I Introduction
Prefetching portions of popular content into cache memories distributed throughout the network in order to enable coded multicast transmissions useful for multiple receivers is regarded as a highly effective technique for reducing traffic load in wireless networks. The fundamental ratememory tradeoff in cacheaided broadcast networks with independent content has been studied in numerous works, including [1, 2, 3, 4], and more recently, with correlated content in [5, 6, 7, 8, 9, 10, 11, 12]. Exploiting content correlations becomes particularly critical as we move from static content distribution towards realtime delivery of rapidly changing personalized data (as in news updates, social networks, immersive video, augmented reality, etc.) in which exact content reuse is almost nonexistent [13]. Such correlations are especially relevant among files of the same category, such as episodes of a TV show or samesport recordings, which, even if personalized, may share common backgrounds and scene objects, and between multiple versions of dynamic data (e.g., news or social media updates).
The works in [5] and [6] consider a singlereceiver singlecache multiplefile network with lossy reconstructions, and characterize the tradeoffs between rate, cache capacity, and reconstruction distortions. The analysis in [5] also considers two receivers and one cache, in which again only local caching gains can be explored. The work in [6] models the caching problem in a way that resembles the GrayWyner network [14]. Our prior works in [7, 8, 9] focus on lossless reconstruction in a setting with an arbitrary number of files and receivers that allows exploring untapped global caching gains under correlated sources. A correlationaware scheme is proposed in [7] and [8], in which content is cached according to both the popularity of files and their correlation with the rest of the library. Cached information is then used as references for the compression of requested files during the delivery phase. Alternatively, our work in [9] addresses the content dependency by first compressing the correlated library. A subset of the files, most representative of the library, are selected as references, referred to as Ifiles, and the remaining files are intercompressed with respect to the selected files and referred to as Pfiles. This results in a compressed library where each file is made up of an Ifile and a Pfile, leading to a multiplerequest caching problem. Differently from previous multiplerequest schemes [15, 16, 4, 17, 18], the demand in [9] has a specific structure dictated by the configuration of the resulting compressed library in terms of Ifiles and Pfiles.
The first informationtheoretic characterization of the ratememory tradeoff in a cacheaided broadcast network with correlated content was studied in [10] for the setting of two files and two receivers, each equipped with a cache. This paper introduces an achievable twostep scheme that exploits content correlations by first jointly compressing the library files using the GrayWyner network [14], and then treating the compressed content as independent files. It is shown in [10] that this strategy is optimal for a large memory regime, while the gap to optimality is quantified for other memory values.
Building on the idea introduced in [9] and [10], concurrent work in [11] proposes a caching scheme for a network with arbitrary number of files and receivers, where the library has a specific correlation structure, i.e., each file is composed of multiple independent subfiles that are common among a fix set of files in the library. This, as in [9], leads to a multiplerequest caching problem where the demand has a particular configuration dictated by the specific library structure.
All previously cited works provide achievable caching schemes without analytically quantifying the gap to optimality, except for the special case of two files and two receivers in [10]. In this paper, by focusing on lossless reconstructions, we extend the informationtheoretic analysis of the broadcast caching network done in [10] to arbitrary number of receivers, each equipped with its own cache. Differently from [7, 8, 9] and [11], we characterize the peak and average ratememory region for files generated by a discrete memoryless source with arbitrary joint distribution, and we propose a class of optimal or nearoptimal twostep schemes, for which preliminary results were presented in [12]. Our main contributions are summarized as follows:

We formulate the problem of efficient delivery of multiple correlated files over a broadcast caching network with arbitrary number of receivers via informationtheoretic tools.

We propose a class of correlationaware twostep schemes, in which the files are first encoded based on the GrayWyner network [14], and in the second step, they are cached and delivered through a correlationunaware multiplerequest cacheaided coded multicast scheme. While most of the literature focuses on equallength files, our multiplerequest scheme in the second step is general enough to account for files compressed at different rates.

