A Fundamental Storage-Communication Tradeoff for Distributed Computing with Straggling Nodes

TL;DR

This paper extends the framework of placement delivery arrays to systems with straggling nodes, providing universal coded computing schemes for MapReduce-like systems that are optimal in terms of storage and communication tradeoffs.

Contribution

It introduces a universal scheme for systems with stragglers based on Comp-PDAs, characterizes the SC tradeoff, and proves optimality of MAN-PDAs for any number of stragglers.

Findings

The scheme is universal and applies to arbitrary functions.

The information-theoretic bound matches the performance of MAN-PDAs.

Optimality of MAN-PDAs holds regardless of the number of stragglers.

Abstract

Placement delivery arrays for distributed computing (Comp-PDAs) have recently been proposed as a framework to construct universal computing schemes for MapReduce-like systems. In this work, we extend this concept to systems with straggling nodes, i.e., to systems where a subset of the nodes cannot accomplish the assigned map computations in due time. Unlike most previous works that focused on computing linear functions, our results are universal and apply for arbitrary map and reduce functions. Our contributions are as follows. Firstly, we show how to construct a universal coded computing scheme for MapReduce-like systems with straggling nodes from any given Comp-PDA. We also characterize the storage and communication loads of the resulting scheme in terms of the Comp-PDA parameters. Then, we prove an information-theoretic converse bound on the storage-communication (SC) tradeoff achieved by universal computing schemes with straggling nodes. We show that the information-theoretic bound matches the performance achieved by the coded computing schemes with straggling nodes corresponding to the Maddah-Ali and Niesen (MAN) PDAs, i.e., to the Comp-PDAs describing Maddah-Ali and Niesen's coded caching scheme. Interestingly, the same Comp-PDAs (the MAN-PDAs) are optimal for any number of straggling nodes, which implies that the map phase of optimal coded computing schemes does not need to be adapted to the number of stragglers in the system. We finally prove that while the points that lie exactly on the fundamental SC tradeoff cannot be achieved with Comp-PDAs that require smaller number of files than the MAN-PDAs, this is possible for some of the points that lie close to the SC tradeoff. For these latter points, the decrease in the requested number of files can be exponential in the number of nodes of the system.

Figures

Equations

$$\mathsf{K},\mathsf{Q},\mathsf{N},\mathsf{D},\mathsf{U},\mathsf{V},\mathsf{W},$$

$$u_d={\varphi }_d(w_1,\dots ,w_{\mathsf{N}})\in {\mathbb{F}}_2^{\mathsf{U}}.$$

$${\varphi }_d(w_1,\dots ,w_{\mathsf{N}})=h_d(f_{d,1}(w_1),\dots ,f_{d,\mathsf{N}}(w_{\mathsf{N}})),$$

$$v_{d,n}\triangleq f_{d,n}(w_n)\in {\mathbb{F}}_2^{\mathsf{V}},\forall n\in [\mathsf{N}];$$

$$u_d={\varphi }_d(w_1,\dots ,w_{\mathsf{N}})=h_d(v_{d,1},\dots ,v_{d,\mathsf{N}}).$$

$${\mathcal{C}}_k\triangleq \{v_{d,n}:d\in [\mathsf{D}],w_n\in {\mathcal{M}}_k\}.$$

$$\mathrm{Pr}\left\{\mathbf{Q}=\mathcal{Q}\right\}=\frac{1}{C_{\mathsf{K}}^{\mathsf{Q}}},\forall \mathcal{Q}\in {\mathbf{\Omega }}_{\mathsf{K}}^{\mathsf{Q}}.$$

$$t_n\ge \mathsf{K}-\mathsf{Q}+1,\forall n\in [\mathsf{N}].$$

$$X_k^{\mathbf{Q}}={\phi }_k^{\mathbf{Q}}\left({\mathcal{C}}_k,{\mathbf{\Gamma }}^{\mathbf{Q}}\right)$$

$$\displaystyle X^{{\mathbf{Q}}}\triangleq\big{\{}X_{k}^{{\mathbf{Q}}}:k\in\mathbf{Q}\big{\}}.$$

