Optimal Cache Placement for Modified Coded Caching with Arbitrary Cache Size

TL;DR

This paper derives the optimal cache placement for the modified coded caching scheme with arbitrary cache sizes, minimizing delivery rate and providing a closed-form solution that generalizes existing schemes.

Contribution

It formulates the cache placement as a linear programming problem and derives a closed-form optimal solution for MCCS with any cache size, extending previous specific-cache solutions.

Findings

Optimal cache placement minimizes average delivery rate.

Closed-form solution generalizes existing cache schemes.

Caching gain varies with user population and cache patterns.

Abstract

We consider content caching between a service provider and multiple cache-enabled users, using the recently proposed modified coded caching scheme (MCCS) that provides an improved delivery strategy for random user requests. We develop the optimal cache placement solution for the MCCS with arbitrary cache size by formulating the cache placement as an optimization problem to minimize the average rate during the delivery phase under random user requests. Through reformulation, we show that the problem is a linear programming problem. By exploring the properties in the caching constraints, we obtain the optimal cache placement solution in closed-form. We verify that the existing cache placement scheme obtained at specific cache sizes is a special case of our solution. Numerical studies show how the caching gain changes as the user population increases, as a result of different cache placement patterns.

Tables

Table 1. TABLE I: The Optimal cache placement ( K = 7 𝐾 7 K=7 , N = 10 𝑁 10 N=10 ).

$M$	Optimal Cache Placement
	$a_0$	$a_1$	$a_2$	$a_3$	$a_4$	$a_5$	$a_6$	$a_7$
0	1.0	0	0	0	0	0	0	0
1	0.3	0.1	0	0	0	0	0	0
2	0	0.086	0.019	0	0	0	0	0
3	0	0	0.043	0.003	0	0	0	0
4	0	0	0.01	0.023	0	0	0	0
5	0	0	0	0.014	0.014	0	0	0
6	0	0	0	0	0.01	0.023	0	0
7	0	0	0	0	0.003	0.043	0	0
8	0	0	0	0	0	0.019	0.086	0
9	0	0	0	0	0	0	0.1	0.3
10	0	0	0	0	0	0	0	1.0

Equations

$$\sum _{l=0}^K\left(\genfrac{}{}{0ex}{}{K}{l}\right)a_l=1.$$

$$\sum _{l=1}^K\left(\genfrac{}{}{0ex}{}{K-1}{l-1}\right)a_l\le \mu .$$

$$\bar{R}={\mathbb{E}}_{\mathbf{d}}\left(\sum _{l=0}^{K-1}\left[\left(\genfrac{}{}{0ex}{}{K}{l+1}\right)-\left(\genfrac{}{}{0ex}{}{K-\tilde{N}(\mathbf{d})}{l+1}\right)\right]a_l\right)$$

$$\text{P1}:\underset{\mathbf{a}}{\min }$$

$$\text{eqref}Constraint1,\text{eqref}Constraint2$$

$$a_l\ge 0,a_l\le 1,l\in \mathcal{K}\cup \{0\}$$

$$\bar{R}=\sum _{\tilde{n}=1}^{\min \{N,K\}}{P}_{\text{u}}(\tilde{n})\sum _{l=0}^{K-1}\left[\left(\genfrac{}{}{0ex}{}{K}{l+1}\right)-\left(\genfrac{}{}{0ex}{}{K-\tilde{n}}{l+1}\right)\right]a_l.$$

$$a_{l^{\ast }}^{\ast }=\frac{l^{\ast }+1-\mu K}{\left(\genfrac{}{}{0ex}{}{K}{l^{\ast }}\right)},a_{l_{\ast }+1}^{\ast }=\frac{\mu K-l^{\ast }}{\left(\genfrac{}{}{0ex}{}{K}{l^{\ast }+1}\right)}.$$

$${\bar{R}}^{\ast }=$$

$$+\left[\left(\genfrac{}{}{0ex}{}{K}{l^{\ast }+2}\right)-\left(\genfrac{}{}{0ex}{}{K-\tilde{n}}{l^{\ast }+2}\right)\right]\frac{\mu K-l^{\ast }}{\left(\genfrac{}{}{0ex}{}{K}{l^{\ast }+1}\right)}).$$

