We here focus on the topology-partially-aware scenario, and we assume in the delivery phase that the cache occupancy vectorL is the realization of a collection of independent random variablesL= (L1,L2, . . . ,LΛ) with expected value ¯L= ( ¯L1,L¯2, . . . ,L¯Λ), where we have used ¯Lλ =E[Lλ], λ∈[Λ].
Fact 1. For the (L,t) topology-partially-aware scenario, the expected delivery time satisfies E[T∗(t,L,L)]¯ ≥T(t,L).¯ (4.61)
Proof.
where (4.63) follows from the independence of the random variables{Lλ} and (4.64) is proven in Appendix B.6.
For the above setting, we will consider 3 different schemes. The first is the single-antenna topology-agnostic scheme of Section 3.3.1 (equivalent to the one in [32]), which does not exploit the knowledge of ¯L for the cache placement, which is done according to MAN cache placement. The second scheme, which we will refer to asTPA ECS scheme (topology-partially-aware equal-cache-size scheme), is the one achieving the delivery time in equation (24a) of [59] for the case when there are no cache-less users. We notice that [59] assumes that all the caches have the same size, which cannot be optimized.
The cache placement of the aforementioned scheme in [59] partitions the set of caches in Ggroups such that all the caches in the same group store the same content, and it applies MAN placement for Gcaches/users. If the cache occupancy vector L is known in advance during placement, the best partition is chosen by leveraging L in order to minimize the delivery time. In the topology-partially-aware scenario considered here, L cannot be known in advance, thus we select the partition according to the average cache occupancy vector ¯L. Delivery is performed by means of the multi-round scheme for the single-antenna setting in Section 3.3.1 (se also [32]). Finally, the third scheme is our proposed topology-partially-aware scheme achieving the delivery time in Theorem 6.
We assume that each random variableLλ follows a poisson distribution with mean ¯Lλ, i.e. Lλ ∼Poiss( ¯Lλ), and we consider the scenario where the expected number of users per cache is ¯L= (20,20,8,6,4,2). Figure 4.2 shows the average delivery time achieved by the three schemes, the memory-rate curve (t, T(t,L)) from Lemma 3, and the average¯
memory-rate curve (t,E[T(t,L)]). It is evident that the proposed scheme with optimized shared caches largely outperforms the other two scheme, thus highlighting the importance of proper memory allocation. The plot also confirms Fact 1, and it shows thatT(t,L) is¯ a good approximation ofE[T(t,L)], i.e. T(t,L)¯ ≈E[T(t,L)].
Multi-Access Shared-Cache Networks
This chapter considers theK-user cache-aided shared-link channel where each user has access to exactlyzneighbouring caches of normalized sizeγ, and where each cache assists exactly z users. For this setting, for two opposing memory regimes, we propose novel caching and coded delivery schemes which maximize the local caching gain, and achieve a coding gain larger thanKγ+ 1 despite the fact that the cache replication factor remains Kγ irrespective of z. Interestingly, when z= K−1Kγ , the derived optimal coding gain is Kγz+ 1, matching the performance of a hypothetical scenario where each user has its own dedicated cache of sizezγ.
Previous Works on Multi-Access Coded Caching
The1 shared-cache setting where each user has access to more than one cache was explored for the first time in [26] which considered a K-user shared-link setting, where each user is assisted by exactly z >1 caches, and where each cache can serve exactlyz users. In this context, the work in [26] provided a caching and delivery strategy whose centralized variant2 achieves a worst-case NDT of
T = K(1−zγ)
Kγ+ 1 (5.1)
reflecting an ability to increase the local caching gain to zγ (meaning that each user sees a fraction zγ of each file), as well as an ability to preserve the coding gain to Kγ+ 1, by virtue of a scheme that servedKγ+ 1 users at a time. For this same setting, in [66] the authors provide explicit designs for K= 4,N = 4,z={2,3}, M = 1 andK = 6,N = 6, z= 3, M = 1, for which matching lower bounds are also developed to prove optimality of the schemes under uncoded cache placement. Finally, the authors provide a scheme for the extreme case of z=K−1.