$${\left\{(v_{d,1},\dots ,v_{d,\mathsf{N}})\right\}}_{d\in {\mathcal{D}}_k^{\mathbf{Q}}}={\psi }_k^{\mathbf{Q}}\left(X^{\mathbf{Q}},{\mathcal{C}}_k,{\mathbf{\Gamma }}^{\mathbf{Q}}\right),$$

$$u_d=h_d(v_{d,1},\dots ,v_{d,\mathsf{N}}),\forall d\in {\mathcal{D}}_k^{\mathbf{Q}}.$$

$$\overline{r}\triangleq \frac{{\sum }_{k=1}^K|{\mathcal{M}}_k|}{\mathsf{N}}.$$

$$\overline{L}=\mathbf{E}\left[\frac{{\sum }_{k\in \mathbf{Q}}|X_k^{\mathbf{Q}}|}{\mathsf{NDV}}\right],$$

$$L_{\mathsf{K},\mathsf{Q}}^{\ast }(r)\triangleq \inf \left\{L:(r,L)\text{is achievable}\right\}.$$

$$\overline{r}$$

$$\displaystyle\left[\begin{array}[]{cc}s&*\\ &s\end{array}\right]~{}\textrm{or}~{}\left[\begin{array}[]{cc}*&s\\ s&*\end{array}\right].$$

$$\displaystyle\mathbf{A}=\left[\begin{array}[]{cccc}*&*&1&2\\ &1&*&3\\ &2&3&*\\ 1&*&*&4\\ 2&*&4&*\\ 3&4&*&*\end{array}\right].$$

$${\theta }_t\triangleq \frac{{\mathsf{S}}_{t}t}{\mathsf{KF}-\mathsf{T}},t\in [\mathsf{K}].$$

$$\displaystyle{\mathbf{A}}^{\{1,2,4\}}=\left[\begin{array}[]{ccc}*&*&2\\ &1&3\\ &2&*\\ 1&*&4\\ 2&*&*\\ 3&4&*\end{array}\right].$$

$$\displaystyle\begin{array}[]{c|ccc}\hline\cr\hline\cr&1&2&4\\ \hline\cr w_{1}&*&*&2\\ w_{3}&*&2&*\\ w_{5}&2&*&*\\ \hline\cr\hline\cr\end{array}$$

$$\displaystyle\begin{array}[]{c|c|c|c|c}\hline\cr\hline\cr\textnormal{Symbol}&1&2&3&4\\ \hline\cr\textnormal{Node }1&v_{2,2}&v_{2,3}^{1}\oplus v_{3,1}^{1}&v_{3,2}&\\ \hline\cr\textnormal{Node }2&v_{1,4}&v_{3,1}^{2}\oplus v_{1,5}^{2}&&v_{3,4}\\ \hline\cr\textnormal{Node }3&\lx@intercol\hfil\textnormal{(straggling)}\hfil\lx@intercol\\ \hline\cr\textnormal{Node }4&&v_{1,5}^{4}\oplus v_{2,3}^{4}&v_{1,6}&v_{2,6}\\ \hline\cr\hline\cr\end{array}$$

$$r_{\mathbf{A}}$$

$$L_{\mathbf{A}}$$

$$r_{\mathbf{A}}$$

$$L_{\mathbf{A}}$$

$${\theta }_g=1,\text{and}{\theta }_t=0,\forall t\in [\mathsf{K}]\backslash \{g\}.$$

$$\displaystyle p_{j,k}\triangleq\left\{\begin{array}[]{ll}*,&\textnormal{if}~{}k\in\mathcal{T}_{j}\\ \kappa(\{k\}\cup\mathcal{T}_{j}),&\textnormal{if}~{}k\notin\mathcal{T}_{j}\end{array}\right..$$

$$L_{{\mathbf{P}}_r}\triangleq \left(1-\frac{r}{\mathsf{K}}\right)\cdot \sum _{l=r+\mathsf{Q}-\mathsf{K}}^{\min \{r,\mathsf{Q}-1\}}\frac{1}{l}\cdot \frac{C_r^l\cdot C_{\mathsf{K}-r-1}^{\mathsf{Q}-l-1}}{C_{\mathsf{K}-1}^{\mathsf{Q}-1}}.$$