$$\text{P2:}\underset{\mathbf{a}}{\min }$$

$${\mathbf{b}}^T\mathbf{a}=1$$

$${\mathbf{c}}^T\mathbf{a}\le \mu$$

$$\mathbf{a}⪰0,\mathbf{a}⪯1.$$

$$\text{eqref}SimpEqualCon:1,\text{eqref}SimpUnequalCon:1,\text{eqref}SimpUnequalCon:2$$

$${\lambda }_1({\mathbf{c}}^T\mathbf{a}-\mu )=0,$$

$${\lambda }_{2,l}{a}_l=0,l\in \mathcal{K}\cup \{0\}$$

$${\lambda }_{3,l}(a_l-1)=0,l\in \mathcal{K}\cup \{0\}$$

$$\mathbf{g}+{\lambda }_1\mathbf{c}-{\mathbf{\lambda }}_2+{\mathbf{\lambda }}_3+\nu \mathbf{b}=0$$

$${\lambda }_1\ge 0,{\mathbf{\lambda }}_2\succcurlyeq 0,{\mathbf{\lambda }}_3\succcurlyeq 0.$$

$$R=\{\begin{array}{l}\frac{K-i}{i+1},\text{for}K-\tilde{N}(\mathbf{d})<i+1\\ \frac{K-i}{i+1}\left(1-\frac{(K-i-1)\dots (K-i-\tilde{N}(\mathbf{d}))}{K(K-1)\dots (K-\tilde{N}(\mathbf{d})-1)}\right),\text{otherwise}.\end{array}$$

$$c_i{a}_i+c_j{a}_j-\mu =0.$$

$$b_i{a}_i+b_j{a}_j=1.$$

$$\frac{b_j\mu -c_j}{b_j{c}_i-b_i{c}_j}$$

$$a_i^{\ast }$$

$$R=$$

$$+\left[\left(\genfrac{}{}{0ex}{}{K}{l^{\ast }+2}\right)-\left(\genfrac{}{}{0ex}{}{K-\tilde{N}(\mathbf{d})}{l^{\ast }+2}\right)\right]\frac{\mu K-l^{\ast }}{\left(\genfrac{}{}{0ex}{}{K}{l^{\ast }+1}\right)}.$$

$$R=$$

$$+\left(\genfrac{}{}{0ex}{}{K}{l^{\ast }+2}\right)\frac{\mu K-l^{\ast }}{\left(\genfrac{}{}{0ex}{}{K}{l^{\ast }+1}\right)}.$$

$$R=\left(\genfrac{}{}{0ex}{}{K}{l^{\ast }+1}\right)\frac{l^{\ast }+1-\mu K}{\left(\genfrac{}{}{0ex}{}{K}{l^{\ast }}\right)}+\left(\genfrac{}{}{0ex}{}{K}{l^{\ast }+2}\right)\frac{\mu K-l^{\ast }}{\left(\genfrac{}{}{0ex}{}{K}{l^{\ast }+1}\right)}.$$

Full text

I Introduction

The last decades have witnessed a dramatic surge of wireless traffic due to the proliferation of mobile devices [1]. Facing the drastic increase of data-intensive new wireless applications and services, future wireless communication networks need to effectively manage the data traffic congestion and meet the requirements of timely content delivery. Using network storage resources for content caching has emerged as a compelling technology to alleviate the network traffic load and reduce the content access latency for users [2, 3]. The availability of local caches at the network edge, either at base stations or users, creates new network resources and opportunities to increase the user service capacity. Caching technologies are anticipated to become key technological drivers for content delivery in future wireless networks.

