Placement Delivery Array Design for Combination Networks with Edge Caching

# Placement Delivery Array Design for Combination Networks with Edge Caching

## Abstract

A major practical limitation of the Maddah-Ali-Niesen coded caching techniques is their high subpacketization level. For the simple network with a single server and multiple users, Yan et al. proposed an alternative scheme with the so-called placement delivery arrays (PDA). Such a scheme requires slightly higher transmission rates but significantly reduces the subpacketization level. In this paper, we extend the PDA framework and propose three low-subpacketization schemes for combination networks, i.e., networks with a single server, multiple relays, and multiple cache-aided users that are connected to subsets of relays. One of the schemes achieves the cutset lower bound on the link rate when the cache memories are sufficiently large. Our other two schemes apply only to resolvable combination networks. For these networks, the new schemes perform closely to the currently best-known coded caching schemes for a wide range of cache sizes while having significantly reduced subpacketization levels.

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## 1 Introduction

Caching is a promising approach to alleviate current network traffics driven by on-demand video streaming. The idea is to pre-fetch contents during off-peak hours before the actual user demands, so as to reduce traffic at peak hours when the demands are made. Therefore, the communication takes place in two phases: content placement at off-peak hours and content delivery at peak hours.

In their seminal work [1], Maddah-Ali and Niesen modeled the content delivery phase by a shared error-free link from the single server to all users, and they showed that delivery traffic in this shared-link setup can be highly reduced through a joint design of content placement and an efficient delivery strategy that exploits multicasting opportunities. The scheme is known as coded caching and has been extended to various settings, e.g., Gaussian broadcast channels [2], multi-antenna fading channels [3, 5, 4], or combination networks [6, 7, 8, 9] as considered in this paper. In a -combination network, a single server communicates over dedicated error-free links with relays and these relays in their turn communicate over dedicated error-free links with users that have local cache memories. Each user is connected to a different subset of relays. Ji et al. first investigated this network [6] for the case dividing (denote by ), and the achievable bound was improved in [7]. In [8], Wan et al. tightened the lower bound under the constraint of uncoded placement, and the achievable bound for the case when the memory size is small. In [9], Zewail et al. showed that the upper bound in [7] is achievable for any -combination network. As the results of our work require a memory size larger than that of [8], we only compare our results with those from [7] and [9].

A key factor that limits the application of all forms of coded caching in practice, is the required high subpacketization level [10], i.e., the number of subpackets must grow exponentially with the number of users. To relax such constraint, [11, 12, 13, 14, 15] proposed new caching schemes that have much lower subpacketization levels at the cost of a certain higher transmission rate. A useful tool for representing these new schemes is the placement delivery arrays (PDA) introduced in [11]. PDAs characterize both the (uncoded) placement and delivery strategies with a single array [11], and thus facilitate the design of good caching schemes.

The main contributions of this paper are summarized as follows:

1. We introduce the combinational PDAs (C-PDA) to represent uncoded placement and delivery strategies for combination networks in a single array. We also determine the rate, memory, and subpacketization requirements of the caching scheme corresponding to a given C-PDA.

2. For the case , we describe how a standard PDA with columns can be transformed into a C-PDA for a -combination network.

3. With this transformation and the previous low-subpacketization schemes for the single-shared link setup, we present two low-subpacketization schemes for -combination networks. The performances of the new schemes are close to the performances of the currently best known schemes that require significantly larger subpacketization level.

4. For arbitrary , we propose a C-PDA for which the corresponding caching scheme achieves the cut-set lower bound for sufficiently large cache sizes.

Notations: We denote the set of positive integers by . For , denote the set by . The Exclusive OR operation is denoted by . For a positive real number , is the least integer that is not less than .

## 2 System Model and Preliminaries

### 2.1 System Model

Consider the -combination network illustrated in Fig. 1, where and are positive integers and . The network comprises a single server, relays:

 H={H1,H2,⋯,Hh},

and users labeled by all the -dimensional subsets of relay indices :

 T△={T:T⊂[h]and|T|=r}. (1)

Each user has a local cache memory of size bits. The relays have no cache memories.

