∈ Ψ

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8 / 16

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

Han 1980 (|Ψ| = 1)

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Def: (Rij)(i ,j)∈E is achievable for XΦ and G = (V ,E )

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(n, (Rij)(i ,j)∈E , δ, ϵ)-code exists

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Def: XΦ is transmissible over N = (V ,E ,C )

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(Rij + τ)(i ,j)∈E is achievable for G = (V ,E ) and any τ > 0

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Theorem (|Ψ| = 1)

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XΦ is transmissible over N⇐⇒ H(XS |XS) ≤ ρt(S) for Ψ = {t} and each ϕ = S ⊂ Φ

The notion of network coding appeared first.

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9 / 16

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

Polymatroid/Co-Polymatroid

E : nonempty finite set

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Def: ρ : 2E → R≥0 is a polymatroid on E

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..

1 0 ≤ ρ(X ) ≤ |X |

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2 X ⊂ Y ⊂ E =⇒ ρ(X ) ≤ ρ(Y )

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3 ρ(X ) + ρ(Y ) ≥ ρ(X ∪ Y ) + ρ(X ∩ Y )

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Def: σ : 2E → R≥0 is a co-polymatroid on E

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1 0 ≤ σ(X ) ≤ |X |

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2 X ⊂ Y ⊂ E =⇒ σ(X ) ≤ σ(Y )

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3 σ(X ) + σ(Y ) ≤ σ(X ∪ Y ) + σ(X ∩ Y )

H(XS |XS) is a co-polymatroid on Φ

ρt(S) = minM:S⊂M,t∈M c(M, M) is a polymatroid on Φ

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10 / 16

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

co-polymatroid σ(S) and polymatroid ρ(S)

Slepian-Wolf is available for proof of Direct Part

{(Rs)s∈Φ|σ(S) ≤∑i∈S

Ri ≤ ρ(S), ϕ = S ⊂ Φ} = ϕ

⇐⇒ σ(S) ≤ ρ(S) , ϕ = S ⊂ Φ

@@

@@

@

@@

@@

@@

-

6

R1

R2

a1b1

a2b2a12b12

a1 ≤ R1 ≤ b1

a2 ≤ R2 ≤ b2

a12 ≤ R1 + R2 ≤ b12

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11 / 16

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

Han 2011

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Theorem (general)

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XΦ is transmissible over N⇐⇒ H(XS |XS) ≤ ρN (S) for each ϕ = S ⊂ Φ

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The proof is much more difficult

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|Ψ| = 1 =⇒ ρN is not a polymatroid

Slepian-Wolf cannot be assumed for proof of Direct Part:

{(Rs)s∈Φ|H(XS |XS) ≤∑i∈S

Ri ≤ ρN (S) , ϕ = S ⊂ Φ}

may be empty

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12 / 16

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

Example 1 for uniform and independent X1,X2 ∈ {0, 1}Φ = {s1, s2}, Ψ = {t1, t2}, cij = 1, (i , j) ∈ E

@@R

��

@@R

��

��

@@R

��

@@R

? ? ?

s1 s2

t1 t2

@@R

��

@@R

��

��

@@R

��

@@R

? ? ?

X1X2 X1X2

X1 X2

X1 X2X1 ⊕ X2

ρN ({s1}) = min(ρt1({s1}), ρt2({s2})) = 1

ρN ({s2}) = min(ρt1({s2}), ρt2({s2})) = 1

ρN ({s1, s2}) = min(ρt1({s1, s2}), ρt2({s1, s2})) = 2

H(X1|X2) = H(X1) = 1 , H(X2|X1) = H(X2) = 1

H(X1X2) = H(X1) + H(X2) = 2

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13 / 16

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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey .

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

Example 2 for binary symetric channel with probability p

@@@R

��

@@@R

��

��

@@@R

��

@@@R

? ? ?

s1 s2

t1 t2

@@@R

��

@@@R

��

��

@@@R

��

@@@R

? ? ?

X1X2 X1X2

X1 X2

X1 X2A(X1 ⊕ X2)

AX1 AX2

ρN ({s1}) = min(ρt1({s1}), ρt1({s2})) = h(p)

ρN ({s2}) = min(ρt1({s2}), ρt2({s2})) = h(p)

ρN ({s1, s2}) = min(ρt1({s1, s2}), ρt2({s1, s2})) = min{1+2h(p), 2}H(X1|X2) = h(p) , H(X2|X1) = h(p)

H(X1X2) = 1 + h(p)

A: m × n, m = nh(p) (Korner-Marton, 1979)

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14 / 16

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

Ahlswede et. al. 2000 (|Φ| = 1)

Propose a coding scheme (α, β, γ-codes) to show thatΦ = {s}R = (Ri ,j)(i ,j)∈E

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Theorem (|Ψ| = 1)

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R is achievable for Xs and G⇐⇒ the capacity of R is no less than H(Xs)

α, β, γ-codes deal with non-DAG cases (with loop).(Ahlswede et. al. 2000 is included by Han 2011 but coversnon-DAG cases)

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15 / 16

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Introduction

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Preliminary

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Han 1980

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Han 2011

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Conclusion

Conclusion

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Contribution

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Short survey of the three papers.

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Future Work

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Extension Han 2011 to the non-DAG case (with loop)

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16 / 16

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eighth asian-european workshop on information theory: fundamental concepts in information theory

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