Low Subpacketization Coded Caching via Projective Geometry for Broadcast and D2D networks

# Low Subpacketization Coded Caching via Projective Geometry for Broadcast and D2D networks

Hari Hara Suthan Chittoor, Prasad Krishnan
Signal Processing and Communications Research Center,
International Institute of Information Technology, Hyderabad, India.
August 23, 2019
###### Abstract

Coded caching was introduced as a technique of systematically exploiting locally available storage at the clients to increase the channel throughput via coded transmissions. Most known coded caching schemes in literature enable large gains in terms of the rate, however at the cost of subpacketization that is exponential in ( being the number of clients, some positive integer). Building upon recent prior work for coded caching design via line graphs and finite-field projective geometries, we present a new scheme in this work which achieves a subexponential (in ) subpacketization of and rate , for large , and the cached fraction being upper bounded by a constant (for some prime power and constant ) . Apart from this asymptotic improvement, we show that through some numerical comparisons that our present scheme has much lower subpacketization than previous comparable schemes, with some increase in the rate of the delivery scheme, for the same memory requirements. For instance, we obtain practically relevant subpacketization levels such as for number of clients. Leveraging prior results on adapting coded caching schemes for the error-free broadcast channel to device to device networks, we obtain a low-subpacketization scheme for D2D networks also, and give numerical comparison for the same with prior work.

coded caching, low subpacketization, broadcast channel, line graph, D2D networks.

## I Introduction

Next generation wireless networks (5G and beyond) present the challenges of serving clients via channels which are not traditional point-to-point communication channels. The study of efficient high throughput communication techniques for broadcast channels, interference channels, multiple access channels, device-to-device (D2D) communication, all obtain relevance in the present wireless communication scenario. The technique of utilizing local storage (which has become quite affordable, thanks to advances in hardware design), either on the client’s device or in a nearby location, for aiding communication services on a broadcast channel was introduced formally in the landmark paper [1], under the title of Coded Caching. In [1], it was shown that a combination of (a) carefully utilizing the local storage or cache available to individual clients, and (b) coded transmissions during the delivery phase, brings tremendous gains in the rate of delivery of information of a broadcast channel. Following [1], generalized cache aided communication techniques have been presented for a number of channel models [2, 3, 4, 5], and in each case has shown to provide gains in the information delivery rate. Specifically, the coded caching problem for D2D networks was considered in [4], and a caching cum coded delivery scheme that was inspired from [1] was presented, which resulted similar rate advantages as [1].

The setup considered in [1] consists of an error-free broadcast channel connecting a single server with clients or receivers. The server has same-sized files, which form the library of files. Each file is divided into equal-sized subfiles ( is known as the subpacketization parameter). Each client has a cache that can store subfiles (i.e., fraction of each file). According to the scheme presented in [1] which takes place in two phases, the caches of the clients are populated by the subfiles during the caching phase (which occurs during off-peak time periods), and during the demand phase (occurring during peak-time periods) coded subfiles are transmitted to satisfy the client demands (each client demands one file in the demand phase). The rate of such a coded caching scheme is defined as the ratio of the number of bits transmitted to the size of each file, which can be calculated as

 Rate R=Number of transmissions in the delivery % phaseNumber of subfiles in a file,

when each transmission is of the same size as the subfiles.

The delivery scheme in [1] consists of transmissions such that in each transmission clients are served. The parameter is known as the global caching gain. The rate achieved is . This rate was shown to be optimal for uncoded cache placement [6]. The subpacketization of the scheme in [1] is , which however becomes exponential in as grows (for constant ) and hence impractical even for tens of clients.

Since then several new coded caching schemes with lower subpacketization have been constructed at the cost of increase in rate, or cache requirement, or the number of users (for instance, [7] and [8]). To the best of the author’s knowledge, these constructions (and others in literature for which ‘large-’ behaviour can be derived) have subpacketization lesser than [1] but still exponential in (for some positive integer ), while having larger rates compared to [1]. In particular, the scheme in [8] achieves global caching gain with subpacketization exponential in with a much smaller exponent than [1], using a combinatorial structure called Placement Delivery Arrays (PDA). The issue of high subpacketization is carried over to the D2D problem also. Improved schemes with lower subpacketization (but higher rates) were also recently constructed for D2D networks in [9, 10].

In this direction of research, a line graph based coded caching scheme was introduced and developed in [11, 12] (co-authored by a subset of the current authors) to construct one of the few explicitly known subexponential (in ) subpacketization schemes. Tools from projective geometries over finite fields were used for this purpose. However the scheme of [12] required a large cache requirement to obtain low subpacketization. This issue was rectified in [11]. The scheme in [11] achieves a rate of ( being the number of users, is some prime power) with subexponential subpacketization when cached fraction is upper bounded by a constant () for some positive integer ).

