Proof of Theorem 6.2.1 - Capacity Reduction from Multiple Unicast to 2-Unicast . 48

Chapter VI: Capacity Reduction from Multiple Unicast to 2-Unicast . 48

6.5 Proof of Theorem 6.2.1

Proof. We use variables without tildes for I and variables with tilde 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).

I

s₁

t_k t₁

s_k W₁ W_k

Z_t

1 Z_t

W_t^N

Z_t^N c W_t^N rate=R^∗_t

(a)

(b)

b₁

c₁

f₁ h₁ a₁

~t₁

~ t

₂

u

₁

u

I

s₁

t₁

a^k b^k

s^k

~ s

I ~

W~₁ W~₂

Z~a₁ Z~a_k

b

Za₁

b

Z^ak

Z~^s₁ Z~s_k

Z~t₁ Z~t_k

b

Z^t₁

b

^tk

B~₁ B~^k

( c )

Figure 6.2: (a) A channel used to illustrate the channel in Lemma 6.4.1. (b) A Slepian-Wolf coding scheme for I to convert a lossy code to a lossless one.

We prove each statement in two parts, first showing that if a rate vector R= (R₁,· · · , R_k)is in the capacity region ofI_λ^bf, then the corresponding rate vector R˜ = (R_sum, R_sum), where R_sum= P

i∈[k]

R_i, is in the capacity region of I,˜ and then showing the converse.

We now present the proof of the assertion R˜ ∈ R( ˜I) ⇔ R ∈ lim

λ→0R(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 are contained in square brackets (i.e., [...]).

R˜ ∈ R( ˜I) ⇒ R ∈ lim

λ→0R(I_λ^bf): Fix any ˜,ρ >˜ 0. We start with an ( ˜R−ρ,˜ ˜,n,˜ 0)-feasible network code C˜ for I. Under the operation of˜ C˜on networkI, let the edge information variables be denoted (capital letters) as in˜ Figure 6.2(c). In particular, let each Z˜_s be the information variable received

by the old source nodes ∈S and eachZ˜_t be the information variable received by the old terminal node t∈T in I.˜

By Lemma 6.5.1, to be stated shortly, for each source-destination pair of I, (s, t)∈ {(s_i, t_i), i∈[k]}, we have

H( ˜Z_s|Z˜_t)≤˜n˜γ, where ˜γ tends to zero as ρ˜and ˜ tend to zero.

Next, we bound the dependence among the variables ( ˜Zs, s∈S). We have I( ˜W₁; ( ˜Z_s, s∈S))

(a)

≥ I( ˜W₁; ( ˜Z_t, t ∈T))

(b)

≥n(R˜ _sum−ρ˜−γ˜⁰)

for some ˜γ⁰ that tends to zero as˜tends to zero, where (a) is due to the data processing inequality and (b) is due to Fano’s inequality. This gives

H( ˜Z_s, s∈S)≥n(R˜ _sum−ρ˜−γ˜⁰) +H( ˜Z_s, s∈S|W˜₁) = ˜n(R_sum−ρ˜−˜γ⁰).

Further, since each link carrying Z˜_s has a capacity of R_s, the support size of eachZ˜s is bounded above by 2^˜^nR^s, givingH( ˜Zs)≤nR˜ s for eachs∈S. Hence

H( ˜Z_s)≥H( ˜Z_s⁰, s⁰ ∈S)− X

s⁰∈S\{s}

H( ˜Z_s⁰)≥˜n(R_s−ρ˜−˜γ⁰),

s∈S

H( ˜Z_s)

−H( ˜Z_s, s∈S)≤n( ˜˜ ρ+ ˜γ⁰).

If we consider ( ˜Z_s, s ∈ S) as source message variables, those source message variables would be n˜δ-dependent for˜ δ˜= ˜ρ+ ˜γ⁰.