We discuss the optimality of the proposed twostep scheme in a twofile and receiver network by characterizing an upper bound on the peak and average ratememory tradeoffs for this class of schemes, and comparing it with a lower bound on the optimal ratememory tradeoffs derived in [19] using a cutset argument on the corresponding cachedemand augmented graph [20]. We identify a set of operating points in the achievable GrayWyner region [14] for which the proposed twostep scheme is optimal over a range of cache capacities, and approximates the optimal rate in the twofile network within half of the conditional entropy for all cache sizes.

We then extend the analysis to the threefile scenario since it captures the essence of the multiplefile case without involving the exponential complexity of the multiplefile GrayWyner network. We show that for two receivers the proposed scheme is optimal for high memory sizes and its gap to optimality is less than half of the joint entropy of two of the sources conditioned on the third source for other memory.

As a means to designing an achievable scheme for the second step of the proposed GrayWynerbased methodology, we also present a novel nearoptimal multiplerequest caching scheme for a network with two receivers and three independent files, where each receiver requests two of the files. The proposed scheme uses coded cache placement to achieve optimality for cache capacities up to half of the library size.
The paper is organized as follows. Sec. II presents the informationtheoretic problem formulation. In Sec. III, we introduce a class of twostep schemes based on the GrayWyner network. The multiplerequest caching problem, arising from the GrayWyner network, is discussed in Sec. IV, and a multiplerequest scheme for the twofile network is proposed and analyzed in detail in Sec. V. Sec. VI combines the multiplerequest scheme proposed in Sec. V with the GrayWyner encoding step, and analyzes the optimality of the overall twostep scheme with respect to a lower bound on the ratememory tradeoff in a twofile network. Extensions to a threefile network are analyzed in Sec. VII. After numerically analyzing the ratememory tradeoff using an illustrative example, the paper is concluded in Sec. IX.
Ii Network Model and Problem Formulation
We consider a broadcast caching network composed of one sender (e.g., base station) with access to a library of uniformly popular files generated by an component discrete memoryless source (NDMS). The NDMS model consists of finite alphabets and a joint pmf over . For a block length , library file is represented by a sequence , where , and , is generated i.i.d. according distribution . The sender communicates with receivers, , over a shared errorfree broadcast link. Each receiver is equipped with a cache of size bits, where denotes the (normalized) cache capacity.
We assume that the system operates in two phases: a caching phase and a delivery phase. During the caching phase, which takes place at offpeak hours when network resources are abundant, receiver caches are filled with functions of the library files, such that during the delivery phase, when receiver demands are revealed and resources are limited, the sender broadcasts the shortest possible codeword that allows each receiver to losslessy recover its requested file. We refer to the overall scheme, in which functions of the content are prefetched into receiver local caches, and are later used to reduce the delivery rate by transmitting coded versions of the requested files, as a cacheaided coded multicast scheme (CACM). A CACM scheme consists of the following components:

Cache Encoder: During the caching phase, the cache encoder designs the cache content of receiver using a mapping
The cache configuration of receiver is denoted by .

Multicast Encoder: During the delivery phase, each receiver requests a file from the library. The demand realization, denoted by , is revealed to the sender, where denotes the index of the file requested by receiver . The sender uses a fixedtovariable mapping
to generate and transmit a multicast codeword over the shared link.

Multicast Decoders: Each receiver uses a mapping
to recover its requested file, , using the received multicast codeword and its cache content as .
The worstcase probability of error of a CACM scheme is given by
(1) 
In this paper, we consider two performance criteria:

The peak multicast rate, , which corresponds to the worstcase demand,
(2) where denotes the length (in bits) of the multicast codeword , and the expectation is over the source distribution.

