Improving Placement Delivery Array Coded Caching Schemes With Coded Placement

Coded caching schemes with low subpacketization and small transmission rate are desirable in practice due to the requirement of low implementation complexity and efficiency of transmission. Placement delivery arrays (PDA in short) can be used to generate coded caching schemes. However, many known coded caching schemes, which have low subpacketizations, realized by PDAs do not fully use the users’ caching content to create multicasting opportunities. So we propose a method to overcome this drawback. As an application, we obtain a new scheme, which has significantly advantages on the tradeoff between memory ratio and transmission rate.


I. INTRODUCTION
The wireless networks have been imposed tremendous pressure on the data transmission during the peak traffic times due to the explosive increasing of mobile services, especially the video streaming. A coded caching scheme, which has been recognized as an efficient solution to reduce this tremendous pressure, was proposed in [1] and has been rapidly used to in various settings such as cache-aided combination networks [2], D2D networks [3], hierarchical networks [4], insecure channel [5], among others.

A. SYSTEMS MODEL
In a centralized (K , M , N ) coded caching system (see Figure 1), a single server containing N independent files with the same length connects to K users over a shared link and each user has a cache memory of size M files, N ≥ K . Denote the N files by W = {W 0 , . . . , W N −1 } and K users by K = {0, . . . , K −1}. An F-division (K , M , N ) coded caching scheme consists of two phases as follows [1]: The associate editor coordinating the review of this manuscript and approving it for publication was Xijun Wang.
• Placement phase: During the off peak traffic times, each file is divided into F equal packets, i.e., W i = {W i,j : j = 0, 1, . . . , F − 1}. Then each user caches some packets (or linear combinations of packets) from the server. If packets are cached directly, it is called uncoded placement; if linear combinations of packets are cached, we call it coded placement. Denote Z k the contents cached by user k. In this phase we assume that the server does not know the users' requests in the following phase.
• Delivery phase: During the peak traffic times, each user randomly requests one file from the files set W independently. The request vector is denoted by d = (d 0 , · · · , d K −1 ), i.e., user k requests the d k -th file W d k , where d k ∈ {0, 1, . . . , N − 1} and k ∈ K. The server broadcasts a coded signal (XOR of some required packets) of size S d packets to users such that each user is able to recover its requested file with the help of its caching contents. In this article, we focus on the worst-case scenario, i.e., all the users require different files. In this case, the transmission rate of a coded caching scheme is defined as the maximal normalized transmission amount among all the requests in the delivery phase, i.e.
Since the implementation complexity of a coded caching scheme increases along with its subpacketization level, it is desirable to design a scheme with the transmission rate and the subpacketization as small as possible.

B. PRIOR WORK
Maddah-Ali and Niesen [1] introduced the first deterministic F-division (K , M , N ) coded caching scheme with F = K KM /N when KM /N is an integer. Obviously, the subpacketization F = K KM /N increases rapidly as K increases, which makes this scheme impractical when K is large. It is well known that there exists a tradeoff between the transmission rate and the subpacketization for a fixed number of users and a fixed memory ratio. Indeed, there are many research papers focus on constructing the coded caching schemes with lower subpacketization while sacrificing some transmission rate, such as [6]- [14], [17], etc.
The first scheme with lower subpacketization compared with the MN scheme was proposed by Shanmugam et al. in [15] by grouping method. Yan et al. [6] proposed an F × K combinatoric structure, which is called a placement delivery array (PDA), to generate an F-division (K , M , N ) coded caching scheme. Consequently, the MN scheme is equivalent a special PDA. Furthermore, for any positive integer q > 1, the schemes with M N = 1 q and 1 − 1 q respectively were obtained by constructing PDAs. It is worth noting that the F and R of the scheme with M N = 1 q achieves the tradeoff proposed in [15]. It implies that this scheme maybe the best and the tradeoff is tight for some parameters. There are many other constructions focusing on further reducing the subpacketization by increasing the transmission rate, such as by the special (6, 3)-free hypergraphs [12], the resolvable combinatorial design and linear block codes [14], the (r, t) Ruzsa-Szeméredi graphs [7], [13], the strong edge coloring of bipartite graphs [9], projective space [16], etc. In [7], Shanmugam et al. discovered that all deterministic F-division coded caching schemes can be recasted into a PDA when K ≤ N . By means of PDA, Cheng et al. [17] showed that the MN PDA has the minimum subpacketization with minimum transmission rate among PDAs. There are also many other results on PDA such as [8], [9], [17]. Very recently there are several schemes with low subpacketization such as [10], [11], which can be represented by PDAs.
In fact, a scheme realized by a PDA has uncoded placement phase, which implies that any two packets cached by a user are independent. As a result, many cached packets are not fully used in some well known coded caching schemes realized by PDAs, i.e., do not generate multicasting opportunities in the delivery phase. Intuitively, for a scheme realized by a PDA, when the subpacketization F is reduced, this drawback always exists.