$$L_{\mathsf{K},\mathsf{Q}}^{\ast }(r)=\left(1-\frac{r}{\mathsf{K}}\right)\cdot \sum _{l=r+\mathsf{Q}-\mathsf{K}}^{\min \{r,\mathsf{Q}-1\}}\frac{1}{l}\cdot \frac{C_r^l\cdot C_{\mathsf{K}-r-1}^{\mathsf{Q}-l-1}}{C_{\mathsf{K}-1}^{\mathsf{Q}-1}},r\in [\mathsf{K}-\mathsf{Q}+1:\mathsf{K}],$$

$$L_{\mathsf{K},\mathsf{Q}}(r)=\left(1-\frac{r}{\mathsf{K}}\right)\cdot \left(\frac{C_{\mathsf{K}-r}^{\mathsf{Q}-1}}{C_{\mathsf{K}-1}^{\mathsf{Q}-1}}+\sum _{l=\max \{1,r+\mathsf{Q}-\mathsf{K}-1\}}^{\min \{r,\mathsf{Q}\}-1}\frac{1}{l}\cdot \frac{C_{r-1}^l{C}_{\mathsf{K}-r}^{\mathsf{Q}-l-1}}{C_{\mathsf{K}-1}^{\mathsf{Q}-1}}\right),$$

Full text

I Introduction

Distributed computing has emerged as one of the most important paradigms to speed up large-scale data analysis tasks. One of the most popular programming models is MapReduce [2] which has been used to parallelize computations across distributed computing nodes, e.g., for machine learning tools [3, 4].

Consider the task of computing $\mathsf{D}$ output functions from $\mathsf{N}$ files through $\mathsf{K}$ nodes. With MapReduce, each output function $\phi_{d}$, for $1\leq d\leq\mathsf{D}$, can be decomposed into

• •

$\mathsf{N}$ map functions $f_{d,1},\ldots,f_{d,\mathsf{N}}$, each depending on exactly one different file; and

• •

a reduce function $h_{d}$ that combines the outputs of the $\mathsf{N}$ map functions.

Each node $k$ is responsible for computing a subset of $\mathsf{\frac{D}{K}}$ output functions through three phases. In the first map phase, a central server stores a subset of files $\mathcal{M}_{k}$ at node $k$, for each $k\in[\mathsf{K}]$. Each node $k$ then computes all the $\mathsf{D}$ intermediate values (IVAs) $f_{1,n}(w_{n}),\ldots,f_{\mathsf{D},n}(w_{n})$ corresponding to each of its stored files $w_{n}\in\mathcal{M}_{k}$. In the subsequent shuffle phase, it creates a signal from its computed IVAs and sends the signal to all the other nodes. Based on the received exchanged signals and the locally computed IVAs, in the final reduce phase it reconstructs all the IVAs pertaining to its own output functions and calculates the desired outputs.

Recently, Li et al. proposed a scheme named coded distributed computing (CDC) to reduce the communication load for data shuffling between the map and reduce phases [5]. The idea is to create multicast opportunities by duplicating the files and computing the corresponding map functions at different nodes. It is shown that the CDC scheme achieves the fundamental storage-communication tradeoff, i.e., it has the lowest communication load for a given storage constraint. This result has been extended in various directions. For example, [8, 6, 7] account also for the computation load during the map phase; [9] studies the computation resource-allocation problem; [10, 11, 12, 13] consider wireless (noisy) networks between computation nodes; [14] considers a model where during the shuffle phase each node broadcasts only to a random subset of the nodes.

In this paper, we consider a setup where during the map phase each node takes a random amount of time to compute its desired map functions[15]. In this case, instead of waiting for all the nodes to finish the assigned computations, which can cause an intolerable delay, data shuffling starts as soon as any set of $\mathsf{Q}$ nodes, $\mathsf{Q}\in[\mathsf{K}]$, terminate their map procedures. The set of $\mathsf{Q}$ nodes that first terminate the map procedure are called active nodes, while the remaining $\mathsf{K-Q}$ nodes are called straggling nodes or stragglers. Note that the stragglers are not identified prior to the beginning of the map phase and the map phase has to be designed without such knowledge.