By Lemma 6.4.1, for any > 0, there exists a blocklength n such that I_λ^cf is (R−ρ, , n,0)-feasible whereρandλtend to zero asρ˜and˜tend to zero. This implies that R ∈ lim

λ→0R(I_λ^bf) as desired. [Note that if C˜were a linear code, then ( ˜Z_s, s ∈ S) would be n˜δ-linearly-dependent with˜ W˜₁ as the underlying random process (See Section 5.2). Lemma 6.4.1 therefore yields a linear code for I_λ^cf which in turn implies that R∈ lim

λ→0R^L(I_λ^bf).]

R˜ ∈ R( ˜I) ← R ∈ lim

λ→0R(I_λ^bf): Fix any , ρ, λ > 0. We start with an (R−ρ, , n,0)-feasible network code C forI_λ^bf. By Lemma 5.3.1(2),I is(R− ρ⁰,0, n⁰, δ⁰)-feasible (under code C⁰) for some δ⁰ and ρ⁰ that tend to zero as , λ, and ρ tend to zero. Let W = (W1,· · · , Wk) be the corresponding set

of nδ⁰-dependent sources. [If R ∈ R^L(I_λ^bf), then R−ρ ∈ R^L(I) for some ρ that tends to zero as λ tends to zero (Theorem 3.2.1). Using the fact that R^L(I) =R^L₀(I) (Theorem 9.1.1), there exists a blocklengthn⁰ such that I is (R−ρ⁰,0, n⁰,0)-linearly-feasible (under code C⁰), where ρ⁰ tends to zero as λ tends to zero.]

Since C⁰ is a zero error code, any source realization in the support set W_sp of Wcan be transmitted to the terminals without error. Using this fact, consider the following communication scheme forI, in which we transmit independent˜ source messages W˜₁ and W˜₂:

1. At source node˜s₂, the source messageW˜₂ ∈Fⁿ

0Rsum

2 is split intokchunks, giving W˜₂ = ( ˜Z_a₁,· · · ,Z˜_a_k), where each

Z˜_i ∈Fⁿ

0Ri

is transmitted on edge (u₂, a_i)and forwarded to nodes b_i and f_i. 2. We set W˜₁ to be in Fⁿ

0(Rsum−kρ⁰−δ⁰)

2 . At source node s˜₁, the encoder f_s_˜₁ :Fⁿ

0(Rsum−kρ⁰−δ⁰)

2 → W_sp

maps theith element inFⁿ

0(Rsum−kρ⁰−δ⁰)

2 to theith element in the support set ofW, with respect to an arbitrary but fixed ordering ofFⁿ

0(Rsum−kρ⁰−δ⁰) 2

and W_sp. Note that this mapping is well defined since log₂|W_sp| ≥H(W)≥

i∈[k]

H(W_i)

−n⁰δ⁰ ≥n⁰(R_sum−kρ⁰−δ⁰).

3. Consider operating C⁰ on the sub-networkI that is contained in I. That˜ is, treating

f_s_˜₁( ˜W₁) = ( ˜W_1,1,· · · ,W˜_1,k)

as the dependent sources and applying encoders of C⁰ on nodes V \T and the decoders on each of the terminals t ∈ T. By assumption of the zero-error code C⁰, each terminal t_i is able to obtain an error-free reconstruction of W˜_1,i.

4. The rest of the network applies a “butterfly” network code. For each i ∈ [k], node t_i forwards W˜_1,i to node b_i, which computes the element- wise binary sum (denoted by operator “+”)

W˜1,i+ ˜W2,i

and forwards it to nodes c_i, f_i, and h_i.

5. Finally, each h_i receives variable W˜_1,i from s_i and extracts W˜_2,i from W˜_1,i+ ˜W_2,i, then transmits it tot˜₂. Similarly, eachf_i computesW˜_1,i and transmits it to ˜t1.