The caching design and analysis have attracted increasing research interests. Many recent works have investigated into the strategies for cache placement and delivery to understand the effect of caching on reducing the network load [4, 5, 6, 7]. Conventional uncoded caching can improve the hit rate [4, 5], but is not efficient when there are multiple caches[8]. Coded caching is first introduced in[6], where a Coded Caching Scheme (CCS) is proposed that combines a cache placement scheme specifying the cached (uncoded) content and a coded multicasting delivery strategy, assuming uniform file and cache characteristics. With a focus on the theoretical limit, the minimum peak traffic load under caching is analyzed and substantially coded caching gain for load reduction is shown. Coded caching has since drawn considerable attentions, with extension to the decentralized cache placement scheme [7], transmitter caching in mobile edge networks [9, 10], and for both transmitter and receiver caching in wireless interference networks [11]. Instead of designing cache placement schemes to reduce the peak rate (load), the optimization of cache placement in the CCS is considered in [12, 13], where using different approaches, the authors have obtained the optimal cache placement strategy to minimize the peak rate.

The original CCS[6] aims to minimize the peak rate in the worst-case scenario where users request distinct files, assuming more files ($N$) than the user population ($K$). For the general cases where multiple users may request the same file, the CCS contains some redundancy in the delivery phase causing additional traffic load. To address this limitation, for the cache size being multiples of $N/K$, a recent study [14] has proposed a Modified Coded Caching Scheme (MCCS) to remove the redundancy of the CCS in the delivery phase, and the minimum average rate has been obtained. The corresponding cache placement has been further shown to be optimal [15]. However, the technique used to verify the optimality of the cache placement is complicated and is developed only for the specific cache sizes (multiples of $N/K$), and cannot obtain the general optimal cache placement solution for the MCCS with an arbitrary cache size.

In this paper, we use the optimization approach to obtain the optimal cache placement solution for the MCCS with any cache size. We formulate the cache placement design into a cache placement optimization problem, aiming to minimize the average rate in the delivery phase, under random user demands. Through reformulation, the optimization problem is shown to be a linear programming problem. By exploring the properties in the problem, we derive the optimal cache placement solution in closed-form. Our result is general to show the optimal cache placement solution for the MCCS with arbitrary user population and cache memory size. We verify that the existing optimal cache placement scheme [14, 15] for cache sizes at multiples of $N/K$ are special cases in our solution. Through simulation, we analyze the performance of the optimal caching scheme and compare it other schemes. We show how the caching gain changes as the user population increases, as a result of different cache placement patterns.

II System Model

Consider a cache-aided transmission system with a server connecting to $K$ users, each with a local cache, over a shared error-free link, as shown in Fig 1. The server has a database consisting of $N$ files, $\{W_{1},\ldots,W_{N}\}$, each of size $F$ bits. Denote ${\cal N}\triangleq\{1,\ldots,N\}$. We assume uniform popularity distribution of these files with $p_{n}=1/N$, for $n\in{\cal N}$. Denote the set of users by ${\cal K}\triangleq\{1,\ldots,K\}$. Each user $k$ has a local cache of capacity $MF$ bits, for $M\in[0,N]$, and we denote its cache size (normalized by file size) by $M$.

The system operates in two phases: cache placement phase and content delivery phase. The cache placement is performed in advance during the off-peak hours without knowing the user file requests, and is changed at a longer time scale. During this phase, under a cache placement scheme, each user $k$ uses a caching function $\phi_{k}(\cdot)$ to map $N$ files into its cached content: $Z_{k}\triangleq\phi_{k}(W_{1},\ldots,W_{N})$. Each user $k$ independently requests one file from the server, with the index of requested file denoted by $d_{k}$, $k\in{\cal K}$. Denote ${\bf d}\triangleq[d_{1},\ldots,d_{K}]$ as the demand vector containing the indices of file requested by all users. In the content delivery phase, based on the demand vector ${\bf d}$ and the cache placement, the server generates coded messages and transmit them to the users over the shared link. Denote the codeword as $X_{{\bf d}}=\psi_{{\bf d}}(W_{1},\ldots,W_{N})$, where $\psi_{{\bf d}}(\cdot)$ is the encoding function for demand ${\bf d}$. Upon receiving the codewords, each user $k$ applies a decoding function $\varphi_{{\bf d},k}(\cdot)$ to obtain the (estimated) requested file $\hat{W}_{{\bf d},k}$ from the received signal and its cached content as $\hat{W}_{{\bf d},k}\triangleq\varphi_{{\bf d},k}(X_{{\bf d}},Z_{k})$. Thus, an entire coded caching scheme can be represented by the caching, encoding and decoding functions.