Distributed computing systems with straggling nodes have mainly been studied in the context of a server-worker framework. In this framework, a central server distributes the raw data to the workers like in the above described map phase, but following this map phase the workers directly communicate their intermediate results to the server, which then produces the final outputs. (Thus, under the server-worker framework, all final outputs are calculated at the server and not at the distributed computing nodes as is the case in MapReduce systems.) Under the server-worker framework, distributed computing systems with straggling nodes have for example been studied in [15, 24, 16, 17, 18, 19, 20, 22, 21, 23, 25], which focused on high-dimensional matrix-by-matrix or matrix-by-vector multiplications, and in [26, 27, 28, 29, 30], which proposed codes for gradient computing.

Fewer works studied MapReduce systems with straggling nodes (hereafter referred to as straggling systems) which are more relevant for the present article. Specifically, Li et al. [31] proposed to incorporate MDS codes into the CDC scheme to cope with straggler nodes. Their construction however works only when the map functions accomplish matrix-by-vector multiplications. Improved constructions were proposed by Zhang et al. by choosing the parameters of MDS code and CDC scheme separately in a more flexible way [32], but also their techniques are applicable only for map functions that are matrix-by-matrix multiplications. In many practical applications such as computations in neural networks and machine learning, the map functions are non-linear and can be very complicated with little structure. This motivates us to investigate the MapReduce framework with straggling nodes for general map and reduce functions. In particular, we will present universal coded computing schemes that can be applied to arbitrary straggling systems, irrespective of the specific map and reduce functions. Moreover, we will show the optimality of our schemes among the class of universal schemes that do not rely on special properties of the map and reduce functions.

More specifically, in this work, we first propose a systematic construction of universal coded computing schemes for straggling systems from any placement delivery array for distributed computing (Comp-PDA) [8]. A placement delivery array (PDA) is an array whose entries are either a special symbol $``*"$ or some integer numbers called ordinary symbols. It was introduced in [33] to represent in a single array both the placement and the delivery of coded caching schemes with uncoded prefetching. In particular, the coded caching schemes proposed by Maddah-Ali and Niesen in [34] can be represented as PDAs [33]. The corresponding PDAs will be referred to as MAN-PDAs, and, as we will see, they play a fundamental role also in coded computing with stragglers. PDAs have further been generalized to other coded caching scenarios such as device-to-device models [35], combination networks [36], networks with private demands [37], and medical data sharing problems [38]. Moreover, several different PDA constructions have been proposed in [39, 40, 42, 41, 43]. In this paper our focus is on a subclass of PDAs, called Comp-PDAs, which were introduced in [8] to describe coded computing schemes for MapReduce systems without straggling nodes. In this paper, we show that Comp-PDAs can also be used to construct coded computing schemes for straggling systems, and we express the storage and computation loads of the obtained schemes in terms of the Comp-PDA parameters.

We then proceed to characterize the fundamental storage-communication (SC) tradeoff for straggling systems by showing that the SC tradeoff curve achieved by coded computing schemes obtained from the MAN-PDAs matches a new information-theoretic converse for universal computing schemes with stragglers. That means, our converse bounds the SC tradeoffs achieved by coded computing schemes that apply to arbitrary map and reduce functions. For special map and reduce functions, e.g., linear functions, it is possible to find tailored coded computing schemes that achieve better SC tradeoffs, than the one implied by our information-theoretic converse, see e.g., [32]. It is worth pointing out that the MAN-PDA based coded computing schemes adopt a fixed storage strategy irrespective of the active set size $\mathsf{Q}$. This implies that the fundamental SC-tradeoff curve remains unchanged even if the active set size $\mathsf{Q}$ has not yet been determined during the map phase. The proposed schemes are thus optimal also in a scenario where the system imposes a strict time constraint proceeds to the shuffle and reduce phases with the random number of nodes that have by then terminated their IVA calculations.