The average multicast rate, , over all possible demands
(3) where the expectation is over demands and source distribution.
Definition 1
A peak ratememory pair is achievable if there exists a sequence of CACM schemes for cache capacity and increasing file size , such that , and .
Definition 2
The peak ratememory region is the closure of the set of achievable peak ratememory pairs and the optimal peak ratememory function, , is
Definition 3
An average ratememory pair is achievable if there exists a sequence of CACM schemes for cache capacity and increasing file size , such that , and .
Definition 4
The average ratememory region is the closure of the set of achievable average ratememory pairs and the optimal average ratememory function, , is
Iii GrayWynerNetworkBased TwoStep Achievable Schemes
In this section, we propose a class of CACM schemes, based on a twostep lossless source coding setup, as depicted in Fig. 1. The first step involves lossless GrayWyner source coding [14], and the second step is a MultipleRequest CACM scheme.
The proposed twostep scheme exploits the correlation among the library content by first compressing the library using the GrayWyner network, described in detail in Sec. IIIA, and depicted for two and three files in Fig. 2. The GrayWyner network represents the library using private descriptions and descriptions that are common to files for , thereby transforming the caching problem with correlated content and receivers requesting only one file, into a caching problem with a larger number of files where receivers require multiple descriptions. Note that this multiplerequest caching problem has a specific class of demands. We assume that the multiplerequest CACM scheme in the second step is agnostic to the correlation among the content generated by the GrayWyner network, i.e., the second step is correlationunaware. The two steps are jointly designed to optimize the performance of the overall scheme, which is referred to as the GrayWyner MultipleRequest CACM (GWMR) scheme. Before formally describing the GWMR scheme, we briefly review the GrayWyner Network.
Iiia GrayWyner Network
The GrayWyner network was first introduced for two files in [14], in which a 2DMS is represented by three descriptions , where is referred to as the common description, and and are the corresponding private descriptions as depicted in Fig. 2(a). The descriptions are such that file can be losslessly recovered from descriptions , and file can be losslessly recovered from descriptions , both asymptotically, as block length . In [14], Gray and Wyner fully characterized the rate region for lossless reconstruction of both files, which is restated in the following Theorem.
Theorem 1 (GrayWyner Rate Region)
The optimal rate region for the twofile GrayWyner network, , is
(4) 
where denotes the closure of set , and the union is over all choices of for some with .
The GrayWyner network can be extended to files such that the GrayWyner encoder observes a DMS and communicates to decoder . The encoder is connected to the decoders through errorfree links, such that there is a link connecting the encoder to any subset of the decoders. In particular, there is one common link connecting the encoder to all decoders, there are links common to any of the decoders, and finally, there are private links connecting the encoder to each decoder. For any nonempty set , description is communicated to all decoders . The GrayWyner rate region, , is represented by the set of all ratetuples for the descriptions, for which file , can be losslessly reconstructed from the descriptions
asymptotically, as . In general, files are encoded into descriptions, such that: i) of the descriptions contain information exclusive to only one file, and ii) the remaining descriptions comprise information common to more than one file. In this paper we will study in detail the threefile GrayWyner network depicted in Fig. 2. The encoded descriptions are such that^{2}^{2}2With an abuse of notation, the subscripts of and denote sets.

,

, , ,

, , ,
and the GrayWyner rate region is represented by the set of all ratetuples
(5) 
for which file , , can be losslessly reconstructed from the descriptions with asymptotically, as .
IiiB GrayWyner MultipleRequest CACM Scheme
The proposed class of twostep schemes consists of:

GrayWyner Encoder: Given the library , the GrayWyner encoder at the sender computes descriptions , where is the set of all nonempty subsets of , using a mapping
for .

MultipleRequest Cache Encoder: Given the compressed descriptions, the correlationunaware cache encoder at the sender computes the cache content at receiver , as .

MultipleRequest Multicast Encoder: For any demand realization revealed to the sender, the correlationunaware multicast encoder generates and transmits the multicast codeword .

MultipleRequest Multicast Decoder: Receiver decodes the descriptions corresponding to its requested file as
where .