The server can directly access a library of files,

 W={W1,W2,⋯,WN},

where each file consists of independent and identically uniformly distributed (i.i.d.) random bits. The server can send bits to each of the relays over an individual error-free link. Here denotes the link rate (or rate for brevity). Each relay can communicate with some of the users. Specifically, user is connected through individual error-free links of rate to the relays with index in , i.e., to relays .

We now describe the communication over the error-free links and the storage operations in detail. The system operates in two consecutive phases.
1. Placement Phase: In this phase, each user directly has access to the file library and can store an arbitrary function thereof in its cache memory, subject to the space limitation of bits. Denote the cached content at user by , and the set of all cached contents by .
2. Delivery Phase: In this phase, each user arbitrarily requests a file from the server, where . The users’ requests are revealed to all parties, i.e., to server, relays, and users. For each , the server sends bits to relay :

 Xi=ϕi(W1,…,WN,Z,d),

for some function . Relay forwards the signal to all connected users.1

At the end of this phase, each user , , decodes its requested file based on all its received signals , its cache content , and demand vector :

 ^WdT=ψT(XT,ZT,d)

for some function .

The optimal worst-case rate is the smallest delivery rate for which there exist some placement and delivery strategies so that the probability of decoding error vanishes asymptotically as at all the users and for any possible demand . has the following cut-set bound [6]:

 R∗(M)≥maxt∈N+,r≤t≤h1tmaxl∈[min{N,(tr)}]⎛⎝l−l⌈Nl⌉M⎞⎠ (2)

Special focus will be given to -combination networks with . In this case, the users can be partitioned into subsets so that in each subset exactly one user is connected to a given relay, see [7].

###### Definition 1 (Resolvable Networks).

A combination network is called resolvable if the user set can be partitioned into subsets so that for all the following two conditions hold: If and , then . . Subsets satisfying these conditions are called parallel classes.

### 2.2 Preliminaries: Shared Link Setup and PDAs

For the purpose of this subsection, consider the original coded caching setup [1] with a single server and users each having a cache memory of bits. The server is connected to the users through a shared error-free link of rate .

Yan et al. [11] proposed to unify the description of uncoded placement and delivery strategies for this shared-link setup in a single array, the placement delivery array (PDA).

###### Definition 2 (Pda,[11]).

For positive integers and , an array , , composed of a specific symbol and ordinary symbols , is called a placement delivery array (PDA), if it satisfies the following conditions: The symbol appears times in each column; Each ordinary symbol occurs at least once in the array; For any two distinct entries and , is an ordinary symbol only if , , i.e., they lie in distinct rows and distinct columns; and , i.e., the corresponding sub-array formed by rows and columns must be of the following form We refer to the parameter as the subpacketization level. Specially, if each ordinary symbol occurs exactly times, is called a - PDA, or -PDA for short.

###### Remark 1.

Deleting some columns of a PDA, results in an array that is still a PDA with the same subpacketization level . In particular, the new array still satisfies condition C of Definition 2.

###### Remark 2.

As shown in [11], the Maddah-Ali & Niesen coded caching scheme (MAN scheme) [1] can also be represented in form of a PDA. Specifically, for , with , it is represented by a - PDA.

Any PDA can be transformed into a caching scheme having the following performance [11]:

###### Remark 3.

A PDA corresponds to a caching scheme for the shared error-free link setup with users that is of subpacketization level , requires cache size , and delivery rate .

Two low-subpacketization schemes were proposed in [11]:

###### Lemma 1 (PDA for NM∈N+, [11]).

For any , there exists a - PDA, with rate and subpacketization level .

###### Lemma 2 (PDA for NN−M∈N+, [11]).

For any , there exists a - PDA, with rate and subpacketization level .

