In this section, we present a converse bound on the optimal delivery time T∗(t,L), which will serve as a proof for Theorem 4. We will also show that, restricting the cache placement scheme to be homogeneous as in Definition 2, the proposed lower bound naturally implies the optimality of the achievable delivery time in Lemma 3, thus proving case (i) of Theorem 5.
In what follows, we denote the set of demands with distinct file requests byDwc, such that Dwc ,{d∈[N]K:dj 6=di, ∀i6=j}. Finally, we will use the notationWτ(i) to refer to the part of file W(i) exclusively stored in the caches in setτ.
Lower boundingT∗(Z,d,L)
We first present a lemma that provides ageneric lower bound on the delivery time T∗(Z,d,L) as a function of the cache permutationσ∈ SΛ.
Lemma 5. Consider the delivery phase of a shared-cache network with a cache placement described by Z, demand vector d and cache occupancy vector L. Then, under the assumption of uncoded cache placement, the optimal delivery time T∗(Z,d,L) can be lower bounded by the quantity
Tlb,σ(Z,d,L),
Λ
X
λ=1 Lσ(λ)
X
`=1
X
τλ⊆[Λ]\{σ(1),...,σ(λ)}
|Wτ(dλσ(λ)(`))|, (4.30) where σ denotes an arbitrary permutation of the set of caches [Λ], i.e. σ∈SΛ.
Proof. The lemma is direct from equation (3.44) in Section 3.4, employed here with N0 = 1. Furthermore, equation (3.44) was stated for a specific permutation denoted by σs; it can be easily verified that it actually holds for every permutation σ∈SΛ, as stated in the above lemma.
Next, aflexible lower bound onT∗(Z,d,L) can be constructed as a weighted average of the Λ! possible lower bounds that stem from (4.30). Thus, it holds that
T∗(Z,d,L)≥ X
σ∈SΛ
wσTlb,σ(Z,d,L), (4.31) where the weights{wσ}satisfyP
σ∈SΛwσ = 1.
Remark 6. The use of the weighted average is key to the construction of a tight bound, and it is one of the key contributions of our work. In fact, in many works that follow the index coding approach originally proposed in [4] (and the similar genie-aided approach used in [5]) to construct lower bounds on coded caching problems, it is common to lower bound the delivery time as a uniform average of several bounds driven by a certain cache permutation σ (see for example [5, 64, 65]). This method has been shown to work well in
settings that are uniform in terms of number of users per cache and sizes of the caches.
However, whenever the system model is affected by some heterogeneity, this approach can easily fail to meet the goal. In our particular setting, we have shown in [43] that the uniform average (i.e. wσ = Λ!1) leads to a loose bound which proved our achievable performance to be optimal within a gap that scales linearly with the normalized total cache sizet. On the other hand, in this work, we show that a careful choice of the weights can be crucial to derive a tight bound. We believe that this approach can be helpful to derive lower bounds for other generic heterogeneous coded caching problems.
In our derivation, for any p∈[Λ]0, the choice of the weightswσ is taken as
where we have used the upper index(p) to highlight the dependency of the value of the weights on the choice of the parameter p. Forp= 0 we definew(0)σ , |S1Λ|.
Lower bound onT∗(t,L)
We now proceed to derive the lower bound on the optimal delivery time T∗(t,L). In this respect, we start by bounding from below — for a fixed cache placementZ — the worst-case delay by the average rate over all demands with distinct requests as
T∗(Z,L),max Combining (4.33) and (4.31), it yields
T∗(Z,L)≥ 1
where in (a) we have used (4.30).
Next, we write the R.H.S of (4.35), which we denote by Tlb,p(Z,L), in the more
where the value ofc(p)τ,n is expressed in the following lemma. Before presenting the lemma, let us introduce the notation ˙Lq , Q|q|
j=1Lq(j) for any subset q ⊆ [Λ] for the sake of readability.
Lemma 6. The value of c(p)τ,n does not depend on the file index n and it takes the form Proof. The proof of this lemma is presented in Appendix B.3.
We can tighten the bound onT∗(Z,L) by selecting the most restricting p, such that T∗(Z,L)≥ max
p∈[Λ]0
Tlb,p(Z,L). (4.38)
From the definition of the optimal delayT∗(t,L) in (4.3), using (4.38) and substituting (4.37) in (4.36) yields
where we have introduced the notationaτ , N1
|Wτ(1)|+|Wτ(2)|+...+|Wτ(N)| . Now, considering the constraint on the total files size and the one on the sum cache size, a lower bound on the optimal delayT∗(t,L) can be obtained from the solution of the following linear program
Let us now focus on the proof of the lower bound in Theorem 4, and later we will consider the proof for the optimality in Theorem 5
Proof of Theorem 4
In what follows, we further bound from below the constructed lower bound in (4.41). First of all, let us introduce some useful notation. We define ˜c(p)τ as ˜c(p)τ ,
Then, we define the subset of cardinality j that minimizes ˜c(p)τ as τj?, i.e., τj? ,
Proof. The proof is relegated to Appendix B.5.
Now, jointly employing (4.42) in (4.40) and using the max-min inequality yields minZ max
We present now a result that will be instrumental for the following step in the derivation.
Proposition 2. The sequence
˜ c(p)τ?
j is a decreasing sequence in j ∈[Λ]0.
Proof. The proof is relegated to Appendix B.4.
We know focus on the inner optimization problem in (4.45) for a fixed p ∈ [Λ]0, and we follow the same steps as in [5] to solve it analytically. In this respect, we know from Proposition 2 that
˜ c(p)τ?
j is a decreasing sequence inj ∈[Λ]0, and thus its convex envelope is a decreasing and convex sequence. Thus, applying Jensen inequality, we obtain that
T∗(t,L)≥ max
p∈[Λ]0
Tlow(p,L)(t) (4.46)
where
Tlow(p,L)(t),Convj∈[Λ]0
˜ c(p)τ?
j
. (4.47)
Theorem 4 simply follows from the fact that
p∈{0,1,...,Λ}max Tlow(p,L)(t)≥ Tlow(¯t,L)(t), (4.48) where ¯t= round(t).
Proof of case (i) in Theorem 5
We recall that, under the assumption of homogeneous caching,t is integer, thus implying that ¯t= round(t) =t. Forp=t, and under the aforementioned assumption, the problem in (4.45) reduces to
min¯aj
˜ c(t)τ?
t ¯at subject to a¯t= 1.
(4.49) It is easy to verify that, for allτ ∈[Λ] :|τ|=t, it holds that
˜
c(t)τ = ˜c(t)τ? t =
P
q∈C[Λ]t+1 t+1Q
j=1
Lq(j) P
q∈Ct[Λ]
t
Q
j=1
Lq(j)
, (4.50)
which, together with (4.49), directly results in T∗(t,L)≥c˜(t)τ?
t , (4.51)
which concludes the proof of case (i) in Theorem 5.