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- 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 1 / 16 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 2 / 16 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 3 / 16 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 4 / 16 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) 5 / 16 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 6 / 16 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} 7 / 16 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 8 / 16 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. 9 / 16 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 10 / 16 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 11 / 16 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 12 / 16 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 13 / 16 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) 14 / 16 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) 15 / 16 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) 16 / 16 Network Coding and PolyMatroid/Co-PolyMatroid:, A Short Survey