C. CONTRIBUTIONS AND ORGANIZATIONS
In this article, the main idea of the proposed scheme is as follows. On the observation that some cached contents of each user are not leveraged in the existing PDA schemes, we propose to use Minimum Distance Separable (MDS) code in the cache placement phase such that each user does not cache these unused cached contents. As a result, the needed memory ratio is reduced. A similar idea to reduce the cache redundancy by using MDS coded cache placement was originally proposed in [2] for the cache-aided combination network problem.
In order to verify the effectiveness of our method, we take the strong edge coloring based scheme in [9] as an example, and obtain a new scheme, which has smaller subpacketization and memory ratio than the original scheme in [9]. Simulations show that under the same memory ratio, the new scheme has smaller transmission rate and more flexible memory ratio than the original scheme in [9].
The rest of this article is organized as follows. The relationship between a PDA and a coded caching scheme is explained in Section II. In Section III we introduce our research motivation and present the main method. As an application, we construct a new scheme in Section IV. Finally, we conclude the paper in Section V.

II. CODED CACHING SCHEMES REALIZED BY PDAs
In this article, we use the following notations unless otherwise stated. We use bold capital letters and curlicue letters to denote arrays and sets respectively. For any positive integers Next we will give the definition of a PDA, and demonstrate the relationship between a PDA and a coded caching scheme.
Definition 1 [6]: For positive integers K , F, Z and S, C3. For any two distinct entries p j 1 ,k 1 and p j 2 ,k 2 , p j 1 ,k 1 = p j 2 ,k 2 = s is an integer only if a. j 1 = j 2 , k 1 = k 2 , i.e., they lie in distinct rows and distinct columns; and b. p j 1 ,k 2 = p j 2 ,k 1 = * , i.e., the corresponding 2 × 2 subarray formed by rows j 1 , j 2 and columns k 1 , k 2 must be of the following form s * * s or * s s * .
Theorem 1 [6]: Using Algorithm 1, an F-division (K , M , N ) coded caching scheme with memory ratio M N = Z F and transmission rate R = S F can be realized by a (K , F, Z , S) PDA.
Using Algorithm 1, we can obtain a 6-division (6, 2, 6) coded caching scheme in the following way.

Then by Lines 3-5, the contents cached by users are
• Delivery Phase: Assume that the request vector is d = (0, 1, 2, 3, 4, 5). By Lines 8-10, the server sends the coded signals in Table 1 at time slots 0 − 11. Remark 1: In a (K , F, Z , S) PDA P = (p j,k ), j ∈ [0, F), k ∈ [0, K ), each column represents one user's caching contents, i.e., if p j,k = * , then user k has cached the j-th packet of all the files. Clearly, it belongs to uncoded placement. The property C1 of Definition 1 implies that all the users have the

III. NEW SCHEMES FROM PDAs
In this section, based on an appropriate PDA, an improved scheme with coded placement is proposed.

A. RESEARCH MOTIVATIONS
Let P be a (K , F, Z , S) PDA. For any integer s ∈ [0, S), assume that the occurrence number of s is r s , say p j u ,k u = s, u ∈ [0, r s ), j u ∈ [0, F) and k u ∈ [0, K ). Consider the subarray formed by rows j 0 , · · · , j r s −1 and columns k 0 , · · · , k r s −1 , which is of order r s × r s since we have j u = j v and k u = k v for all 0 ≤ u = v < r s from the property C3 of Definition 1. Furthermore, we have p j u ,k v = * for all 0 ≤ u = v < r s from the property C3. Then this subarray is equivalent to the following r s × r s array s * · · · * * * s · · · * * . . . . . . . . . . . . . . . * * · · · s * * * · · · * s with respect to row/column permutation. According to Line 9 in Algorithm 1, the signal u∈[0,r s ) W d ku ,j u is transmitted by the server at time slot s, which is simultaneously useful for r s users. So the occurrence number r s is called the coded caching gain at time slot s. Clearly we prefer to design a scheme with the coded caching gain at each time slot as large as possible. Furthermore, the stars in P (s) guarantee that each user can recover its requested packet from the transmitted signal, so they are useful for generating the coded caching gain. In other words, for each star in P (s) , say p j u ,k v = * where 0 ≤ u = v < r s , the packet W d ku ,j u , which is cached by user k v and simultaneously requested by user k u , is helpful for user k v to recover its requested packet W d kv ,j v , so it generates a multicasting opportunity. Therefore, for a star entry p j,k = * , we call it useful if it occurs in P (s) for some integer s ∈ [0, S), otherwise we call it useless.
In fact, given a PDA, if there exist some useless stars, we can delete these useless stars and then further reduce the subpacketization and the memory ratio without reducing the coded caching gain at each time slot. Now let us see the array P in (1) again. It is easy to check that the stars at p j,5−j (j ∈ [0, 6)) of P are useless. We delete all the useless stars and obtain the following array. According to P in (3), we can modify the scheme in Example 1 as follows.
Using the caching strategy in Lines 3-5 in Algorithm 1, all the users cache the following contents.
Clearly the memory ratio of each user is 1 5 , which is smaller than the memory ratio 2 6 in Example 1. • Delivery Phase: We also assume that the request vector is d = (0, 1, 2, 3, 4, 5). By Lines 8-10, the server also sends the coded signals listed in Table 1