In the present work, we go further than the scheme of [11]. The tools remain the same; we use a line graph based technique combined with projective geometries over finite fields. The contributions and organization of the current work are as follows. After briefly going over the line graph coded caching approach in [12] (Section II), we present our new scheme in Section III. In Section IV, we show that for large and the cached fraction (for some constant ), we show that our scheme achieves rate and subpacketization for large , thus improving upon [11]. Further we also compare in Table I, by giving some numerical values to our scheme’s parameters with those of [8] and [11], and show that the subpacketization achieved is several orders of magnitude lesser compared to [11] (which itself is orders of magnitude less than [8]). However the rate (equivalently, the gain) of our present scheme can be few orders of magnitude greater (equivalently, lesser) than [8] and roughly the same as [11]. Finally, in Section V, we extend the present scheme for the error-free broadcast channel to D2D networks, utilizing a result from [13]. This results in a D2D coded caching scheme with lower subpacketization than some known schemes before. In Table II we perform a numerical comparison of the new D2D scheme with those of [10, 4].

Notations and Terminology: denotes the set of positive integers. We denote the set by for some positive integer . For sets , the set of elements in but not in is denoted by . The finite field with elements is . The dimension of a vector space over is given as . For two subspaces , their subspace sum is denoted by . Note that (the direct sum) if . The span of two vectors , is represented as . We give some basic definitions in graph theory. The sets denote vertex set and edge set of a (simple undirected) graph respectively, where . The square of a graph is a graph having and an edge if and only if either or there exists some such that . The complement of a graph is denoted as . A set is called a clique of if every two distinct vertices in are adjacent to each other. A single vertex is also considered as a clique by definition. A clique cover of is a collection of disjoint cliques such that each vertex appears in precisely one clique.

## Ii The line graph based coded caching of [12] and its relation to PDAs

Consider a coded caching system consisting of a server with files . Let be any set such that . We shall use to indicate the set of users. Let be any set such that . The subfiles of a file are denoted by where and takes values in some Abelian group.

In [12], a line graph based framework was proposed to study the coded caching problem, which we now describe.

###### Definition 1.

(Line graph, -caching line graph) [12] A graph consisting of vertices (for some ), such that , (for some sets such that ) is said to be a caching line graph (or simply, a line graph) if

1. The set of vertices forms a clique of size , for each (We refer to these cliques as the user cliques).

2. The set of vertices forms a clique of size (for some fixed ), for each (We refer to these cliques as the subfile cliques).

3. Each edge in lies between vertices within a user clique or within a subfile clique.

When there is a clique cover of (the complement of the square of ) consisting of disjoint -sized cliques, then the line graph is called as a -caching line graph.

By the above conditions P1-P3, it holds that (these unions being disjoint). Therefore we can write . Furthermore, it follows that .

It was shown in [12] (refer Section IV Proposition of [12]) that such a line graph corresponds to a caching system in which there are users (indexed by ) and subfiles (indexed by ), where the user caches subfiles if and does not cache them otherwise. We therefore have that each user does not cache subfiles of each file, and hence the uncached fraction is . Further each subfile of any file is not cached in of the users.

It is also shown in [12] that for the caching phase as defined by the line graph , a delivery scheme is given by a clique cover of .

###### Remark 1.

For a -caching line graph, it is shown in Theorem of [12] that the parameters of the caching and delivery scheme come out naturally, with (and thus the uncached fraction being ). Further the -sized cliques of result in a delivery scheme with rate . This is illustrated in the following example.

###### Example 1.

Consider a coded caching system defined by the graph as shown in the left of Fig. 1. This graph corresponds to a coded caching setup with (since it has 4 user cliques, each of size ) and (since it has 4 subfile cliques, each of size ). Thus, there are users, each user does not cache subfiles (for instance, user does not cache the subfiles indexed by and and caches ). Each subfile is cached in users (for instance, subfile is cached in users ). The cached fraction is . The graph is shown on the right with 4 cliques, indicated by vertices of a same color. Corresponding to each clique of , there is one transmission in the delivery scheme. For instance, corresponding to the clique , there is a transmission , where signifies the file demanded by client . Note that this enables the client to decode and client to decode , as they each have the other subfile in their cache. Similarly, the entire set of transmissions will enable decoding of all the missing subfiles at all clients.

In the forthcoming sections, we construct a new caching line graph based scheme using projective geometry over finite fields, building on the results of [12, 11], and show that these results outperform prior known schemes in terms of the subpacketization , while trading it off with some increase in the rate of the delivery scheme. Towards that end, we now recall a following structural lemma (which will be used in the next section) proved in [11] which gives the conditions under which an edge exists in .

