Achievable Scheme for Kγ = 2

In this section we present a caching and delivery scheme for the case ofKγ= 2. The scheme preserves the full local caching gain as in [26], and achieves a coding gain that strictly exceedsKγ+ 1.

5.3.1 Cache Placement Scheme

In the cache placement phase, we first split each fileW⁽ⁿ⁾ intoS = ^K(K−2z+2)₄ subfiles W_T⁽ⁿ⁾ for each pair T ,{T1,T2} from the set

Ψ,{T :T1∈[K−z],T2∈[T1+z: 2 : min{K−z+T1, K}]} (5.5) where in the above we used the notation [a:b:c] to denote an ordered set of integers⁵, fromatoc, in additive steps ofb. After splitting the files, each cachek is filled as follows Zk={W_T⁽ⁿ⁾ | ∀n∈[N],∀T 3k}. (5.6) Verifying the memory constraint To show that the cache placement satisfies the per-cache memoryM = ^2N_K, we focus without loss of generality on cache 1, and note that the

5Note thatbmay be negative.

number of subfiles (per file) in this cache is ^K−2z₂ + 1. Recalling that each such subfile is of size 1/S, yields

N ^K−2z₂ + 1

K(K−2z+2) 4

= 2N K =M thus proving that the memory constraint is satisfied.

5.3.2 Delivery Scheme

The delivery scheme has two phases, where the first phase transmits XORs composed of 4 subfiles, while the second phase transmits XORs composed ofKγ+ 1 = 3 subfiles.

Phase 1

Recall from above that any subfile W_T^(dk), k ∈ T ∈ Ψ (T1 ∈ [k : 1 : hk+z−1i]), is already available at one of the caches seen by the requesting userk∈[K]. For each user k∈[K], the aim of this first phase is to serve the subfiles in the set

n W_T^(dk)

1,T₂ | ∀{T1,T2} ∈Ψ :T1 ∈ h[k−z+1:1:k−1]i∪h[k+z:1:k+2z−2]io

. (5.7)

For anyk∈[K], let us define the following two sets Ω_k,1,[k+ 1 : 1 :k+z−1]

Ω_k,2,[k−z+ 1 : 1 :k−1] (5.8) and the set

Bk,j ,[Ω_k,1(j) +z: 2 :Uk,j] (5.9) where

Uk,j,

hΩ_k,2(j)−zi, if Ω_k,2(j)−z <0

K, otherwise. (5.10)

and where in the above we used the notation Γ(j) to denote the j-th element of an ordered set Γ. Next, for anyk∈[K] and any j∈[z−1] we form the following XOR

X(k, j, m) =W_hΩ^(dhk−z+1i)

k,1(j)i,Bk,j(m)⊕W_hΩ^(dk)

k,2(j)i,Bk,j(m)

⊕W_hΩ^{(dhBk,j(m)−z+1i)}

Bk,j(m),1(j)i,k ⊕W_hΩ^(dBk,j(m))

Bj,k(m),2(j)i,k. (5.11)

Creating the above XORs for every m∈[|Bk,j|] and every k∈[K], spans the entire set of requested files in (5.7), and what we show below is that each component subfile (in the XORs) can be successfully decoded by its corresponding user.

Decoding Consider any XOR as in (5.11) and let us focus on the subfilesW_Ω^(dhk−z+1i)

k,1(j),Bk,j(m)

andW_Ω^(dk)

k,2(j),Bk,j(m) which are desired by users hk−z+ 1i andk, respectively. By the cache placement phase, we notice that the subfilesW_T^{(dhBk,j(m)−z+1i)}

Bk,j(m),z,1(j),k andW_T^(dBk,j(m))

bj,k,z(m),z,2(j),k

are both cached in cache k, thus enabling both users hk−z+ 1i and k to subtract these subfiles fromX(k, j, m). Next, we also notice that userhk−z+ 1i can cache out W_Ω^(dk)

k,2(j),Bk,j(m)since Ω_k,2(j)∈ ∪^i∈Chk−z+1iZⁱ. Similarly, userkcan removeW_Ω^(dhk−z+1i)

k,1(j),Bk,j(m)

fromX(k, j, m) because Ωk,1(j)∈ ∪^i∈CkZi. Hence, we conclude that any XOR in (5.11) is decodable by both users k andhk−z+ 1i. In the same way, it can be shown that usershB_k,j(m)−z+ 1i andB_k,j(m) can successfully decode their own requested subfiles.