In both the CCS [6, 7] and the MCCS [14], each file $W_{n}$ is partitioned into non-overlapping subfiles with equal size, one for each specified user subsets. During the cache placement phase, user $k$ caches those subfiles for the user subsets containing user $k$. In the delivery phase, the server delivers the missing subfiles of the requested file not in a user’s local cache, using a coded multicasting delivery scheme.

III Problem formulation

A key design issue in a coded caching scheme is the cache placement. Existing coded caching schemes describe specific ways of file partitioning for the cache placement, when cache size $M$ is multiple of $N/K$. Instead of this design approach, we formulate the coded caching problem as a cache placement optimization problem for a given cache size $M$, to minimize the average rate (load) over the shared link, where the delivery strategy is specified by the MCCS.

III-A Cache Placement

To formulate the problem, each file $W_{n}$ is partitioned into $2^{K}$ non-overlapping subfiles, one for each unique user subset ${\cal S}\subseteq{\cal K}$. Since we assume that file lengths and popularity and the cache sizes are all uniform, a symmetric cache placement is adopted by treating all files equally. Thus, all the files are partitioned in the same way. That is, let $W_{n,{\cal S}}$ denote the subfile of $W_{n}$ for user subset ${\cal S}$. Its size satisfies $|W_{1,{\cal S}}|=\cdots=|W_{N,{\cal S}}|$, for all ${\cal S}\subseteq{\cal K}$. In addition, the size of these subfiles only depends on the size of user subset $|{\cal S}|$ under the symmetric cache placement.

Note that there are ${K\choose l}$ different user subsets with the same size $l$, for $l=0,\ldots,K$, where $l=0$ corresponds to the empty set $\emptyset$. Let ${\cal S}^{l}_{i}$ denote user subset $i$ of size $l$, i.e., $|{\cal S}^{l}_{i}|=l$, for $i=1,\ldots,{K\choose l}$. Let ${\cal A}^{l}\triangleq\{{\cal S}^{l}_{i},i=1,\ldots,{K\choose l}\}$ denote cache subgroup $l$ containing all user subsets of size $l$, for $l=0,\ldots,K$. Thus, all user subsets are partitioned into $K+1$ cache subgroups based on the subset size. Accordingly, all subfiles are partitioned into $K+1$ subgroups: ${\cal W}^{l}=\{W_{n,{\cal S}}:{\cal S}\in{\cal A}^{l},n\in{\cal N}\}$, $l=0,\ldots,K$, where the subfiles in the same group ${\cal W}^{l}$ have the same size.

Define the normalized subfiles size $a_{l}\triangleq|W_{n,{\cal S}}|/F$, as a fraction of the file size $F$, for all $n\in{\cal N}$, ${\cal S}\in{\cal A}^{l}$. Let ${\bf a}=[a_{0},\ldots,a_{K}]^{T}$ denote the cache placement vector (common to all files) describing the size of subfiles to be cached in each cache subgroup. Note that $a_{0}$ represents the fraction of a file that solely exists in the server. In the cache placement phase, user $k$ caches all the subfiles in ${\cal W}^{l}$, $l=1,\ldots,K$, that are for user subsets containing user $k$. In other words, user $k$ caches $\{W_{n,{\cal S}}:{\cal S}\in{\cal A}^{l}~{}\text{and}~{}k\in{\cal S},n\in{\cal N}\}$, $l=1,\ldots,K$.