## 3 C-PDAs for Combination Networks

A PDA is especially useful for a combination network, if for any coded packet, all the intended users are connected to the same relay. This allows the server to send each coded packet only to this single relay. The following definition ensures the desired property.

###### Definition 3.

Let with , and . A PDA is called -combinational, for short C-PDA, if its columns can be labeled by the sets in in a way that for any ordinary symbol , the labels of all columns containing symbol have nonempty intersection.

The following example presents a C-PDA for and , and explains how this C-PDA leads to a caching scheme for the -combination network in Fig. 1.

###### Example 1.

Let and . The following table presents a C-PDA combined with a labeling of the columns that satisfies the condition in Definition 3.

The above C-PDA implies the following caching scheme for the combination network in Fig. 1.
1. Placement phase: Each file is split into packets (i.e., the number of rows of the C-PDA), i.e., . Place the following cache contents at the users:

 Z{1,2}=Z{3,4}={Wn,1,Wn,4: n∈[N]} Z{1,3}=Z{2,4}={Wn,2,Wn,5: n∈[N]} Z{1,4}=Z{2,3}={Wn,3,Wn,6: n∈[N]}

2. Delivery phase: Table 2 shows the signals the server sends to the four relays when users request files , respectively. Each of the coded signals consists of bits, and thus the required rate is .

Table 2 also indicates the users that are actually interested by each coded signal. In the problem definition, we assumed that each relay forwards its entire received signal to all its connected users. From Table 2, it is obvious that it would suffice to forward only a subset of the bits to each user.

We now present a general way to associate a C-PDA to a caching scheme for a -combination network where are positive integers with .

Placement phase: Label the columns of the C-PDA with the set so that the condition in Definition 3 is satisfied. Placement is the same as for standard PDAs. That means, split each file into subpackets each consisting of bits. Place subfiles into the cache memory of user , if the C-PDA has entry “” in row and the column corresponding to label . This placement strategy requires a cache size of .

Delivery phase: The server first creates the coded signals pertaining to each ordinary symbol in the same way as for standard PDAs. It then delivers the coded signal created for each ordinary symbol to one of the relays whose index is contained in the labels of all columns containing . The average rate required on the server-to-relay links is .

###### Remark 4.

When in the described scheme the server sends the same number of bits to each relay, then the following theorem follows immediately from the above description. In fact, in this case subpacketization level is sufficient. Otherwise, the rate on each server-to-relay link has to be made equal by first splitting each file into subfiles and then applying a caching scheme with the same C-PDA but a different shifted version of the column labels to each of the subfiles.

To describe the shift mentioned in Remark 4, for each , we define the function:

 Δi(t)={t+i−1,if t+i−1≤ht+i−1−h,if t+i−1>h,t∈[h].

Notice that is a bijection from onto , the inverse map is denoted by .

###### Theorem 1.

Given a C-PDA. For any combination network with , the following upper bound is achieved by a scheme of subpacketization level not exceeding :

{IEEEproof}

By the description of the delivery scheme above, the ordinary symbols can be partitioned into subsets , where contains the symbols of which the associated coded signals are sent to relay . Now consider the following scheme:

Split each file into subfiles, each of size bits. Similarly, split each memory into sectors, each of size bits. For each , create a new C-PDA and apply its corresponding scheme to the -th subfile of all files with the help of the -th memory sector of all users. The new C-PDA is formed as follows:

Denote . For , define

 T(i)={Δi(t1),Δi(t2),⋯,Δi(tr)}

Then is a permutation of . The new C-PDA is obtained by relabeling the column of the original C-PDA by . Obviously, the new labeling also satisfies the conditions in Definition 3, and accordingly, the relay is able to send the coded packets associated with the symbols in . Thus, the number of packets each relay sends is

 h∑i=1|SΔ−1i(j)|=h∑i=1|Si|=S

Each packet is of size bits. Hence, the rate of this scheme is . Apparently, the subpacketization level of this scheme is .

## 4 Transforming PDAs into Larger C-PDAs

We present a way of constructing C-PDAs for resolvable -combination networks (i.e., when ) from any smaller PDA that has columns. We start with an example.