Since the butterfly network code introduces no error, this scheme yields a code C˜that is ( ˜R−kρ⁰−δ⁰,0, n⁰,0)-feasible for I. Since˜ ρ⁰ and δ⁰ tend to zero as , ρ and λ tend to zero, R˜ ∈ R(I). [Note that if we start with a linear code C⁰ that is feasible for independent sources, the corresponding support set then becomes W_sp = Q

i∈[k]Fⁿ

0(Ri−ρ⁰)

2 , the encoding function f_s_˜₁ described in Step 2) of the scheme can then be made linear. Further, since the butterfly network code described in steps 4) and 5) is also linear, this yields a linear network codeC˜for I. Thus˜ R˜ ∈ R^L(I).]

Lemma 6.5.1. (Based on [14]) Let R˜ = (Pk

i=1R_i,Pk

i=1R_i). If I˜ is ( ˜R−

ρ,˜,n,˜ 0)-feasible, then for alli∈[k], H( ˜Z_s_i|Z˜_t_i)≤˜n˜γ, where γ˜ goes to zero as

ρ and ˜goes to zero.

Proof. This proof follows the proof idea from [14] and uses variables from Figure 6.2(c). We bound I( ˜B_i; ˜W₂) as follows,

I( ˜Bi; ˜W2)

=I( ˜B_i; ˜W₁,W˜₂)−I( ˜B_i; ˜W₁|W˜₂) By chain rule of mutual information.

=I( ˜B_i; ˜W₁,W˜₂)−I( ˜B_i,W˜₂; ˜W₁) Independence of W˜₁ and W˜₂

=I( ˜B_i; ˜W₁,W˜₂)−I( ˜B_i,Zb_t_i,W˜₂; ˜W₁) Yb_i is a function of W˜₂,B˜_i

≤nR˜ _i−I(Zb_t_i; ˜W₁)

= ˜nR_i−I((Zb_t_j, j ∈[k]); ˜W₁)

+I((Zb_t_j, j ∈[k]\ {i}); ˜W₁|Zb_t_i) By chain rule of mutual information.

≤P

j∈[k]nR˜ _j By decoding condition

−((P

j∈[k]nR˜ _j)−n˜δ˜_i) = ˜n˜δ_i and Fano’s inequality.

Next, we bound I( ˜B_i; ˜W₂,Z˜_s_i), I( ˜B_i; ˜W₂,Z˜_s_i)

≥I( ˜B_i; ˜W₂|Z˜_s_i)

=I( ˜B_i,Z˜_s_i; ˜W₂) Independence of W˜₂ and Z˜_s_i.

=I( ˜B_i,Z˜_s_i,Zb_a_i; ˜W₂) Zb_a_i is a function of B˜_i,Z˜_s_i

≥I(Zb_a_i; ˜W₂)

=I((Zb_a_j, j ∈[k]); ˜W₂)

−I((Zbaj, j ∈[k]\ {i}); ˜W2|Zbai) By chain rule of mutual information.

≥(P

j∈[k]

nR_j−n˜δ˜_i) By decoding condition

− P

j∈[k]\{i}

nRj = ˜nRi−˜nδ˜i and Fano’s inequality.

Finally we bound H( ˜Y_s_i|Y˜_t_i), H( ˜Z_s_i|Z˜_t_i)

=H( ˜Z_s_i|Z˜_t_i,W˜_˜_s₂,( ˜Z_a_j, j ∈[k])) By independence of

( ˜W_˜_s₂,( ˜Z_a_j, j ∈[k]) with ( ˜Z_s_i,Z˜_t_i)

=H( ˜Z_s_i|Z˜_t_i,W˜₂,( ˜Z_a_j, j ∈[k]),B˜_i) Since B˜_i is a function of Z˜_a_i,Z˜_t_i.

≤H( ˜Z_s_i|B˜_i,W˜₂) Conditioning reduces entropy.

=H( ˜Z_s_i|W˜₂)−I( ˜B_i; ˜Z_s_i|W˜₂) Expansion of I( ˜B_i; ˜Z_s_i|W˜₂).

=H( ˜Zsi|W˜2)

−I( ˜B_i; ˜W₂,Z˜_s_i) +I( ˜B_i; ˜W₂) By chain rule of mutual information.