Phase 2

We start by defining the set δ=

"

z: 2 :

$K−2−z 3

%#

as well as the following set of triplets Θ =

(

(θ₁, θ₂, θ₃)|θ₁ =δ(j), θ₂ = 2δ(j) +z+ 2(i−1), θ₃=δ(j) + 2(i−1), j∈[|δ|], i∈

K−3δ(j)−z 2

)

. (5.12)

For each tripletθ∈Θ and for each p∈[K], we generate the following two XORs Yp(θ,1) =W_hθ^(dp)

2−θ₁+p−1i,hθ2+p−1i⊕W_p,hθ^{(dhθ2−θ1−z+pi)}

2+pi

⊕Whz−2+pi,hz−2+θ^{(dhθ2+p−1i)} 3+pi (5.13)

Yp(θ,2) =W_hθ^(dp)

2−θ1+pi,hθ2+pi⊕W_p,hθ^{(dhθ2−θ1−z+1+pi)}

2+pi

⊕Whz+p−1i,hz+θ^{(dhθ2+p−1i)} 3+p−1i. (5.14) As we did for phase 1, below we show that each subfile in the above XORs from (5.13) and (5.14) can be decoded successfully by their requesting user.

Decoding For anyp∈[K], we will prove that the subfiles inYp(θ,1) can be decoded by their intended users. The decodability proof forY_p(θ,2) will then follow directly.

User p can cache out subfilesW_p,hp+θ^{(dhθ2−θ1−z+pi)}

2i and Whz−2+pi,hz−2+θ^{(dhθ2+p−1i)} 3+pi from Yp(θ,1) since their subscripts p ∈ ∪^i∈CpZ_i and hz−2 +pi ∈ ∪^i∈CpZ_i correspond to the caches

that user p is connected to. Next, we notice that user hθ₂−θ₁−z+pi is connected to cache hθ₂−θ₁+p−1i and thus it can cache out subfile W_hθ^(dp)

2−θ1+p−1i,hθ2+p−1i. The same user hθ₂−θ₁−z+pi=hδ(j) +p+ 2(i−1)i has access to subfiles with subscripts in the set h[δ(j) +p+ 2(i−1) : 1 : δ(j) +p+ 2(i−1) +z−1]i. A relabelling of hθ₃+p+z−2i to hδ(j) +z+p−2 + 2(i−1)i highlights that user hθ₂−θ₁−z+pi can also removeWhz−2+pi,hz−2+θ^{(dhθ2+p−1i)} 3+pi from Y_p(θ,1), and hence obtains its desired subfile W_p,hθ^{(dhθ2−θ1−z+pi)}

2+pi successfully. Finally, we recall that userhθ2+p−1i has access to caches Chθ2+p−1i =h[θ₂+p−1 : 1 :θ₂+p−2 +z]i and hence can successfully decode its desired subfile since it can cache out subfiles W_hθ^(dp)

2−θ₁+p−1i,hθ2+p−1i and W_p,hθ^{(dhθ2−θ1−z+pi)}

2+pi .

Performance of the algorithm

We observe that each generated XOR serves a different set of 3 or 4 subfiles, which can all be decoded. Therefore, we can conclude that the achieved sum DoF is between 3 and 4.

From the description of the algorithm, it can be seen that the proposed delivery scheme is valid for any demand vectord. Following the construction, we can readily count the total number of XORs transmitted during phase 1 and phase 2 to respectively be X1 and X₂ from (5.2), which concludes the proof of the achievable delay in Theorem 7.