GrayWyner Decoder: Receiver decodes its requested file using the descriptions recovered by the multicast decoder, as , via the GrayWyner decoder
The GrayWyner encoder and decoder correspond to the first step, namely the encoder and decoder of the GrayWyner network, and the multiplerequest cache encoder, multiplerequest multicast encoder, and multiplerequest multicast decoder comprise the second multiplerequest CACM (MR) step of the proposed twostep scheme.
Note that the performance of the class of schemes described above depends on the operating point of the GrayWyner network . For a given , the performance of the overall twostep scheme is dictated by the peak and average multicast rates of the MR scheme, which similar to (2) and (3), are defined as
(6)  
(7) 
respectively.
Furthermore, the worstcase probability of error of the class of twostep schemes depends on the probability of error of the GrayWyner source coding step and the probability of error of the multiplerequest CACM step. Since , and is a GrayWyner description of with , it is guaranteed that GrayWyner decoding is asymptotically lossless with . Hence, the probability of error of a twostep scheme is approximately upper bounded by the probability of error of the MR scheme, given by
Definition 5
For a given ratetuple , an MR peak ratememory pair is achievable if there exists a sequence of MR schemes with rate , for cache capacity and increasing file size , such that , and .
Definition 6
For a given ratetuple , the MR peak ratememory region is the closure of the set of achievable MR peak ratememory pairs , and the MR peak ratememory function is defined as
In the class of twostep schemes, we refer to the scheme operating at the ratetuple that minimizes the MR peak ratememory function as the GWMR scheme. The peak ratememory pair achieved by this scheme is the GWMR peak ratememory function defined below.
Definition 7
The GWMR peak ratememory function is given by
Definition 8
For a given ratetuple , an MR average ratememory pair is achievable if there exists a sequence of MR schemes, with rate , for cache capacity and increasing file size , such that , and .
Definition 9
For a given ratetuple , the MR average ratememory region is the closure of the set of achievable MR average ratememory pairs , and the MR average ratememory function is defined as
Definition 10
The GWMR average ratememory function is given by
In the remainder of this paper, we first present the MR scheme in the second step for a general setting (Sec. IV), and then describe and analyze the performance of the overall GWMR scheme for the case of two files and receivers (Secs. V and VI), and for the case of three files and two receivers (Sec. VII).
Iv MultipleRequest CACM Scheme
In this section, we focus on the second step of the GWMR scheme depicted in Fig. 1, namely the MR scheme when the GrayWyner network operates at .
Recall that the GrayWyner network converts the file library into descriptions, each required for the lossless reconstruction of a set of the files in the original library. The MR scheme arranges the descriptions generated by the GrayWyner encoder into groups, , referred to as sublibraries. Sublibrary contains the descriptions that are common to exactly files. We refer to sublibrary as the commontoall sublibrary, and to , which contains all the private descriptions, as the private sublibrary. The MR scheme accounts for populating the receiver caches with content from sublibraries and serving the demand realizations placed in the original library, which translate into a new class of demands from the compressed sublibraries. More specifically, each receiver demand corresponds to a demand for a set of descriptions from each of the sublibraries (hence the term multiplerequest), such that the original library file requested by receiver maps to descriptions from sublibrary , . Even though receivers request single files from the original library independently and according to a uniform demand distribution, the GrayWyner encoding process leads to a nonuniform multiplerequest demand for descriptions (files) from the compressed sublibraries.