≤2˜nδ˜_i

Here each δ˜_i goes to zero as ρ˜ and ˜ goes to zero. It suffices to set γ˜ = max

i∈[k] 2˜δ_i.

C h a p t e r 7

REDUCTION FROM NETWORK CODING TO INDEX CODING

The materials of this chapter are published in part as [49].

The index coding problem [5] is a special case of the network coding problem that can be interpreted as a “broadcast with side information” problem: a broadcast node has access to all sources and wishes to communicate with several terminals, each having and desiring to reconstruct potentially different sets of sources.

A code reduction from acyclic network coding to index coding is derived in [7].

Thus, any efficient scheme that solves all index coding problems would yield an efficient scheme that solves all acyclic general network coding problems.

Although the connection between network coding and index coding presented in [7] is very general, it does not resolve the question of whether the network coding capacity region can be obtained by solving the capacity region of a corresponding index coding problem.

In this work, we show that the capacity reduction from acyclic network coding to index coding is equivalent to the AERS. Section 7.1 describes the mapping Φ₃ from an acyclic network coding instance to an index coding instance. We describe the main result in Section 7.2 and give its proof in Section 7.3.

7.1 Reduction Mapping Φ3

We begin by describing the reduction from I to I˜ employed in [6], modified slightly here in order to fit our model. Note that in the reduction of [6], the instanceI˜depends only onI (and not on the parametersnandas permitted by Definition 1).

Given network coding instanceI = (G, S, T)with topologyG= (V, E, C)and given any rate vector R, we define index coding problem I˜ = ( ˜S,T ,˜ H,˜ ˜c_B) and rate vectorR˜ as follows. The source setS˜contains one source node s˜_s for each source node s ∈ S and one source node ˜se for each edge e that is not a

~ s_s

1 s~_s

~ t_t

1 ~t_t

~ s_e

1 ~s_e

~ t_e

1 t~_e

~ s_e

~ t_e

Source Nodes S˜={˜s_s₁,s˜_s₂,s˜_e₁,s˜_e₂,s˜_e₃} Destination Nodes ˜t={˜t_t₁,t˜_t₂,˜t_e₁,˜t_e₂,t˜_e₃}

Side H˜˜tt1 ={˜ss2,s˜e1} Information H˜˜t2 ={˜s_s₁,s˜_e₂} Sets H˜_e₃ ={˜s_s₁,˜s_s₂}

H˜_e₁ ={˜s_e₃},H˜_e₂ ={˜s_e₃} Broadcast Capacity ˜c_B = 3

Rate R˜ = (R,1,1,1)

Figure 7.1: The index coding network corresponding to the “butterfly” like network in Figure 2.2 in Section 2.2.

source edge, giving

S˜={˜s_s :s ∈S} ∪ {˜s_e:e∈E_S^c}.

Similarly, terminal set T˜ has one terminal ˜t_t for each terminal t ∈T and one terminal ˜t_e for each edge e that is not a source edge, giving

T˜ ={˜t_t:t∈T} ∪ {˜t_e:e∈E_S^c}.

The “has” set H˜˜t for terminal˜t varies with the terminal type. When˜t= ˜t_e for some edge e∈ E_S^c, H˜˜t includes the source nodes s˜_e⁰ for all edges e⁰ incoming to e and source nodes s˜s for all source edge e⁰ = (s,In(e))∈ E incoming to e inG; when˜t= ˜tt for some terminalt∈T,H˜˜tincludes the source nodess˜e⁰ for all edgese⁰ incoming totand source nodes˜s_sfor all source edgee⁰ = (s, t)∈E incoming to t inG. Thus

H˜˜t=

( {˜s_e⁰ :Out(e⁰) =In(e)} ∪ {˜s_s : (s,In(e))∈E} if ˜t= ˜t_e for some e∈E_S^c {˜s_e :Out(e) =t} ∪ {˜s_s : (s, t)∈E} if ˜t= ˜t_t for somet ∈T. The bottleneck capacity ˜c_B is set to the sum of all finite edge capacities in I, giving

c_B = X

e∈E_Sc

c_e. The rate vector R is mapped to

R˜ = (R,(c_e :e∈E_S^c)).

