Proof of Theorem 7.2.1

Chapter VII: Reduction from Network Coding to Index Coding

7.3 Proof of Theorem 7.2.1

Proof. We first give a high level description of the proof. Throughout this proof, we use “untilded” variables forI and “tilded” variables forI. It suffices˜ (Theorem 5.5.1) to show the following two “if-and-only-if” statements:

R˜ ∈ R( ˜I)⇔R∈ lim

λ→0R(I_λ^bf) and R˜ ∈ R^L( ˜I)⇔R∈ lim

λ→0R^L(I_λ^bf).

To prove each of the above two statements, we present two proof “directions”.

In the first direction, we show that if a rate vector R= (R₁,· · · , R_k)is in the capacity region of I_λ^bf, then the corresponding rate vector R˜ = (R,(c_e, e ∈ E_S^c)) is in the capacity region of I. We show the converse in the second˜ direction.

The proof uses the idea of code reduction, in which we transform an (R(1− ρ), , n) network code C for I_λ^bf into a rate ( ˜R(1−ρ),˜ ˜,n)˜ network code C˜for I˜ and vice versa. Theρ-loss in rate tends to zero as˜ ρ, tends to zero and the blocklengthn˜tends to infinity. By taking the closure of these rates, we get the desired result. We now present the proof of the assertion R˜ ∈ R( ˜I) ⇔R ∈

λ→0limR(I_λ^bf). The proof for linear capacity follows from that presented since it uses the same reduction network I˜ and our code reductions preserve code linearity. Parallel arguments for the proof for linear capacity will be contained in square brackets (i.e., [...]).

First direction: R˜ ∈ R( ˜I)⇒ R ∈ lim

λ→0R(I_λ^bf). Define W˜ _S = ( ˜W_s, s ∈S) and W˜ E_Sc = ( ˜We, e ∈ ES^c). Fix any , ρ > 0, we start with a code C˜that is ( ˜R(1−ρ),˜ ˜,n)-feasible for˜ I. For any non-source node˜ v ∈S, denote by

W˜ _Hv =

( ˜W_e⁰, e⁰ ∈E :Out(e⁰) =v),( ˜W_s⁰, s⁰ ∈S : (s⁰, v)∈E_S)

the vector of source and edge messages in the has set H˜_v of v in I. Let the˜ broadcast encoder be denoted by

f˜_B( ˜W_S,W˜_ESc)

and the decoders be denoted by,

˜

g˜te( ˜XB,WH_In(e))

˜

g˜tt( ˜X_B,W_Ht).

We design a codeC that operates onI_λ^bf in two phases, reusing these functions:

1. In the first phase, the super-node s_su broadcasts an overhead message Xα = ˜fB(W, FE_Sc(W))

to nodes in S of I_λ^bf, where F_ESc :Y

s∈S

F^nR2^{˜ s(1−˜ρ)} → Y

e∈E_Sc

F^nc2^{˜ e(1−˜ρ)}

is a suitable function that maps a realization of W˜ _S to a realization of W˜ _ESc.

2. In the second phase, the sources W are transmitted through the rest of I_λ^bf by having the following encoding function for each e ∈E_S^c,

f_e(Z_In(e)) = ˜g˜te(X_α, Z_In(e)).

Each terminal t implements the decoding function Wc_t= ˜g˜tt(X_α, Z_t).

Note that if the original index code operates without error on a message realization ( ˜WS,W˜ E_Sc) = ( ˜wS, FE_Sc( ˜wS)), then by induction on the topological order of G, C will also be able to operate without error on message realization W= ˜w_S.

By Lemma 7.3.1, stated shortly, there exists a functionf_ESc so that the super- node inI_λ^bf can broadcast the overhead messageX_αusing only a small alphabet Σ and that the coding scheme described above operates with error at most 2˜, where n˜⁻¹log|Σ| goes to zero as n˜ goes to infinity and ρ˜ goes to zero.

Therefore, ∀λ,ρ,˜ ˜ > 0, I_λ^bf is (R(1−ρ),˜ 2˜,˜n)-feasible for large enough ˜n, which also means that R ∈ lim

λ→0R(I_λ^bf). [Note that if C˜ were linear, then Lemma 7.3.2 implies a linear encoder FE_Sc (of a rate that tends to zero as ρ˜ tends to zero) for the broadcast facilitator, which will yield a linearC. Hence R∈ lim

λ→0R^L(I_λ^bf).]