###### Example 2.

Reconsider Example 1, where and , and notice that for this resolvable network (see Definition 1), a possible partition of is and . Consider now the PDA of the Maddah-Ali & Niesen scheme with users:

 A=⎡⎢⎣∗121∗323∗⎤⎥⎦.

One can verify that the C-PDA in Table 1 is obtained from above PDA by replicating each column of first horizontally and then each column of the resulting array also vertically, and by then replacing the 3 replicas of each ordinary symbol with 3 new (unused) symbols. The column labels are obtained by labeling the first two columns of with the two elements of , the following two columns with the elements of , and the last two columns with the elements of .

We now present the general transformation method. We use the following notations. For a given user , let indicate the parallel class that belongs to, i.e., iff . Let be the -th smallest element of . For example, if , then . Likewise, denote the inverse map by , i.e., iff .

###### Transformation 1.

Given a PDA . Let the following -by- array be the outcome applied to PDA for parameters :

 C=⎡⎢ ⎢ ⎢ ⎢ ⎢⎣c1,T1c1,T2⋯c1,TKc2,T1c1,T2⋯c2,TK⋮⋮⋱⋮cr,T1cr,T2⋯cr,TK⎤⎥ ⎥ ⎥ ⎥ ⎥⎦,

where are the elements of the user set in (1), and is a single-column array of length , with -th entry

 ci,j,Tk={∗,if ~cj,δ(Tk)=∗~cj,δ(Tk)+(T−1k[i]−1)~S,if ~cj,δ(Tk)≠∗. (3)
###### Theorem 2.

Let be positive integers so that , and . Transformation 1 with parameters transforms any PDA into a C-PDA, where With the resultant C-PDA, subpacketization level is sufficient to achieve the rate .

{IEEEproof}

We prove the theorem by verifying obtained by applying Transformation 1 to is a C-PDA. Specially, we need to verify the conditions C, C, C and specify the labeling of the columns that satisfies the conditions in Definition 3.

Firstly, C holds since each contains symbols, and the -th column is composed of .

Secondly, by (3), the ordinary symbol set of is since can be any of , and can be any of . Thus, C holds.

Thirdly, to verify C-a., we need to prove that if two distinct entries

 ci,j,Tk=ci′,j′,Tk′=s∈[S], (4)

they lie in distinct rows and columns. Notice that (4) implies

 ~cj,δ(Tk)+(T−1k[i]−1)~S=~cj′,δ(Tk′)+(T−1k′[i′]−1)~S, (5)

which is equivalent to

 ~cj,δ(Tk) =~cj′,δ(Tk′)∈[~S], (6) T−1k[i] =T−1k′[i′]. (7)

Recall that entry lies in row and column . Since and are distinct entries, at least one of and holds. We argue that both hold.

1. Assume , then by (6), and the fact that is a PDA, .

2. Assume , but , then by (6), . Thus or . But both contradict (7), since .

3. Assume and , then by (6) and the fact that is a PDA, , thus .

4. Assume , but , then , which contracts (7).

Therefore, lie in distinct rows and columns. Moreover, by the above arguments, . Hence by (6) and because is a PDA,

 ~cj,δ(Tk′)=~cj′,δ(Tk)=∗.

Therefore, by (3),

 ci,j,Tk′=ci′,j′,Tk=∗,

i.e., C holds.

Finally, we label the -th column of by . By (7), if , then

 T−1k1[i1]=T−1k2[i2]=⋯=T−1kg[ig]≜l, (8)

which indicates

 l∈g⋂n=1Tkg≠∅. (9)

In addition, (8), (9) and (5) indicate that the relay is able to forward the signals associated with the symbols in . This implies that the server sends equal number of bits to each relay. With Remark 4, the subpacketization level is sufficient to achieve the rate .