For a given caching scheme, each original file should be able to be reconstructed by combining all its subfiles. For each file, among the partitioned subfiles, there are ${K\choose l}$ subfiles with size $a_{l}$ (for all user subsets with ${\cal S}=l$). Thus, we have the file partitioning constraint

[TABLE]

For the local cache at each user, note that among all user subsets of size $l$, there are total ${K-1\choose l-1}$ different user subsets containing the same user, for $l=1,\ldots,K$. Since each file is partitioned based on user subsets, it means that for each file, the total number of subfiles a user can possibly cache is $\sum_{l=1}^{K}{K-1\choose l-1}$; Considering the subfile size $a_{l}$ for each cache subgroup $l$, this amounts to $\sum_{l=1}^{K}{K-1\choose l-1}a_{l}$ bits that can be cached by the user for each file. Define $\mu\triangleq M/N$ as the normalized cache size. We have the local cache size constraint at each user as

[TABLE]

III-B Content Delivery under the MCCS

The recently proposed MCCS [14] provides a new delivery strategy that removes this redundancy existed in the CCS for further rate reduction. The delivery scheme in the CCS is by multicasting a unique coded message to each user subset ${\cal S}\in{\cal A}^{l+1}$, $l=0,\ldots,K-1$, formed by bitwise XOR operation of subfiles (of the same size $a_{l}$) as: $\bigoplus_{k\in{\cal S}}\!W_{d_{k},{\cal S}\backslash\{k\}}$. Each user in subset ${\cal S}$ can retrieve the subfile of its requested file. Assuming the worst case of distinct file requests, coded messages for all user subsets are delivered. There are ${K\choose l+1}$ user subsets in ${\cal A}^{l+1}$, to which coded messages of size $a_{l}$ are delivered. The overall peak rate is $\sum_{l=0}^{K-1}{K\choose l+1}a_{l}$.

When the file requests are not distinct, the coded delivery in the original CCS contains some redundant subfiles. Let $\widetilde{N}({\bf d})$ denote the distinct requests for demand vector ${\bf d}$. Based on the MCCS, it forms a leader group that contains exactly $\widetilde{N}({\bf d})$ distinct requests. Denote ${\cal D}$ the leader group s.t. $|{\cal D}|=N_{e}({\bf d})$. Then any group that has intersection with the leader group is called non-redundant group, which is denoted by ${\cal S}$. We can know that the number of non-redundant group is ${K\choose l+1}-{K-\tilde{n}\choose l+1}$. From the decentralized MCCS describe in [14] users will be able to reconstruct the files they requested once the coded messages of all the non-redundant groups are delivered.

III-C Cache Placement Optimization for the MCCS

Our objective is to minimize the expected rate $\bar{R}$ by optimizing the cache placement. From delivering strategy described in Section III-B, we know that $\bar{R}$ is equal to the expected size of all the coded messages of the non-redundant groups. There are ${K\choose l+1}$ user subsets with size $l+1$, and among them, ${K-\widetilde{N}({\bf d})\choose l+1}$ are redundant subgroups, of which coded messages to them are redundant for users to recover the subfiles for their requested files. Removing these redundant transmissions, the expected rate is given by

[TABLE]

where the expectation is taken with respect to ${\bf d}$. Following the common practice, we define ${n\choose k}=0$ when $n<0$ or $k>n$. The cache placement optimization problem is formulated as

[TABLE]

where constraints (4) are the requirements for the subfile size.

IV The Optimal Cache Placement for the MCCS

For the uniform file popularity, the probability of having $\tilde{n}$ distinct requests is $P_{\text{u}}(\tilde{n})={\bf S}(K,\tilde{n}){N\choose\tilde{n}}\frac{\tilde{n}!}{N^{K}}$, for $\tilde{n}=1,\ldots,\min\{N,K\}$, where ${\bf S}(\cdot,\cdot)$ is the Stirling number of the second [16]. Based on this, we can express the expected rate $\bar{R}$ in (3) as

[TABLE]

It is clear that $\bar{R}$ is linear in $a_{l}$’s. In addition, all the constraints in P1 are also linear in $a_{l}$’s. Thus, P1 is a linear programming problem with respect to ${\bf a}$. In the following, we solve P1 to obtain the optimal cache placement solution. The result is given bellow.