Second direction: R˜ ∈ R( ˜I) ⇐ R ∈ lim

λ→0R(I_λ^bf). Fix any , ρ > 0. We start with an(R(1−ρ), , n)-feasible code for I_λ^bf. Denote this code by

C_λ = ({f_e}e∈E_λ,{g_t}t∈T),

where E_λ is the set of edges in instances I and I_λ^bf, respectively. Let I˜ be the corresponding index coding instance for I and letI˜_λ be the index coding instance obtained from I˜ by adding an extra λ to the capacity of the bottle- neck link. Let{F_e}e∈E_λ be the set of global encoding functions corresponding to C_λ and let α be the bottleneck edge of the broadcast facilitator in I_λ^bf. Following [7], we construct an index code C˜for I˜_λ by reusing the code for I, concatenating it with a linear outer code as follow.

1. Let “+” denote the element-wise binary addition operator. We decom- pose the broadcast encoder f˜B(WS,WE_Sc) = ˜XB into components and

define each component,

X˜B = ( ˜XB,e, e∈ES^c ∪ {α}) X˜_B,e =







W˜_e+F_e( ˜W_S) e∈E_S^c Fα( ˜WS) e=α.

2. At the decoders, for each edge terminal ˜t = ˜t_e, each decoder ˜g˜t first computesF_e( ˜W_S)using its side information and the broadcast message,

F_e( ˜W_S) = f_e(( ˜X_B,e⁰ + ˜W_e⁰, e⁰ :Out(e⁰) =In(e)),W˜ _HIn(e))

and then finally obtains cW˜_e= ˜X_B,e+F_e( ˜W_S). Similarly, for eacht˜= ˜t_t that demands W˜_s, each decoder d˜˜t outputs a reconstruction of W˜_t by applying the decoders from Cλ

c˜

W_s =g_t(( ˜X_B,e⁰ + ˜W_e⁰, e⁰ :Out(e⁰) = t),W˜ _Ht).

Note that this index code operates without error on inputs( ˜W_S,W˜ _ESc) = ( ˜w_S,w˜_ESc)if and only if the original network code operates without error on inputs W= ˜w_S.

Therefore, for all λ, ρ, >0, there exists blocklength n such that I˜_λ is( ˜R(1− ρ), , n)-feasible. Thus, R˜ ∈ R( ˜I_λ). By Lemma 7.3.3, stated shortly, we have R(1˜ −_c^λ

B)∈ R( ˜I) for allλ. Since capacity regions are closed,R˜ ∈ R( ˜I). [If we start with a linear C_λ, the resulting C˜_λ will also be linear since the outer code described above is linear. ThusI˜λ is( ˜R(1−ρ), , n)-linearly-feasible. By Theorem 3.2.1, I˜ is ( ˜R(1−ρ), , n)-linearly-feasible where˜ ρ˜tends to zero as ρ and λ tends to zero, which implies R˜ ∈ R^L( ˜I).]

Lemma 7.3.1. ([7, Claim 1]) Let I = (G, S, T) be a network coding instance and let I˜= ( ˜S,T ,˜ H,˜ ˜c_B) be its corresponding index coding instance according to Φ₃. For any index code that is ( ˜R(1−ρ),˜ ˜,n)-feasible on˜ I, there exists a˜ function

FE_Sc :Y

s∈S

F^nR2^{˜ s(1−˜ρ)} → Y

e∈E_Sc

F^nc2^{˜ e(1−˜ρ)}

and a setΣ⊂F^˜n˜2^cB satisfying |Σ| ≤4˜n(1−ρ)(˜ P

s∈SR_s)2^n˜ρ˜cB such that at least a (1−2˜) fraction of source realizations w˜_S ∈Q

s∈SF^nR2^{˜ s(1−˜ρ)} satisfy f˜B( ˜wS, FE_Sc( ˜wS))∈Σ

and the index code operate without error on message realization ( ˜W_S,W˜_ESc) = ( ˜w_S, F_ESc( ˜w_S)).