5.3.3 Illustrative Example

In this subsection we offer an example that may help to better understand the scheme.

We consider the setting with parameters K = 10,Kγ = 2, and z= 2. For the sake of simplicity, we will use 0 to represent the index 10 and also, when describing a double indexi, j, we will omit the comma.

In the placement phase, we first split each file, according to (5.5), into S = 20 equally-sized subfiles with indices

Ψ ={13,15,17,19,24,26,28,20,35,37,39,46,48,40,57, 59,68,60,79,80}

and we then fill the caches according to (5.6) as follows

Z1 ={W₁₃⁽ⁿ⁾, W₁₅⁽ⁿ⁾, W₁₇⁽ⁿ⁾,W₁₉⁽ⁿ⁾, ∀n∈[N]} Z2 ={W₂₄⁽ⁿ⁾, W₂₆⁽ⁿ⁾, W₂₈⁽ⁿ⁾,W₂₀⁽ⁿ⁾, ∀n∈[N]}

...

Z9 ={W₁₉⁽ⁿ⁾, W₃₉⁽ⁿ⁾, W₅₉⁽ⁿ⁾,W₇₉⁽ⁿ⁾, ∀n∈[N]} Z0 ={W₂₀⁽ⁿ⁾, W₄₀⁽ⁿ⁾, W₆₀⁽ⁿ⁾,W₈₀⁽ⁿ⁾, ∀n∈[N]}.

We notice that the cache placement guarantees an empty intersection Zk∩ Zhk+1i=∅of any z= 2 neighboring caches, and thus a full local caching gain (zγ= 0.4).

In the delivery phase we consider the worst-case demand vector d= (1,2, . . . ,9,0).

We will list the XORs of phase 1 and phase 2, but before doing that, let us offer some intuition on the design of the XORs of the first phase.

Let us consider a pair of users (say, users 0 and 1) that “see” a common cache (in this case, cache 1). For these two users we will create a generic XOR

W_σ⁽⁰⁾_1,σ2 ⊕W_˜σ⁽¹⁾

1,˜σ2 (5.15)

which will be combined with another XOR W_τ⁽³⁾_1,τ2 ⊕W_˜τ⁽⁴⁾

1,˜τ2 (5.16)

which is meant for another pair of users, say 3 and 4, that again share a common cache (cache 4). Combining the two XORs yields a new XORX=Wσ⁽⁰⁾1,σ2 ⊕W_˜σ⁽¹⁾

1,˜σ2 ⊕Wτ⁽³⁾1,τ2 ⊕ W_τ˜⁽⁴⁾_1,˜τ2 of 4 subfiles. To guarantee decoding for all, we will set σ1 = ˜σ1 = 4 to let user 3 and user 4 “cache out” from X the subfiles in (5.15) and similarly we set τ₁ = ˜τ₁ = 1 in order to let users 0 and 1 cache out the XOR in (5.16). Next, we chooseσ₂ = 2 so that user 1 can remove subfileW_4,2⁽⁰⁾ fromX and ˜σ₂= 0 to let user 0 remove subfileW_4,0⁽¹⁾ fromX. A similar choice of τ₂ and ˜τ₂ will result in⁶

X =W₂₄⁽⁰⁾⊕W₄₀⁽¹⁾⊕W₁₅⁽³⁾⊕W₁₃⁽⁴⁾. (5.17) The list of XORs sent during phase 1 is given below.

