Kamakura, Kanagawa, JAPAN May 17-19, 2013 Editors: Hiroyoshi Morita A. J. Han Vinck Te Sun Han Akiko Manada

1. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
Network Coding and PolyMatroid/Co-PolyMatroid:
A Short Survey
Joe Suzuki
Osaka University
May 17-19, 2013
Eighth Asian-European Workshop on Information Theory
Kamakura, Kanagawa
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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

2. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
Road Map
From Multiterminal Information Theory to Network Coding
Why Polymatroid/Co-Polymatroid?
Comparing three papers
Future Problems
.
1 T. S. Han ”Slepian-Wolf-Cover theorem for a network of channels”,
Inform. Control, vol. 47, no. 1, pp.67 -83 1980
.
2 R. Ahlswede , N. Cai , S. Y. R. Li and R. W. Yeung ”Network
information flow”, IEEE Trans. Inf. Theory, vol. IT-46, pp.1204
-1216 2000
3 Han Te Sun “Multicasting Multiple Correlated Sources to Myltiple
Sinks over a Noisy Channel Network”, IEEE Trans. on Inform.
Theory, Jan. 2011
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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

3. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
Network N = (V , E, C)
G = (V , E): DAG
V : finite set (nodes)
E ⊂ {(i, j)|i ̸= j, i, j ∈ V } (edge)
Φ, Ψ ⊂ V , Φ ∩ Ψ = ϕ (source and sink nodes)
Source Xn
s = (X
(1)
s , · · · , X
(n)
s ) (s ∈ Φ): stationary ergodic
XΦ = (Xs)s∈Φ, XT = (Xs)s∈T (T ⊂ Ψ)
Channel C = (ci,j ), ci,j := lim
n→∞
1
n
max
Xn
i
I(Xn
i , Xn
j ) (capacity)
statistically independent for each (i, j) ∈ E
strong converse property
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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

4. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
Existing Results assuming DAGs
Sinks
Sources single multiple
single Ahlswede et. al. 2000
multiple Han 1980 Han 2011
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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

5. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 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) := min
M:S⊂M,t∈ ¯M
c(M, ¯M)
for each ϕ ̸= S ⊂ Φ, t ∈ Ψ
ρN (S) := min
t∈Ψ
ρt(S)
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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

6. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
Example 1
Φ = {s1, s2}, Ψ = {t1, t2}, cij = 1, (i, j) ∈ E
dd‚ © dd‚ ©
© dd‚ © dd‚
c c c
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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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

7. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
Example 2
Φ = {s1, s2}, Ψ = {t1, t2}, 0 < p < 1, cij is replaced by
h(p) := −p log2 p − (1 − p) log2(1 − p) for −→
dd‚ © dd‚ ©
© dd‚ © dd‚
c c c
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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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

8. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
(n, (Rij )(i,j)∈E , δ, ϵ)-code
Xs: possible values Xs can take
fsj : Xn
s → [1, 2n(Rsj −δ)
] for each s ∈ Φ, (s, j) ∈ E
hsj = ψsj ◦ wsj ◦ φsj ◦ fsj : Xn
s → [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{ˆXΦ,t ̸= Xn
Φ} ≤ ϵ
gt :
∏
k:(k,t)∈E
[1, 2n(Rkt −δ)
] → Xn
Φ for each t ∈ Ψ
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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

9. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
Han 1980 (|Ψ| = 1)
Def: (Rij )(i,j)∈E is achievable for XΦ and G = (V , E)
.
.(n, (Rij )(i,j)∈E , δ, ϵ)-code exists
Def: XΦ is transmissible over N = (V , E, C)
.
.
(Rij + τ)(i,j)∈E is achievable for G = (V , E) and any τ > 0
Theorem (|Ψ| = 1)
XΦ is transmissible over N
⇐⇒ H(XS |X¯S ) ≤ ρt(S) for Ψ = {t} and each ϕ ̸= S ⊂ Φ
The notion of network coding appeared first.
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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

10. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
Polymatroid/Co-Polymatroid
E: nonempty finite set
Def: ρ : 2E → R≥0 is a polymatroid on E
.
.
.
1 0 ≤ ρ(X) ≤ |X|
.
2 X ⊂ Y ⊂ E =⇒ ρ(X) ≤ ρ(Y )
.
3 ρ(X) + ρ(Y ) ≥ ρ(X ∪ Y ) + ρ(X ∩ Y )
Def: σ : 2E → R≥0 is a co-polymatroid on E
.
1 0 ≤ σ(X) ≤ |X|
2 X ⊂ Y ⊂ E =⇒ σ(X) ≤ σ(Y )
3 σ(X) + σ(Y ) ≤ σ(X ∪ Y ) + σ(X ∩ Y )
H(XS |X¯S ) 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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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

11. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 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
d
d
d
d
d
d
d
d
d
d
d
E
T
R1
R2
a1b1
a2
b2
a12
b12
a1 ≤ R1 ≤ b1
a2 ≤ R2 ≤ b2
a12 ≤ R1 + R2 ≤ b12
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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

12. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
Han 2011
Theorem (general)
.
.
XΦ is transmissible over N
⇐⇒ H(XS |X¯S ) ≤ ρN (S) for each ϕ ̸= S ⊂ Φ
The proof is much more difficult
.
.
|Ψ| ̸= 1 ̸=⇒ ρN is not a polymatroid
Slepian-Wolf cannot be assumed for proof of Direct Part:
{(Rs)s∈Φ|H(XS |X¯S ) ≤
∑
i∈S
Ri ≤ ρN (S) , ϕ ̸= S ⊂ Φ}
may be empty
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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

13. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
Example 1 for uniform and independent X1, X2 ∈ {0, 1}
Φ = {s1, s2}, Ψ = {t1, t2}, cij = 1, (i, j) ∈ E
d
d‚
©
d
d‚
©
©
d
d‚
©
d
d‚
c c c
s1 s2
t1 t2
d
d‚
©
d
d‚
©
©
d
d‚
©
d
d‚
c c c
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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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

14. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
Example 2 for binary symetric channel with probability p
d
dd‚
©
d
dd‚
©
©
d
dd‚
©
d
dd‚
c c c
s1 s2
t1 t2
d
dd‚
©
d
dd‚
©
©
d
dd‚
©
d
dd‚
c c c
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) (K¨orner-Marton, 1979)
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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

15. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
Ahlswede et. al. 2000 (|Φ| = 1)
Propose a coding scheme (α, β, γ-codes) to show that
Φ = {s}
R = (Ri,j )(i,j)∈E
Theorem (|Ψ| = 1)
.
.
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 covers
non-DAG cases)
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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey

16. Introduction Preliminary Han 1980 Han 2011 Ahlswede et. al. 2000 Conclusion
Conclusion
Contribution
.
.
Short survey of the three papers.
Future Work
.
.
Extension Han 2011 to the non-DAG case (with loop)
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Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey