Placement Delivery Array Design for Combination Networks with Edge Caching
Abstract
A major practical limitation of the MaddahAliNiesen 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 socalled 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 lowsubpacketization schemes for combination networks, i.e., networks with a single server, multiple relays, and multiple cacheaided 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 bestknown coded caching schemes for a wide range of cache sizes while having significantly reduced subpacketization levels.
1 Introduction
Caching is a promising approach to alleviate current network traffics driven by ondemand video streaming. The idea is to prefetch contents during offpeak 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 offpeak hours and content delivery at peak hours.
In their seminal work [1], MaddahAli and Niesen modeled the content delivery phase by a shared errorfree link from the single server to all users, and they showed that delivery traffic in this sharedlink 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], multiantenna 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 errorfree links with relays and these relays in their turn communicate over dedicated errorfree 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:

We introduce the combinational PDAs (CPDA) 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 CPDA.

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

With this transformation and the previous lowsubpacketization schemes for the singleshared link setup, we present two lowsubpacketization 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.

For arbitrary , we propose a CPDA for which the corresponding caching scheme achieves the cutset 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:
and users labeled by all the dimensional subsets of relay indices :
(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,
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 errorfree 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 errorfree links of rate to the relays with index in , i.e., to relays .
We now describe the communication over the errorfree 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 :
for some function . Relay forwards the signal to all connected users.
At the end of this phase, each user , , decodes its requested file based on all its received signals , its cache content , and demand vector :
for some function .
The optimal worstcase 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 cutset bound [6]:
(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 errorfree link of rate .
Yan et al. [11] proposed to unify the description of uncoded placement and delivery strategies for this sharedlink 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 subarray 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.
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 errorfree link setup with users that is of subpacketization level , requires cache size , and delivery rate .
Two lowsubpacketization schemes were proposed in [11]:
Lemma 1 (PDA for , [11]).
For any , there exists a  PDA, with rate and subpacketization level .
Lemma 2 (PDA for , [11]).
For any , there exists a  PDA, with rate and subpacketization level .
3 CPDAs 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 CPDA, 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 CPDA for and , and explains how this CPDA leads to a caching scheme for the combination network in Fig. 1.
Example 1.
Let and . The following table presents a CPDA combined with a labeling of the columns that satisfies the condition in Definition 3.
The above CPDA 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 CPDA), i.e., . Place the following cache contents at the users:
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 .
Signal  Symbol  Coded Signal  Intended Users 

11  
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 CPDA to a caching scheme for a combination network where are positive integers with .
Placement phase: Label the columns of the CPDA 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 CPDA 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 servertorelay 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 servertorelay link has to be made equal by first splitting each file into subfiles and then applying a caching scheme with the same CPDA 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:
Notice that is a bijection from onto , the inverse map is denoted by .
Theorem 1.
Given a CPDA. For any combination network with , the following upper bound is achieved by a scheme of subpacketization level not exceeding :
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 CPDA 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 CPDA is formed as follows:
Denote . For , define
Then is a permutation of . The new CPDA is obtained by relabeling the column of the original CPDA 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
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 CPDAs
We present a way of constructing CPDAs 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 MaddahAli & Niesen scheme with users:
One can verify that the CPDA 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 :
where are the elements of the user set in (1), and is a singlecolumn array of length , with th entry
(3) 
Theorem 2.
Let be positive integers so that , and . Transformation 1 with parameters transforms any PDA into a CPDA, where With the resultant CPDA, subpacketization level is sufficient to achieve the rate .
We prove the theorem by verifying obtained by applying Transformation 1 to is a CPDA. 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 Ca., we need to prove that if two distinct entries
(4) 
they lie in distinct rows and columns. Notice that (4) implies
(5) 
which is equivalent to
(6)  
(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.
Therefore, lie in distinct rows and columns. Moreover, by the above arguments, . Hence by (6) and because is a PDA,
Therefore, by (3),
i.e., C holds.
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 CPDA, and this CPDA 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 lowsubpacketization PDAs in Lemmas 1 and 2. This yields the first lowsubpacketization CPDAs and caching schemes for resolvable combination networks.
Theorem 3 (CPDA 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 “lowsubpacketization”.)
Proof.
Theorem 4 (CPDA 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.
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
(10) 
Similarly, when for some integer , then
(11) 
As a consequence, when or for some integer , then
(12) 
As a consequence, when or for some integer , then
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
(13) 
Example 3.
For and , let the 1subsets of sorted as , the CPDA is:
Lemma 3.
The array denoted by (13) is a CPDA. 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
(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 CPDA , 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