The coding scheme for resolvable combination networks in [7] can be represented in form of a C-PDA, and this C-PDA can be obtained by applying Transformation 1 to the PDA of the MAN scheme (Remark 2). Theorem 2 thus allows to recover the following result from [7]:

###### Corollary 1.

For a -combination network, where , when , there exists a caching scheme that requires rate and has subpacketization level .

We apply Transformation 1 to reduced versions (so as to have the right number of columns, see Remark 1) of the low-subpacketization PDAs in Lemmas 1 and 2. This yields the first low-subpacketization C-PDAs and caching schemes for resolvable combination networks.

###### Theorem 3 (C-PDA construction from Lemma 1).

Let . For any -combination network with cache sizes , the following uppper bound is achieved by a scheme with subpacketization level : (Here, subscript “LSub” stands for “low-subpacketization”.)

###### Proof.

By Lemma 1, there exists a PDA with columns. Delete any of the columns. Since each ordinary symbol occurs in distinct columns, some ordinary symbols can be completely deleted whenever . In this case, the reduced PDA has rate smaller than . Then the theorem is concluded from Theorem 1 and 2. ∎

###### Theorem 4 (C-PDA construction from Lemma 2).

Let . For any -combination network with cache sizes , the following upper bound is achieved by a scheme with subpacketization level :

###### Proof.

Similarly to the proof of Theorem 3, except that deleting columns does not delete any of the ordinary symbols, as each of them occurs times. The theorem is concluded from Theorem 1 and 2. ∎

We compare the new schemes with the currently best known scheme for in [7]. We start with a comparison of the required rates. When for some integer , then

 KMrKMr+Nh≤RTRR% LSub1≤1. (10)

Similarly, when for some integer , then

 KMrKMr+Nh≤RTRR% LSub2≤1. (11)

As a consequence, when or for some integer , then

 lim––––K→∞RTRRLSub1=1orlim––––K→∞RTRRLSub2=1. (12)

On the other hand, for large values of , by Corollary 1 and [11, Lemma 4]:

 FTR∼√N2hr2πKM(N−M)⋅eKrh(MNlnNM+(1−MN)lnNN−M),

and

 FLSub1 ≤r⋅eKrh⋅MNlnNM, FLSub2 ≤rMN−MeKrh⋅(1−MN)lnNN−M.

As a consequence, when or for some integer , then

 lim––––K→∞FTRFLSub1=∞orlim––––K→∞F% TRFLSub2=∞.

From (10)-(12), the new scheme suffers a loss compared to the scheme in [7], which diminishes as becomes large. However, the is of order and or is of order or when or respectively. The new scheme saves a factor of order or , which goes to infinity exponentially with .

## 5 Achieving the Cutset Bound with Low Subpacketization Level

Throughout this section, denote positive integers with . But does not necessarily divide .

Let denote all the subsets of of size . Define to be the -by- dimensional array with entry in row and column , where

 bj,T={∗,if Sj⊄TT∖Sj,if Sj⊂T (13)
###### Example 3.

For and , let the 1-subsets of sorted as , the C-PDA is:

{1,2} {1,3} {1,4} {2,3} {2,4} {3,4} ∗ ∗ ∗ 3 4 ∗ 2 ∗ 4 ∗ 2 3
###### Lemma 3.

The array denoted by (13) is a C-PDA. It implies a caching scheme for -combination networks that achieves rate with subpacketization level .

###### Proof.

It’s easy to check that, (13) is a subcase of the construction in [12, Theorem 2], thus is a PDA. With (13), whenever , then

 s∈g⋂n=1Tkn≠∅. (14)

Thus, satisfies the condition in Definition 3. Moreover, by (14), each relay sends a single coded packet associated with the symbol , which indicates that the rate is evenly allocated into relays. By Remark 4, the subpacketization level is . ∎

The caching scheme corresponding to C-PDA , allows to determine the optimal rate of a combination network for large cache sizes.

###### Theorem 5.

For an -combination network: This can be achieved with subpacketization level when .

###### Proof.

By Lemma 3, the array implies a caching scheme with memory size