An example is shown in Figure 7.1, which gives the index coding network I˜ corresponding to the butterfly network from Figure 2.2 of Chapter 2.

7.2 Main Result

The authors in [7] pose the question of whether code reductionΦ₃ can be used to derive a corresponding capacity reduction. Theorem 7.2.1 gives a partial solution to that question.

Theorem 7.2.1(Capacity Reduction from Network Coding to Index Coding).

1. Linear capacity characterization for network coding reduces to linear capacity characterization for index coding. That is, under mappingΦ₃, for any acyclic network coding instance I and rate vector R,

R˜ ∈ R^L( ˜I)⇔R∈ R^L(I).

2. Capacity characterization for network coding reduces to capacity characterization for index coding under mapping Φ₃ if and only if the AERS holds. That is, for any acyclic network coding instanceI and rate vector R,

R˜ ∈ R( ˜I)⇔R∈ R(I)

if and only if the AERSholds.

Since the question of whether the edge removal statement is always true or sometimes false is unresolved, the question of capacity reduction between network coding and index coding remains open in the general case. However, our result provides another way to understand the capacity region of network coding problems via the edge removal statement.

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˜ _H_v =

( ˜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˜_E_Sc)

and the decoders be denoted by,

g˜te( ˜XB,WH_In(e))

g˜tt( ˜X_B,W_H_t).

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_E_Sc :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˜ _E_Sc.

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_E_Sc 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 bottleneck 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˜ _H_In(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˜ _H_t).

Note that this index code operates without error on inputs( ˜W_S,W˜ _E_Sc) = ( ˜w_S,w˜_E_Sc)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^c^B satisfying |Σ| ≤4˜n(1−ρ)(˜ P

s∈SR_s)2^n˜^ρ˜^c^B 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˜_E_Sc) = ( ˜w_S, F_E_Sc( ˜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_E_Sc :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^c^B 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_E_Sc)∈Σ.

Proof. Let

f˜_B( ˜w_S,w˜_E_Sc) = ˜w_Sf˜_S+ ˜w_E_Scf˜_E_Sc, wheref˜_S and f˜_E_Sc 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˜_E_Sc ⊆[˜n˜c_B(1−ρ)]˜ such that{F˜_E_Sc(i)}_i∈B˜_ESc forms a basis forRS( ˜f_E_Sc). Let B˜_S1 ⊆[P

s∈SnR˜ _s(1−ρ)]˜ such that

[

i∈B˜S1

{f˜_S(i)}

∪

[

i∈B˜E

{f˜_E_Sc(i)}

forms a basis for RS( ˜f_S) +RS( ˜f_E_Sc). Since the index code is assumed to be zero-error,f˜_E_Sc is full rank, we must have|B˜_S1| ≤˜nρ˜˜c_Bor we would have more thann˜˜c_B independent vectors inF^n˜2^˜^c^B. 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_E_Sc).

We observe that rank( ˜f_S1)≤n˜ρ˜˜c_B. SinceRS( ˜f_S2)⊆RS( ˜f_E_Sc), we can find a matrix F_E_Sc 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 developed 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.

8.1 Entropic Regions

An approach to characterize the network coding capacity region is to find all information inequalities. For a long time, the knowledge of information inequalities are limited to the basic inequalities due to Shannon [1] (Shannon inequalities). Shannon inequalities has the advantage of being able to be veri- fied by a linear program [46]. Discovery of non-Shannon inequities [51] calls for the definition of the entropic region, which in principle is capable of verifying all information inequalities. Unfortunately, this region cannot be computed explicitly.

In the following, we first define the entropic region Γ^∗(·). For consistency, we

employ notation from [15], [16]. For a given network coding problem I, let N(I) ={W_s, s∈S;X_e, e∈E}={W_S,X_E}

be a collection of discrete random variables corresponding to the source message random variables and the edge information random variables. When the underlying network I is clear, we simply denote N(I) by N. Let HN be the

|2^{|N |} −1|-dimensional Euclidean space where h ∈ HN consists of entries h_A labeled by A ⊆ N, for A 6=∅. A vector h ∈ HN is called an entropic vector (or an entropic function) if there exists a set of|N |random variables such that

∀A⊆ N, A6=∅,

h_A =H(A).