###### Lemma 1.

[11] Let . The edge if and only if and .

Now we recall the definition of placement delivery array(PDA) presented in [8].

###### Definition 2 (Placement delivery array [8]).

For positive integers and an array , composed of a specific symbol “” and integers is called a placement delivery array (PDA), if it satisfies the following conditions:

1. The symbol “” appears times in each column.

2. Each integer occurs at least once in the array.

3. For any two distinct entries and we have , an integer, only if

1. i.e., they lie in distinct rows and distinct columns; and

2. .

If each integer occurs exactly times, is called a regular PDA, or -PDA for short.

Most known coded caching schemes in literature correspond to PDAs. We now show that any caching line graph is equivalent to a PDA.

###### Lemma 2.

is a -caching line graph when and there is a partition of with cliques of size each if and only if there exist a regular PDA.

###### Proof:

We will prove the only if part. Let be a caching line graph as given in the lemma statement. From the condition P2 of Definition 1, we have that the -sized disjoint cliques of partition . Since , thus is an integer. Also we know that there is a clique cover of consisting of -sized disjoint cliques. As , the set can be partitioned into number of -sized cliques of . It is clear that . Let , and . Now consider an array . So is a array such that rows represent subfile cliques and columns represent user cliques. The entries of are defined as follows

 af,k={∗ if (k,f)∉V(L)s if (k,f)∈Cs for some s∈[KDd]

Now we will check the conditions C1-C3 of Definition 2.

1. Consider an arbitrary . By condition P1 of Definition 1, . Therefore “” appears times in each column of .

2. From the definition of , it is clear that each integer occurs at least once in the array.

3. Consider such that for some . From Lemma 1, it is easy to see that and . Therefore .

Therefore satisfies all the conditions of Definition 2. Hence is a PDA. The proof of if part follows similarly. ∎

## Iii A new projective geometry based scheme

Towards presenting our new scheme, we first review some basic concepts from projective geometry.

### Iii-a Review of projective geometries over finite fields [14]

Let such that is a prime power. Let be a -dim (we use “dim” for dimensional) vector space over a finite field . Consider an equivalence relation on (where represents the zero vector) whose equivalence classes are -dim subspaces (without ) of . The -dim projective space over is denoted by and is defined as the set of these equivalence classes. For , let denote the set of all -dim subspaces of . It is known that (Chapter in [14]) is equal to the q-binomial coefficient , where (where ). In fact, gives the number of -dim subspaces of any -dim vector space over . Further, by definition,

Let . Let denotes the number of distinct -dim subspaces of . Therefore .

The following lemma and corollary will be used repeatedly in this paper.

###### Lemma 3.

Let such that . Consider a -dim vector space over and a fixed -dim subspace of . The number of distinct (un-ordered) -sized sets such that and is .

###### Proof:

First we find the number of such that is a -dim subspace of . To pick such a we define, for some . Such a can be picked in ways. However for one such fixed , there exist number of , where ) such that . Thus the required number of unique is . Similarly for every such we can select with the condition that is -dim subspace of in ways. So the number of distinct ordered sets is . By induction the number of distinct ordered sets is . We know that the number of permutations of a -sized set is . Therefore the number of distinct (un-ordered) sets satisfying the required conditions is . This completes the proof. ∎

###### Corollary 1.

Consider two subspaces of a -dim vector space over such that . The number of distinct such that is .

We now proceed to construct a caching line graph using projective geometry.

### Iii-B A new caching line graph using projective geometry

Consider such that . Consider a -dim vector space . Let be a fixed -dim subspace of . Consider the following sets of subspaces, where each such subspace contains .

 V ≜{V∈PGq(k−1,t−1):W⊆V}. R ≜{R∈PGq(k−1,t):W⊆R}. S ≜{S∈PGq(k−1,m+t−1):W⊆S}. U ≜{U∈PGq(k−1,m+t+1):W⊆U}.

Now, consider the following sets, which are used to present our line graph and the corresponding coded caching scheme.

 X ≜{{V1,V2}:V1,V2∈V,V1+V2∈R}. (1)
 Y ≜{{V1,V2,⋯,Vm+1}:∀Vi∈V,m+1∑i=1Vi∈S}. (2) Z ≜{{V1,V2,⋯,Vm+3}:∀Vi∈V,m+3∑i=1Vi∈U}. (3)

To construct a caching line graph , we need to satisfy the conditions P1-P3 in Definition 1. Following the notations in Section II, let and We construct systematically by first initializing by its user-cliques. The user-cliques are indexed by . For each create the vertices corresponding to the user-clique indexed by as . Thus, Now, for each we construct the subfile clique of associated with as . We thus see that Now, if we show that the user cliques (and equivalently, subfile cliques) are of the same size each, then the properties P1-P3 will be satisfied by . By invoking the notations from Section II, we have (number of user-cliques), and subpacketization (the number of subfile cliques).