X(1,1,1) =W₂₄⁽⁰⁾⊕W₄₀⁽¹⁾⊕W₁₅⁽³⁾⊕W₁₃⁽⁴⁾ X(1,1,2) =W₂₆⁽⁰⁾⊕W₆₀⁽¹⁾⊕W₁₇⁽⁵⁾⊕W₁₅⁽⁶⁾ X(1,1,3) =W₂₈⁽⁰⁾⊕W₈₀⁽¹⁾⊕W₁₉⁽⁷⁾⊕W₁₇⁽⁸⁾ X(2,1,1) =W₃₅⁽¹⁾⊕W₁₅⁽²⁾⊕W₂₆⁽⁴⁾⊕W₂₄⁽⁵⁾ X(2,1,2) =W₃₇⁽¹⁾⊕W₁₇⁽²⁾⊕W₂₈⁽⁶⁾⊕W₂₆⁽⁷⁾ X(2,1,3) =W₃₉⁽¹⁾⊕W₁₉⁽²⁾⊕W₂₀⁽⁸⁾⊕W₂₈⁽⁹⁾ X(3,1,1) =W₄₆⁽²⁾⊕W₂₆⁽³⁾⊕W₃₇⁽⁵⁾⊕W₃₅⁽⁶⁾ X(3,1,2) =W₄₈⁽²⁾⊕W₂₈⁽³⁾⊕W₃₉⁽⁷⁾⊕W₃₇⁽⁸⁾ X(3,1,3) =W₄₀⁽²⁾⊕W₂₀⁽³⁾⊕W₁₃⁽⁹⁾⊕W₃₉⁽⁰⁾ X(4,1,1) =W₅₇⁽³⁾⊕W₃₇⁽⁴⁾⊕W₄₈⁽⁶⁾⊕W₄₆⁽⁷⁾ X(4,1,2) =W₅₉⁽³⁾⊕W₃₉⁽⁴⁾⊕W₄₀⁽⁸⁾⊕W₄₈⁽⁹⁾ X(5,1,1) =W₆₈⁽⁴⁾⊕W₄₈⁽⁵⁾⊕W₅₉⁽⁷⁾⊕W₅₇⁽⁸⁾ X(5,1,2) =W₆₀⁽⁴⁾⊕W₄₀⁽⁵⁾⊕W₁₅⁽⁹⁾⊕W₅₉⁽⁰⁾

6This intuition can be generalized forz >2 and used as a baseline in the general description of the scheme presented in Section 5.3.2.

X(6,1,1) =W₇₉⁽⁵⁾⊕W₅₉⁽⁶⁾⊕W₆₀⁽⁸⁾⊕W₆₈⁽⁹⁾ X(7,1,1) =W₈₀⁽⁶⁾⊕W₆₀⁽⁷⁾⊕W₁₇⁽⁹⁾⊕W₇₉⁽⁰⁾.

In phase 2, theX2 = 20 transmissions from (5.13) and (5.14) are cyclically generated as shown below.

Y1(θ,1)=W₄₆⁽¹⁾⊕W₁₇⁽³⁾⊕W₁₃⁽⁶⁾, Y₂(θ,1)=W₅₇⁽²⁾⊕W₂₈⁽⁴⁾⊕W₂₄⁽⁷⁾ Y₃(θ,1)=W₆₈⁽³⁾⊕W₃₉⁽⁵⁾⊕W₃₅⁽⁸⁾

...

Y₀(θ,1)=W₃₅⁽⁰⁾⊕W₆₀⁽²⁾⊕W₂₀⁽⁵⁾ and

Y₁(θ,2)=W₅₇⁽¹⁾⊕W₁₇⁽⁴⁾⊕W₂₄⁽⁶⁾ Y₂(θ,2)=W₆₈⁽²⁾⊕W₂₈⁽⁵⁾⊕W₃₅⁽⁷⁾ Y₃(θ,2)=W₇₉⁽³⁾⊕W₃₉⁽⁶⁾⊕W₄₆⁽⁸⁾

...

Y0(θ,2)=W₄₆⁽⁰⁾⊕W₆₀⁽³⁾⊕W₁₃⁽⁵⁾.

In the end, we haveS = 20,X₁ = 15 and X₂ = 20 which gives T = X1+X2

S = 35 20 and a coding gainK(1−zγ)/T = 3.43.