The set of entropic vectors corresponding to N is denoted by Γ^∗(N). When the set N is implied, we simplify Γ^∗(N) toΓ^∗.

Quasi-uniform and Individually-uniform Entropic Region

In this section, we introduce two subsets of the entropic region, namely the quasi-uniform and the individually-uniform entropic region. The quasi-uniform entropic region is employed in [17] to show that the Yeung outer bound is tight when sources are collocated. Our definition of individually-uniform entropic region is inspired by [17]. The individually-uniform entropic region enables us to derive an implicit characterization of the zero-error capacity region (Chap- ter 9).

We first give the definitions of quasi-uniform entropic vectors and entropic region. A set of random variable{X₁, ..., X_n}is quasi-uniform if for any subset α⊆[n], (X_α) = (X_i, i∈α)is uniformly distributed over its support (sp(X_α)), or equivalently,

H(X_α) = log|sp(X_α)|. (8.1)

DefineΓ^∗_Q(N) to be the set of all entropic vectors that correspond to random vectors N that are quasi-uniform. When the set N is implied, we simplify Γ^∗_Q(N) toΓ^∗_Q.

The requirement of (8.1) allows a quasi-uniform variableXi to be transmitted across an edge of capacity H(X_i) by sending the indices of sp(X_i). This is a useful property in mapping an entropic vector directly to a network code in the proof of Theorem 8.3.1.

In the derivation of the zero-error network coding region, one crucial step in the proof is to map a particular network code to a vector in the entropic region.

Since not all network code corresponds to a quasi-uniform random variable, we relax the condition in (8.1) and define individually uniform variables. A set of random variables {X₁, ..., X_n} is individually-uniform if for any i∈[n], the random variableX_i is uniform over its support, or equivalently,

H(X_i) = log|sp(X_i)|.

DefineΓ^∗_U(N) to be the set of all entropic vectors that correspond to random vectors N that are individually-uniform. When the set N is implied, we simplifyΓ^∗_U(N) toΓ^∗_U.

The definition of individually-uniform random variables relaxes that of quasi- uniform random variables by allowing variables with joint distributions that are not uniform. Since Γ^∗_Q ⊆ Γ^∗_U ⊆ Γ^∗, [17, Proposition 2] implies that Γ^∗_U is also dense inΓ^∗ (and Γ^∗) in the sense that for every elementh∈Γ^∗ and every >0there is an element h⁰ ∈Γ^∗_U within Euclidean distance of h.

8.2 The Yeung Outer Bound R_out

We define the following sub-spaces that will be used to describe the Yeung outer bound [15] and the zero-error capacity region in Chapter 9. We denote linear sub-spaces of HN byL_i and the intersection of L₁, L₂, ..., L_m byL_12...m. ForA, B ⊂ N, definehA|B=hA∪B−h_B.

• The sub-space

L₁(I) =

h∈ HN :h_W_S −X

s∈S

h_W_s = 0

describes the set of entropic vectors that corresponds to independent source message variables {h_W_s, s∈S}.

• Fore∈E, define Z_In(e) = (Xe⁰, e⁰ ∈E,Out(e⁰) = In(e)). The sub-spaces L2(I) =

h ∈ HN :∀e= (s, v)∈E, s∈S, h_X_e|Ws = 0 L₃(I) =

h ∈ HN :∀e∈E,In(e)∈/ S, h_X_e|Z_In(e) = 0

describe the set of entropic vectors whose edge information variables match the topology of I. That is, the edge information sent on edge e must be a function of the edge variables entering In(e) or of the source message variable originating at In(e).

Dalam dokumen A Study of Communication Networks through the Lens of Reduction (Halaman 63-69)