Theorem 1**.**

For any cache size $M\leq N$ and $\mu=M/N$, the optimal cache placement to minimize the expected rate $\bar{R}$ in P1 is ${\bf a}^{*}=[0,\ldots,a^{*}_{l^{*}},a_{l^{*}+1}^{*},\ldots,0]^{T}$, where $\mu K-1\leq l^{*}<\mu K$, and

[TABLE]

*The minimum expected rate is *

[TABLE]

We will detail the proof of Theorem 1 in Section IV-A.

Remark: Using a different approach, [15] has shown that the cache placement scheme proposed by [14] for the cache size at $M\in\{0,\frac{N}{K},\frac{2N}{K},\ldots,N\}$ is optimal for minimizing the expected rate. However, the approach is more complicated, and cannot be generalized to obtain the optimal cache placement for any arbitrary cache size $M$ between those points. Our result in Theorem 1 provides the optimal cache placement solution in a closed-form for the MCCS, for any given $M$, $K$ and $N$. Also, note that our optimization approach can be applied to derive the optimal cache placement solution for the original CCS for the peak load minimization, which has been obtained in [12]. However, the approach used there cannot be applied to solve the cache placement problem for the MCCS considered in this work.

IV-A Proof of Theorem 1

We first reformulate problem P1, and then solve it using the KKT conditions [17]. Define ${\bf g}\triangleq[g_{0},\ldots,g_{K}]^{T}$, where $g_{l}\triangleq{K\choose l+1}-\sum_{\tilde{n}=1}^{\min\{N,K\}}P_{\text{u}}({\tilde{n}}){K-\tilde{n}\choose l+1}$, $l=0,\ldots,K$, ${\bf b}\triangleq[b_{0},\ldots,b_{K}]^{T}$, where $b_{l}\triangleq{K\choose l}$, and ${\bf c}\triangleq[c_{0},\ldots,c_{K}]^{T}$, where $c_{l}\triangleq{K-1\choose l-1}$. Then, we can rewrite P1 as

[TABLE]

To solve P2, define the Lagrange multipliers $\lambda_{1}$, $\boldsymbol{\lambda}_{2}=[\lambda_{2,0},\ldots,\lambda_{2,K}]^{T}$, $\boldsymbol{\lambda}_{3}=[\lambda_{3,0},\ldots,\lambda_{3,K}]^{T}$ for constraints (9) and (10), respectively, and $\nu$ for constraint (8). The KKT conditions are given as follows

[TABLE]

IV-A1 Optimal file partitioning strategy

We first introduce Lemmas 1-3 which help reduce the complexity in finding the solution. The corresponding proofs are omitted due to the space limitation.

Lemma 1**.**

At the optimality, inequality (9) is attained with equality, i.e., the cache storage $\mu$ is always fully utilized under the optimal cache placement vector ${\bf a}^{*}$ for P1.

Lemma 2**.**

When $\mu=0$, the optimal ${\bf a}^{*}=[1,0,\ldots,0]^{T}$ with the minimum expected rate $\bar{R}^{*}=E(\widetilde{N}({\bf d}))$; when $\mu=1$, the optimal ${\bf a}^{*}=[0,\ldots,0,1]^{T}$ with the minimum expected rate $\bar{R}^{*}=0$.

Lemma 2 describes the two extreme cases of having no cache memory ($\mu=0$) and sufficient cache size to hold all $N$ files ($\mu=1$). In the following, we only need to discuss the case when $\mu\in(0,1)$.

By exploring the properties of KKT conditions (12)-(14), we show below the condition on ${\bf a}$ for $\lambda_{1}$ and $\nu$ having feasible solutions.

Lemma 3**.**

For $\mu\in(0,1)$, the optimal cache placement vector $\mathbf{a^{*}}$ has at most two non-zero elements.