Our proposed MR scheme treats each sublibrary independently during both caching and delivery phases. Specifically, the descriptions from each sublibrary are cached and delivered as follows: i) the description in the commontoall sublibrary , which is required for the reconstruction of all files, and hence requested by all receivers, is cached according to the Least Frequently Used (LFU)^{3}^{3}3LFU is a local caching policy that, in the setting of this paper, leads to all receivers caching the same part of the file. strategy and delivered through naive (uncoded) multicasting, ii) sublibrary is cached and delivered according to any singlerequest correlationunaware CACM scheme (such as [24, 25, 26] for unequallength descriptions, and [1, 27, 28, 29, 30, 31, 32] for equallength descriptions), and iii) sublibrary , is cached and delivered according to any correlationunaware CACM scheme in which each receiver requests descriptions. Schemes where each receiver requests more than one file have been analyzed in previous works, e.g., [15, 16, 4, 17, 18]. However, in addition to being limited to settings with equallength files, they have been designed for arbitrary demand combinations and could therefore be suboptimal for the specific class of demands considered in our MR scheme. One of the challenges addressed in the next sections is the design of nearoptimal schemes for delivering the descriptions requested by each receiver from sublibrary , . Since the ultimate goal is to characterize the performance of the overall twostep GWMR scheme, and the optimal GrayWyner rate region is only known for two files [14], in the following sections, we first describe the proposed MR scheme and analyze the performance of the associated GWMR scheme for the case of two files and receivers in Secs. V and VI, respectively. In addition, to illustrate how extensions to more files could be done, we focus on the setting with three files and two receivers in Sec. VII.
It is worth noticing that the setting of our MR scheme, where the descriptions generated by the GrayWyner network are modeled as independent subfiles common to a fix set of files in the original library, is a generalization of the problem considered in [11], where, differently from our setting, subfiles are assumed to have equal length. Our results, for two files and receivers, and for three files and two receivers, if specialized to equallength descriptions (subfiles) are shown to yield optimal or nearoptimal schemes for the problem formulation studied in [11].
V MultipleRequest CACM Scheme for Two Files and Receivers
This section describes in more detail the MR scheme introduced in the previous section for a network with two files and receivers. Let denote the operating point of the GrayWyner network, where denotes the rate of the common description , and and denote the rate of the private descriptions and , respectively. As described in Sec. IV, the MR scheme for two files arranges the three descriptions generated by the GrayWyner network into a commontoall (or simply common) sublibrary , and a private sublibrary . Each receiver demand corresponds to requesting two descriptions: one from the common sublibrary , and one from the private sublibrary . The specific caching and delivery strategies adopted for each sublibrary are provided in Sec. VB.
In order to analyze the performance of the proposed MR scheme we also provide lower bounds on the MR peak and average ratememory functions in Sec. VA, and compare the achievable rates of the proposed MR scheme with these lower bounds in Sec. VC.
Va Lower Bounds on and
Theorem 2
In the twofile receiver network, for a given cache capacity and ratetuple , a lower bound on , the MR peak ratememory function, is given by
A lower bound on , the MR average ratememory function, is given by
VB Proposed MR Scheme
As described in Sec. IV, the proposed MR scheme for two files treats the descriptions in and as independent content and operates as follows: the cache capacity is optimally divided among the two sublibraries, caching is done independently from each sublibrary, and the content requested from each sublibrary is delivered independently, i.e., with no further coding across them. While the common description in is cached according to the LFU strategy and delivered through uncoded multicasting, the private descriptions in can be cached and delivered according to any correlationunaware CACM scheme available in literature (e.g. [1, 27, 28, 29, 30, 31, 32]), properly generalized to a setting with unequallength files. In the following, for a given generalized correlationunaware CACM scheme adopted for sublibrary , we describe how to optimally allocate the memory to each sublibrary, and we then characterize the peak and average rate achieved by the corresponding MR scheme.
Let and denote the lower convex envelope of the peak and average rates achieved by the scheme adopted for sublibrary , respectively.
Under the peak rate criterion, the cache allocation that minimizes the overall delivery rate (i.e., the sum rate of each sublibrary) is as follows. Let
(8) 
denote the cache encoder threshold, where denotes the left partial derivative with respect to . For a given , the cache encoder allocates the memory to each sublibrary as follows:

If , the common description is not cached at either receiver, and descriptions from are cached according to the caching strategy of the adopted correlationunaware CACM scheme.

If , the first bits of description are cached at both receivers (as per LFU caching), and descriptions are cached according to the scheme adopted for sublibrary over the remaining memory .

If , the common description is fully cached at both receivers, and descriptions are cached according to the adopted correlationunaware CACM scheme over the remaining cache capacity .
The optimality of the cache allocation described above is proved in Appendix A, and its graphical representation is depicted in Fig. 3. This representation can be understood as “waterfilling of two leaky buckets”, which is related to the wellknown waterfilling optimization problem [33]. For small cache capacities up to , all the memory is allocated to the private sublibrary. As the cache size increases, it is optimal to first store the common description until it is fully cached, and for cache capacities larger than , the residual memory is allocated to storing the private descriptions.
Remark 1
Differently from the singlecache setting analyzed in [5], where it is always optimal to cache the common description, in our case, when the cache capacity is smaller than , it is optimal to first cache the private descriptions. This is due to the fact that for small cache sizes delivering the privare descriptions with coded multicast transmissions is more effective in reducing the overall reate compared to mutlicasting the commom description, and therefore it is preferable to prioratize caching from the private sublibrary. As the cache size increases, this difference in rate reduction diminishes to the extent that assigning memory to the common sublibrary is more effective in reducing the rate compared to the private sublibrary. Therefore, it is preferable to fully store the common description prior to allocating additional memory for storing the private descriptions.
Under the average rate criterion, the optimal cache allocation among sublibraries and is similar to the peak rate scenario described previously, with respect to a cache encoder threshold
(9) 
where is the average delivery rate achieved by the correlationunaware CACM scheme adopted for sublibrary .
The following theorem provides the peak and average rates achieved by the proposed MR scheme for the optimal cache allocation described above.
Theorem 3
In the twofile receiver network, for a given cache capacity , ratetuple , and a given adopted correlationunaware CACM scheme with peak rate the peak rate achieved by the MR scheme, , is given by
(10) 
where is defined in (8). Denoting by the average rate achieved by an adopted correlationunaware CACM scheme, the average delivery rate achieved by the corresponding MR scheme, , is similar to that in (10) but with respect to and .
Proof 2
The proof is given in Appendix A.
From Theorem 3 it is observed that the performance of the proposed MR scheme depends on the correlationunaware CACM scheme adopted for the private sublibrary. In order to analyze the optimality of the proposed MR scheme in Sec. VC, for sublibrary , we resort to a scheme that properly combines near optimal schemes available in literature, generalized to unequal file lengths, as described next.
VB1 Peak Rate
The scheme adopted for the private sublibrary is based on memory sharing among generalizations of the correlationunaware CACM schemes proposed in [28] and [29] to files with unequal lengths. Specifically, for small cache sizes, the adopted scheme uses the scheme in [28] (first introduced in [30] for a specific cache capacity), which prefetches coded content during the caching phase, while for large cache sizes, it uses the scheme in [29] with uncoded cache placement. For ease of exposition, the specific details of the scheme adopted for the private sublibrary is given in Appendix C, where we also derive an upper bound on its achievable rate, , given in (39). By combining with Theorem 3, an upper bound on the peak rate achieved by the proposed MR scheme is given in the following theorem.
Theorem 4
In the twofile receiver network, for a given cache capacity and ratetuple , an upper bound on , the peak rate achieved with the proposed MR scheme, is given by
(11) 
where
(12) 
Proof 3
The proof follows from Theorem 3 and from adopting the scheme described in Appendix C for the private sublibrary. An upper bound on the peak rate achieved with this scheme, , is given in (39) and when replaced in (8), the cache encoder threshold becomes . Combining (10) with (39), and using the fact that , (12) is obtained after algebraic manipulation.