1In this paragraph we mentioning only the works that had been published before the submission of our results to the IEEE Information Theory Workshop 2019. After our contributions were published, a significant set of results on multi-access coded caching appeared. We briefly mention these papers in the concluding section of this chapter.
2The work in [26] proposed a decentralized cache placement scheme, whose performance is slightly reduced over the easy-to-extend-to centralized variant whose delivery time we recorded above.
Users
Caches Shared Link
1 file per channel use Infinite capacity link
Server
1 2 3 4 5
1 2 3 4 5
Figure 5.1 – Multi-Access setting forK= 5 andz= 2.
5.1 System Model and Problem Definition
We consider a network whereK users are connected via an error-free shared link to a server storing N (N ≥K) files W(1), W(2), . . . , W(N). Each user has access to z out of K helper caches3, each of size M =N γ (units of file), where γ ∈ {K1,K2, . . . ,1}. We will use Zk to denote the content in cachek, and we will assume that each user has an unlimited capacity link to the caches it is connected to. Reflecting the assumption that each user has access to exactlyz caches and each cache connects to exactlyz users, we will consider the symmetric topology where each userk∈[K] is associated to caches
Ck,hk, k+ 1, . . . , k+z−1i ⊆[K]
where in the above we use the notation hMi={m |m for m≤K ; m−K for m >
K, m∈Z+, ∀m∈ M}.A pictorial representation of the studied setting can be found in Figure 5.1.
The system works in two phases: the cache placement phase and the delivery phase.
The first consists of filling the caches with the content of the library without knowledge of the users’ demands. In the delivery phase, each user krequests a file from the library.
We denote the index of such file by dk and we collect all the indices of the requested files in the demand vector d= (d1, d2, . . . , dK). The work focuses on the worst case where all users request different files. Upon reception of the demand vectord, the server will transmit a message X that spans T units of time. Each user k will use X and their own available cache content∪i∈CkZi to decode the desired file W(dk). Our objective is to provide a caching and delivery scheme that reduces the delivery delayT.
3Notice that the assumption of having as many users as caches can be relaxed to a more general case where the system has more users than caches, with a potential non uniform distribution of the users among the caches. Under these circumstances, the schemes described here can be combined with the topology-agnostic scheme (for the single antenna setting) presented in Chapter 3.
X1 =
This section presents the main results. The proof of the following theorem follows from the scheme described in Section 5.3.
Theorem 7. In the coded caching setting where each user is connected to z consecutive caches, when Kγ= 2, the delivery time satisfies
K−2z
4 < T(2, z)≤ K−2z
3 , (5.3)
and its exact value is T(2, z) = X1+XS 2, where4 X1, X2 and S are given in (5.2).
Remark 7. The scheme consists of transmissions serving either Kγ+ 1 = 3 users or 4 users at a time. Figure 5.2 shows, for several values of z, the achievable DoF as a function of K. We can see how the DoF always exceeds Kγ+ 1 = 3and can approach 4 for some values of K and z.
We now characterize the optimal worst-case delivery time (under the assumption of uncoded cache placement) for the case of z= K−1Kγ , for any Kγ .
Theorem 8. In the addressed K-user caching network with access to an integer number z= K−1Kγ of caches of normalized sizeγ, the optimal delivery time, under the assumption of uncoded cache placement, takes the form
T∗ = K(1−γz) Kγz+ 1 = 1
K (5.4)
corresponding to a DoF ofKγz+ 1users served at a time.
The scheme that achieves the above performance is described in Section 5.4. The optimality — under uncoded cache placement — follows directly from the fact that the achieved performance matches the optimal performance (cf. [52], [5]) of a dedicated-cache coded caching setting (identical to that in [2]), where each cache has an augmented size equal tozγ.
4The above expression holds for the case whereS(K−2z)−4X1 is non negative and divisible by 3.
50 100 150 200 250 300 350 400 K
3 3.2 3.4 3.6 3.8 4 4.2
DoF
z=2 z=4 z=6 z=8 z=10
Scheme from [26] for any z
Figure 5.2 – Achievable sum DoF as a function of K, for severalz. Case ofKγ= 2.