Lemma 3 implies that the number of non-zero elements of an optimal caching vector ${\bf a}^{*}$ can only be one or two (it cannot be zero due to constraint (8)):

Case 1) One non-zero element: In this case, there exists $a_{l}\neq 0$ for some $l\in\{0,\ldots,K\}$, and $a_{j}=0$ for $\forall j\neq l$, $j\in\{0,\ldots,K\}$. From (8), we have $b_{l}a_{l}=1$. Thus, we have $a_{l}={1}/{b_{l}}={1}/{{K\choose l}}$. To find the cache size that leads to this solution, note that since there is only one non-zero subfile size, from Lemma 1, we have $c_{l}a_{l}=\mu$. Thus, the relation of the normalized cache size and index $l$ is given by $\mu={c_{l}}/{b_{l}}={{K-1\choose l-1}}/{{K\choose l}}={l}/{K}$.

Thus, if for some $l\in\{0,\ldots,K\}$ satisfies $\mu={l}/{K}$, the optimal cache placement is ${\bf a}^{*}=[0,\ldots,0,a^{*}_{l^{*}},0\ldots,0]^{T}$, where $a^{*}_{l^{*}}={1}/{{K\choose l^{*}}}$. For given demand ${\bf d}$, the corresponding rate $R$ can be computed based on the redundancy to be removed in the delivery phase, and we have

[TABLE]

As a result, the expected rate is $\bar{R}=\sum_{\tilde{n}=1}^{\min\{N,K\}}P_{\text{u}}({\tilde{n}})R.$

Remark: The optimal solution with one non-zero element in ${\bf a}^{*}$ corresponds to equal file partitioning, where all subfiles have equal size. The optimal ${\bf a}^{*}$ obtained above exactly matches the cache placement scheme proposed in [14] for cache size at points $M\in\{0,\frac{N}{K},\frac{2N}{K},\ldots,N\}$. We see that it is a special case in our general cache placement optimization problem.

Case 2) Two non-zero elements: In this case, there exist some $i$ and $j$, such that $a_{i},a_{j}\neq 0$, and $a_{l}=0$, $\forall l\neq i,j$, $l\in\{0,\ldots,K\}$. With only two non-zero variables $a_{i}$ and $a_{j}$, from Lemma 1, we can rewrite (11) as

[TABLE]

Also from (8), we have

[TABLE]

From (17) and (18), we have $a_{i}=\frac{b_{j}\mu-c_{j}}{b_{j}c_{i}-b_{i}c_{j}}$ and $a_{j}=\frac{b_{i}\mu-c_{i}}{b_{i}c_{j}-b_{j}c_{i}}$. Since $a_{i}$ and $a_{j}$ are both non-zero, the two solutions only exists when

[TABLE]

Assume $0\leq i<j\leq K$. Since ${c_{i}}/{b_{i}}={{K-1\choose i}}/{{K\choose i}}$ is an increasing function of $i$, we have $b_{j}c_{i}-b_{i}c_{j}<0$. Based on (19), we have $b_{i}\mu-c_{i}>0$ and $b_{j}\mu-c_{j}<0$, which means $i$ and $j$ should satisfy ${c_{i}}/{b_{i}}<\mu<{c_{j}}/{b_{j}}$.

Lemma 4**.**

For $i<j$ and $i,j\in\{0,\ldots,K\}$ satisfying $\mu\in({c_{i}}/{b_{i}},{c_{j}}/{b_{j}})$, the expected rate $\bar{R}$ is a decreasing function of $i$ and an increasing function of $j$.

From Lemma 4, we have the conclusion that the minimum expected rate $\min_{0\leq i<j\leq K}\bar{R}$ can only be obtained when $j=i+1$. Any other relation would result in larger $\bar{R}$. Following this, for $0\leq i\leq K-1$ satisfying ${c_{i}}/{b_{i}}<\mu<{c_{i+1}}/{b_{i+1}}$, we have the optimal $a^{*}_{i}$ and $a^{*}_{i+1}$ as

[TABLE]

The corresponding expected rate is $\bar{R}=g_{i}a_{i}^{*}+g_{i+1}a_{i+1}^{*}$. Substituting the values of $g_{i}$, $b_{i}$ and $c_{i}$ into the above expressions, let $l^{*}=i$, we have the following conclusion: For $\mu K-1<l^{*}<\mu K$, the optimal cache placement ${\bf a}^{*}=[0,\ldots,a^{*}_{l^{*}},a_{l^{*}+1}^{*},\ldots,0]^{T}$ where $a^{*}_{l^{*}}$ and $a_{l^{*}+1}^{*}$ are given as in (6).