Specializing the upper bound on given in (11) to receivers results in a tight upper bound^{4}^{4}4By a tight upper bound we mean that for equation (11) holds with equality. for this setting as per the following corollary.
Corollary 1
In the twofile tworeceiver network, for a given cache capacity and ratetuple , the proposed MR scheme achieves the following peak rate
VB2 Average Rate
In this case, for the private sublibrary, we adopt a generalization of the closetooptimal correlationunaware CACM scheme in [29] to unequallength files. The specific details of the adopted scheme are provided in Appendix E, where we also derive an upper bound on its achievable rate, , given in (52). By combining with the average counterpart of (10) given in Theorem 3, an upper bound on the average rate achieved with the proposed MR scheme is given in the following theorem.
Theorem 5
In the twofile receiver network, for a given cache capacity and ratetuple , an upper bound on , the average rate achieved with the proposed MR scheme, is given by
(13) 
for and defined in Theorem 4, and with
(14) 
Proof 4
The proof follows from Theorem 3 and from adopting the scheme described in Appendix E for the private sublibrary. An upper bound on the average rate achieved with this scheme, , is given in (52) and when replaced in (9), the cache encoder threshold becomes . Eq (12) is obtained by combining the average counterpart of (10) given in Theorem 3 with (52).
Specializing the upper bound on given in (13) to the setting with receivers leads to the following corollary.
Corollary 2
In the twofile tworeceiver network, for a given cache capacity and ratetuple , the proposed MR scheme achieves the following average rate
(15) 
VC Optimality of the Proposed MR Scheme
In this section, we compare the performance of the proposed MR scheme charaterized in Sec. VB with the lower bounds given in Sec. VA. The following theorems provide the memory regions for which the proposed MR scheme is optimal or nearoptimal.
VC1 Peak Rate
Theorem 6
In the twofile receiver network and for a given ratetuple , when
the proposed MR scheme is optimal under the peak rate criterion among all multiplerequest schemes, i.e., . For all other cache capacities,
Proof 5
The proof is given in Appendix B.
Corollary 3
In the twofile tworeceiver network, for any cache capacity and ratetuple , the proposed MR scheme is optimal under the peak rate criterion among all multiplerequest schemes, i.e.,
Proof 6
The optimality of the scheme with respect to the MR peak ratememory function follows from setting in Theorem 6, for which the achievable rate meets the lower bound for the entire region of the memory.
VC2 Average Rate
Theorem 7
In the twofile receiver network and for a given ratetuple , when
the proposed MR scheme is optimal under the average rate criterion among all multiplerequest schemes, i.e, For all other cache capacities,
Proof 7
The proof is given in Appendix D.
Corollary 4
In the twofile tworeceiver network, for any cache capacity and ratetuple , the proposed MR scheme is optimal under the average rate criterion among all multiplerequest schemes, i.e.,
Proof 8
The optimality of the scheme with respect to the MR average ratememory function follows from setting in Theorem 7, for which the achievable rate meets the lower bound for the entire region of the memory.
Remark 2
Corollaries 3 and 4 imply that in the twofile tworeceiver network, caching and delivering content independently across the sublibraries, as described in Sec. VB, is sufficient to achieve optimality among all MR schemes, rendering coding across the multicast codewords pertaining to each sublibrary unnecessary.
Vi Optimality of the GWMR Scheme for Two Files
In this section, we first provide lower bounds on the optimal peak and average ratememory functions, and , given in Definitions 2 and 4, respectively, for the twofile receiver network described in Sec. II. Then, by using the results of the MR scheme for two files (Sec. V), we evaluate the performance of the proposed GWMR scheme (Sec. III), by comparing the presented lower bounds with the peak and average rates achieved by the proposed GWMR,
(16)  
(17) 
respectively, where and are the peak and average rates achieved by the MR scheme (Sec. VB).
Via Lower bounds on and
Theorem 8
In the twofile receiver network with library distribution , for a given cache capacity , a lower bound on , the optimal peak ratememory function, is given by
A lower bound on , the optimal average ratememory function, is given by
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