B. NEW SCHEMES
Following the proposal in subsection III-A, the following result can be obtained. Proof: Assume that P is a (K , F, Z , S) PDA where each column has Z useless stars. Deleting the Z useless stars in each column, we obtain a new array P = (p j,k ), j ∈ [0, F), k ∈ [0, K ). Clearly each column of P has Z blanks, Z − Z stars and F − Z integers.
Based on P , we modify the placement strategy in Algorithm 1 as follows: The server divides each file into F − Z equal-sized packets and then encodes them using an (F, F − Z ) maximum distance separable (MDS) code in an appropriate operation field [18]. Let the resulting encoded packets be denoted by W n,0 , W n,1 , . . ., W n,F−1 for each file W n , n ∈ [0, N ). Using the caching strategy in Lines 3-5 in Algorithm 1, each user k caches Z k = {W n,j | p j,k = * , j ∈ [0, F), n ∈ [0, N )}. Clearly the memory ratio of each user is In the delivery phase, we also use the delivery strategy in Algorithm 1 as follows: For any request vector d, using Lines 7-11 of Algorithm 1, each user can get exactly F − Z required coded packets by the property C3 of Definition 1 and Remark 1. From the property of an (F, F − Z ) MDS code, each user can recover its requested file. So the transmission rate is R = S F−Z . Furthermore, the coded caching gain at each time slot is the same as that of the original scheme realized by P since the occurrence number of each integer is unchanged.
Given an appropriate PDA P, using Theorem 2, we can obtain a new scheme which has lower subpacketization and memory ratio while keeping the coded caching gain at each time slot unchanged.

IV. NEW SCHEMES BASED ON THE PDAs IN [9]
In this section, we take the strong edge coloring based scheme in [9], [19] as an example. From the results on optimality in [20], one can check that the scheme in [9] has the smallest transmission rate among all the schemes with the same placement strategy. However, we will show that there are useless stars in the PDA corresponding to the scheme in [9], and use Theorem 2 to obtain an improved scheme.
A. CONSTRUCTIONS FROM [9], [19] For the readers' convenience, the construction of the PDA corresponding to the strong edge coloring based scheme in [19] is given here.
Construction 1 ( [9], [19]): For any positive integers H , b, r, λ satisfying 0 < r, b < H , λ < min{r, b} and r There exist VOLUME 8, 2020 • an Proof: Let P be the PDA generated by (5). For any We claim that such p B,A = * is useless. Otherwise if the star at p B,A is useful, then there must exist two subsets, say A ∈ K \ {A} and B ∈ F \ {B} satisfying |A ∩ B | = λ and |A ∩ B| = λ, such that

C. PERFORMANCE ANALYSES
In this section, we assume the parameters H and r are fixed. The scheme from Lemma 1 is denoted by the original scheme, and the scheme with smaller transmission rate among the schemes from Theorem 3 and Theorem 4 is denoted by the new scheme. Since it is hard to propose a theoretic comparison between the original scheme and the new scheme, the trade off between the transmission rate and the memory ratio is shown in Figure 2 and Figure 3 for H = 10, r = 5 and H = 15, r = 6 respectively. From Figure 2 and Figure 3, it is easy to see that the transmission rate of the original scheme is much larger than that of the new scheme when the memory ratio is small. Moreover, the new scheme has more flexible memory ratio than the original scheme.

V. CONCLUSION
In this article, we focused on coded caching schemes with low subpacketizations. After we observed that there are some cached packets not fully used in some well known coded caching schemes, i.e., do not generate multicasting opportunities in the delivery phase, we modified the uncoded placement of the scheme realized by an appropriate PDA to coded placement to overcome this drawback. Finally based on the scheme in [9], we obtain a new scheme which has smaller transmission rate and more flexible memory ratio than the original scheme in [9].