We now find the values of , the size of user clique and the size of subfile clique .

###### Lemma 4.
 K=|X| =q2[k−t+11]q[k−t1]q. F=|Y| =[k−t+1m+1]qm∏i=0(qm+1−qi)(m+1)!(q−1)(m+1). |CX| =1(m+1)!q(m+1)(m+4)2m+1∏i=1[k−t−i1]q. |CY| =q(2m+3)2[k−m−t1]q[k−m−t−11]q.

for any , .

###### Proof:

(Finding ): Finding is equivalent to counting the number of distinct sets (such that and ) which gives distinct . By Lemma 3 we have, the number of distinct sets , such that and is . It is easy to check that . By Corollary 1 we have, the number of distinct such that for some fixed is . Therefore for each there exist distinct (where ) such that . Therefore we can write

.

(Finding ): Finding is equivalent to counting the number of distinct sets (such that and ) which gives distinct . By Lemma 3 we have, the number of distinct sets , such that and , is . It is easy to check that . By Corollary 1 we have, the number of distinct such that for some fixed is . Therefore for each there exist distinct (where ) such that . Therefore we can write

 F =m∏i=0(θ(k)−θ(t−1+i))(m+1)!q(m+1)(t−1) =m∏i=0(qk−qt−1+i)(m+1)!q(m+1)(t−1)(q−1)m+1 =q(m+1)(t−1)(m∏i=0qi)(m∏i=0(qk−t+1−i−1))(m+1)!q(m+1)(t−1)(q−1)m+1 =m∏i=0(qk−t+1−i−1)m∏i=0(qm+1−i−1)(m∏i=0(qm+1−i−1))(m∏i=0(qi))(m+1)!(q−1)m+1 =[k−t+1m+1]qm∏i=0(qm+1−qi)(m+1)!(q−1)m+1.

(Finding ): Consider an arbitrary . We have , for some . We know that . Now, finding is equivalent to counting the number of distinct sets (such that ) which gives distinct . By Lemma 3 we have, the number of distinct sets such that is . It is easy to check that . By Corollary 1 we have, the number of distinct such that for some fixed is . Therefore for each there exist distinct (where ) such that . Therefore we can write

 |CX| =m∏i=0(θ(k)−θ(t+1+i))(m+1)!q(m+1)(t−1) =m∏i=0(qk−qt+1+i)(m+1)!q(m+1)(t−1)(q−1)m+1
 =q(m+1)(t+1)(m∏i=0qi)(m∏i=0(qk−t−1−i−1))(m+1)!q(m+1)(t−1)(q−1)m+1 =q2(m+1)qm(m+1)2(m+1)!m+1∏i=1(qk−t−i−1)(q−1)m+1 =1(m+1)!q(m+1)(m+4)2m+1∏i=1[k−t−i1]q.

(Finding ): Consider an arbitrary . We have , for some . We know that . Now, finding is equivalent to counting the number of distinct sets (such that ) which gives distinct . By Lemma 3 we have, the number of distinct sets such that is . It is easy to check that . By Corollary 1 we have, the number of distinct such that for some fixed is . Therefore for each there exist distinct (where ) such that . Therefore we can write

 |CY| =1∏i=0(θ(k)−θ(t+m+i))2q2(t−1)=1∏i=0(qk−qt+m+i)2q2(t−1)(q−1)2 =qt+m(qk−t−m−1)qt+m+1(qk−t−m−1−1)2q2(t−1)(q−1)2 =q2m+32[k−m−t1]q[k−m−t−11]q.

This completes the proof. ∎

###### Remark 2.

It can be checked that by construction

Note that by Lemma 4, we have the size of the subfile cliques of as (for any ), and this is same for each . Similarly the user-cliques all have the same size . Hence satisfies properties P1-P3. We now show that has a clique cover with -sized disjoint cliques for some . Therefore is in fact a -caching line graph, giving rise to the main result in this section which is Theorem 1.

### Iii-C Delivery Scheme from a clique cover of ¯¯¯¯¯¯L2

We first describe a clique of and show that such equal-sized cliques partition . This will suffice to show the delivery scheme as per Theorem of [12] (summarized in Remark 1 in Section II in this work).

We now present a clique of size in (where represents binomial coefficient). Recall the definition of from (3).

###### Lemma 5.

Consider . Then is a clique in .

###### Proof:

First note that is well defined as and hence (belongs to user clique ). To show that the set of vertices of