Lemma 7.3.2. Let I = (G, S, T) be a network coding instance and let I˜ = ( ˜S,T ,˜ H,˜ ˜c_B) be its corresponding index coding instance according to Φ₃. For any linear index code that is ( ˜R(1−ρ),˜ 0,˜n)-feasible on I, there exists a linear˜ transformation matrix FE_Sc,

F_ESc :Y

s∈S

F^˜nR2 ^s(1−˜ρ)→ Y

e∈E_Sc

F^˜nc2 ^e(1−˜ρ),

and a linear subspace Σ⊂ F^˜n˜2^cB satisfying dim(Σ)≤ n˜ρ˜˜c_B such that all w˜_S ∈ Q

s∈SF^nR2^{˜ s(1−˜ρ)} satisfy

f˜_B( ˜w_S,w˜_SF_ESc)∈Σ.

Proof. Let

f˜_B( ˜w_S,w˜_ESc) = ˜w_Sf˜_S+ ˜w_EScf˜_ESc, wheref˜_S and f˜_ESc are P

s∈SnR˜ _s(1−ρ)˜ ×n˜˜c_B and ˜n˜c_B(1−ρ)˜ ×n˜˜c_B matrices over F2, respectively.

For any matrix M, denote by M(i) row i of M and by RS(M) the rowspace of M. For vector spaces V and W, let V +W ={v+w|v ∈ V, w∈ W}. Let B˜_ESc ⊆[˜n˜c_B(1−ρ)]˜ such that{F˜_ESc(i)}_i∈B˜_ESc forms a basis forRS( ˜f_ESc). Let B˜_S1 ⊆[P

s∈SnR˜ _s(1−ρ)]˜ such that

[

i∈B˜S1

{f˜_S(i)}

∪

[

i∈B˜E

{f˜_ESc(i)}

forms a basis for RS( ˜f_S) +RS( ˜f_ESc). Since the index code is assumed to be zero-error,f˜_ESc is full rank, we must have|B˜_S1| ≤˜nρ˜˜c_Bor we would have more thann˜˜c_B independent vectors inF^n˜2^˜cB. Therefore,f˜_S(i)can be decomposed as follows:

f˜_S = ˜f_S1+ ˜f_S2, where for each j ∈[P

s∈SnR˜ s], f˜S1(j)is in the linear span of{f˜S(i)}_i∈B˜S1 and f˜_S2(j)∈RS( ˜f_ESc).

We observe that rank( ˜f_S1)≤n˜ρ˜˜c_B. SinceRS( ˜f_S2)⊆RS( ˜f_ESc), we can find a matrix F_ESc that satisfies

FE_Scf˜E_Sc =−f˜S2.

We therefore have for any w˜_S ∈Q

s∈SF^nR2^{˜ s(1−˜ρ)}

f˜B( ˜wS,w˜SFE_Sc) = ˜wSf˜S+ ˜wSFE_Scf˜E_Sc = ˜wSf˜S1.

Lemma 7.3.3. [24, Lemma 4] For any 0< κ < 1and network coding instance I, let I(κ) be obtained from I by multiplying the capacity value of each edge of I by κ. Then for any rate vector R, R∈ R(I) implies Rκ∈ R(I(κ)).

C h a p t e r 8

THE TIGHTNESS OF THE YEUNG NETWORK CODING OUTER BOUND

The materials of this chapter are published in part as [50].

An outer bound [15, Theorem 15.9] and an exact characterization [16] of the capacity region for network coding are known, whether these regions differ remains an open problem. These bounds are derived based on the notion of entropic vectors and the entropic region Γ^∗ (Section 8.1). So far, and there exists no full characterization on the entropic region Γ^∗, there is no known algorithm to evaluate these bounds. Hence these results are known as implicit characterizations.

Throughout the paper, we shall refer to the network coding outer bound de- veloped in [15, Theorem 15.9] as the Yeung outer bound. The Yeung outer bound is tight when all sources are colocated [17] and for single-source network coding. Whether the outer bound is tight in general remains open. Computa- tionally efficient outer bounds are developed in [18].

The tightness of the Yeung outer bound can be expressed in a form that is similar to the definition of a capacity reduction (Theorem 8.3.1). We apply tools from Chapter 5 and show that the Yeung outer bound is tight if and only of the AERS holds. Before describing these bounds, we begin by defining the entropic region.