Remark: The optimal cache placement indicates that, each file is split into two parts with sizes $(l^{*}+1-\mu K)$ and $(\mu K-l^{*})$. Then each part is further partitioned into subfiles of equal sizes, with the first partitioned into ${K\choose l^{*}}$ subfiles (for cache subgroup $l^{*}$), and the second partitioned into ${K\choose l^{*}+1}$ subfiles (for cache subgroup $l^{*}+1$). User $k$ will cache these two types of subfiles for all user subsets including $k$.

Given any $\widetilde{N}({\bf d})$, the rate $R$ depends on the amount of redundancy removed in the delivery phase in following cases:

• •

$K-\widetilde{N}({\bf d})\geq l^{*}+2$: There are redundant coded messages for both user subsets of size $l^{*}$ and $l^{*}+1$, and we have

[TABLE]

• •

$K-\widetilde{N}({\bf d})=l^{*}+1$: The redundant coded message can only be found for user subsets of size $l^{*}$, and we have

[TABLE]

• •

$K-\widetilde{N}({\bf d})<l^{*}+1$: No redundant message for any user subsets, and the rate is

[TABLE]

The minimum expected rate is given by $\bar{R}=\sum_{\tilde{n}=1}^{\min\{N,K\}}P_{\text{u}}({\tilde{n}})R$. From (• ‣ IV-A1)-(22), we arrive at the expression in (1).

Cases 1 and 2 give the optimal cache placement for any $\mu\in(0,1)$. Combining these with the solutions for $\mu=0$ and $\mu=1$ in Lemma 2, we have the results in Theorem 1.

V Numerical Results

Consider a system with $N$ files of equal size, $K$ users with the same cache size $M$. First, we show in Table I the values of the optimal cache placement vector ${\bf a}^{*}$ for different cache sizes $M=0,\ldots,10$, for $K=7$ and $N=10$. Beside the two extreme cases of $M=0$ or $10$ (all the files are either in the servers, or stored at the local cache), for $M$ in between, we see that ${\bf a}^{*}$ always has two non-zero elements (*i.e., *two different subfile sizes for two cache subgroups), and their locations shift to the cache subgroup of larger size $l$ as $M$ increases.

In Fig. 2, for fixed $N=10$ and $M=2$, we study how the expected rate $\bar{R}$ changes with the increasing number of users $K$. We compare the optimal cache placement for the MCCS, with other schemes, including the centralized CCS[6], and the decentralized CCS[7] and MCCS[14]. The optimal cache placement solution achieves the minimum expected rate, and thus outperforms all the other schemes. For both the CCS and the MCCS, the expected rate under the optimal solution increases with $K$ with a certain pattern. The rate increment slows down when $K$ reaches $iN/M$, for $i=1,2,\ldots$, but becomes higher after $K$ passes those points. This suggests the caching gain increases with $K$ from $(i-1)N/M$ to $iN/M$, with the highest gain achieved at $iN/M$. Note that for a given normalized cache size $\mu$, $K$ determines the caching subgroup sizes and the number of user subsets for coded multicasting under the optimal cache placement. For $K=iN/M$, the optimal ${\bf a}^{*}$ only has one cache subgroup (*i.e., *equal file partitioning), while at the two sides of this point, two different caching subgroups are used.

VI Conclusion

In this paper, we formulate the general cache placement design for the MCCS under uniform file popularity as a cache placement optimization problem to minimize the expected rate during the delivery phase, for any number of users, files, and cache size. Through the optimization approach and by exploring the property of the optimization problem, we obtained the general optimal cache placement solution in closed-form for the MCCS.