In a final part of the manuscript, we study the complexity of optimal (or near-optimal) coded computing schemes. In fact, a major practical limitation of the SC-optimal coded computing schemes based on MAN-PDAs is that they can only be implemented if the number of files $N$ in the system grows exponentially with the number of nodes $\mathsf{K}$. However, as we show in this paper, in most cases, MAN-PDAs achieve their corresponding fundamental SC pairs with smallest possible number of files, i.e., with smallest file complexity, among all Comp-PDA based coded computing schemes. The mentioned practical limitation is thus not a weakness of the MAN-PDAs, but seems inherent to PDA-based coded computing schemes for stragglers. Interestingly, the problem can be circumvented by slightly backing off from the SC-optimal tradeoff curve. We show that the coded computing schemes corresponding to some of the Comp-PDAs in [33] achieve SC pairs close to the fundamental SC tradeoff curve but with significantly smaller number of files than the optimal MAN-PDAs. More precisely, we fix an integer $q$, let the number of nodes $\mathsf{K}$ be a multiple of $q$, and the storage load $r$ be such that $\frac{r}{\mathsf{K}}\in\{\frac{1}{q},\frac{q-1}{q}\}$ holds. We compare the Comp-PDA in [33] and the MAN-PDA for such pairs $(\mathsf{K},r)$ while we let both of them tend to infinity proportionally. This comparison shows that the ratio of the minimum required number of files of the Comp-PDA in [33] and the MAN-PDA vanishes as $O\left(e^{\mathsf{K}(1-\frac{1}{q})\ln\frac{q}{q-1}}\right)$, while the ratio of their communication loads approaches $1$.

We summarize the contributions of this paper:

We establish a general framework for constructing universal coded computing schemes for straggling systems from Comp-PDAs, and evaluate their SC pairs in terms of the Comp-PDA parameters. 2. 2.

We derive the fundamental SC tradeoff for any universal straggling system by means of an information theoretic converse that matches the SC pairs achieved by coded computing schemes with stragglers based on the MAN-PDAs. 3. 3.

We prove that, while in most cases points on the fundamental SC tradeoff curve can be achieved only with the same file complexity as MAN-PDA based schemes, points close to the fundamental SC tradeoff curve can be achieved with significantly smaller file complexities.

The remainder of this paper is organized as follows. Section II formally describes our model, and Section III reviews the definitions of PDAs and Comp-PDAs; Section IV presents the main results of this paper; Sections V to VII present the major proofs of our results; and Section VIII concludes this paper.

Notations

For positive integers $n,k$ such that $n\geq k$, we use the notations $[n]\triangleq\{1,2,\ldots,n\}$, and $[k:n]\triangleq\{k,k+1,\ldots,n\}$. The binomial coefficient is denoted by $C_{n}^{k}\triangleq\frac{n!}{k!(n-k)!}$ for $n\geq k\geq 0$; we set $C_{n}^{k}=0$ when $k<0$ or $k>n$. For $k\leq n$, we use $\mathbf{\Omega}_{n}^{k}$ to denote the collection of all subsets of $[n]$ of cardinality $k$, i.e., $\mathbf{\Omega}_{n}^{k}\triangleq\{\mathcal{T}\subseteq[n]:|\mathcal{T}|=k\}$. The binary field is denoted by $\mathbb{F}_{2}$ and the $n$ dimensional vector space over $\mathbb{F}_{2}$ is denoted by $\mathbb{F}_{2}^{n}$. We use $|\mathcal{A}|$ to denote the cardinality of the set $\mathcal{A}$, while for a signal $X$, $|X|$ is the number of bits in $X$. The order of set operations is from left to right. Finally, $\mathbbm{1}(\cdot)$ denotes the indicator function that evaluates to $1$ if the statement in the parenthesis is true and it evaluates to [math] otherwise.

II System Model

A $(\mathsf{K,Q})$ straggling system is parameterized by the positive integers

[TABLE]

as described in the following. The system aims to compute $\mathsf{D}$ output functions $\phi_{1},\ldots,\phi_{\mathsf{D}}$ through $\mathsf{K}$ distributed computing nodes from $\mathsf{N}$ files. Each output function $\phi_{d}:\mathbb{F}_{2}^{\mathsf{NW}}\rightarrow\mathbb{F}_{2}^{\mathsf{U}}$ ($d\in[\mathsf{D}]$) takes as inputs the length $\mathsf{W}$ files in the library $\mathcal{W}=\{w_{1}\ldots,w_{\mathsf{N}}\}$, and outputs a bit stream of length $\mathsf{U}$, i.e.,

[TABLE]

Assume that the computation of the output functions $\phi_{d}$ can be decomposed as:

[TABLE]

where

• •

the map function $f_{d,n}:\mathbb{F}_{2}^{\mathsf{W}}\rightarrow\mathbb{F}_{2}^{\mathsf{V}}$ maps the file $w_{n}$ into a binary stream of length $\mathsf{V}$, called intermediate value (IVA), i.e.,

[TABLE]

• •

the reduce function $h_{d}:\mathbb{F}_{2}^{\mathsf{NV}}\rightarrow\mathbb{F}_{2}^{\mathsf{U}}$, maps the IVAs $\{v_{d,n}\}_{n=1}^{\mathsf{N}}$ into the output stream

[TABLE]

Notice that a decomposition into map and reduce functions is always possible. In fact, trivially, one can set the map and reduce functions to be the identity and output functions respectively, i.e., $f_{d,n}(w_{n})=w_{n},$ and $h_{d}=\phi_{d},~{}\forall~{}n\in[\mathsf{N}],~{}d\in[\mathsf{D}]$, in which case $\mathsf{V}=\mathsf{W}$. However, to mitigate the communication cost during the shuffle phase, one would prefer a decomposition such that the length of the IVAs is as small as possible while still allowing the nodes to compute the final outputs. The computation is carried out through three phases, namely, the map, shuffle, and reduce phases.

Map Phase: Each node $k\in[\mathsf{K}]$ stores a subset of files $\mathcal{M}_{k}\subseteq\mathcal{W}$, and tries to compute all the IVAs from the files in $\mathcal{M}_{k}$, denoted by $\mathcal{C}_{k}$:

[TABLE]

Each node has a random amount of time to compute its corresponding IVAs. To limit latency of the system, the coded computing scheme proceeds with the shuffle and reduce phases as soon as a fixed number of $\mathsf{Q}\in[\mathsf{K}]$ nodes have terminated the map computations. These nodes are called active nodes, and the set of all active nodes is called active set, whereas the other $\mathsf{K}-\mathsf{Q}$ nodes are called straggling nodes. For simplicity, we consider the symmetric case in which each subset $\mathcal{Q}\subseteq[\mathsf{K}]$ of size $|\mathcal{Q}|=\mathsf{Q}$ is active with same probability. Let the random variable $\mathbf{Q}$ denote the random active set. Then,

[TABLE]

In our model, we also assume that the map phase has been designed in a way that all the files can be recovered111In this paper, we thus exclude the “outage” event in which some active sets cannot compute the given function due to missing files. from any active set of size $\mathsf{Q}$. Hence, for any file $w_{n}\in\mathcal{W}$, the number of nodes storing this file $t_{n}$ must satisfy

[TABLE]

The output functions $\phi_{1},\ldots,\phi_{\mathsf{D}}$ are then uniformly assigned222Here we assume for simplicity that $\mathsf{Q}$ divides $\mathsf{D}$. Note that otherwise we can always add empty functions for the assumption to hold. to the nodes in $\mathbf{Q}$. Let $\mathcal{D}_{k}^{\mathbf{Q}}$ be the set of indices of output functions assigned to a given node $k\in\mathbf{Q}$. Thus, \mathbf{\Gamma}^{\mathbf{Q}}\triangleq\big{\{}\mathcal{D}_{k}^{\mathbf{Q}}\big{\}}_{k\in\mathbf{Q}} forms a partition of $[\mathsf{D}]$, and each set $\mathcal{D}_{k}^{\mathbf{Q}}$ is of cardinality $\frac{\mathsf{D}}{\mathsf{Q}}$. Denote the set of all the partitions of $[\mathsf{D}]$ into $\mathsf{Q}$ equal-sized subsets by $\mathbf{\Delta}$. 2. 2.

Shuffle Phase: The nodes in $\mathbf{Q}$ proceed to exchange their computed IVAs. Each node $k\in\mathbf{Q}$ multicasts a signal

[TABLE]

to all the other nodes in $\mathbf{Q}$. For each $k\in\mathbf{Q}$, here $\varphi_{k}^{{\mathbf{Q}}}\;\colon\;\mathbb{F}_{2}^{|\mathcal{C}_{k}|\mathsf{V}}\times\mathbf{\Delta}\rightarrow\mathbb{F}_{2}^{|X_{k}^{\mathbf{Q}}|}$ denotes the encoding function of node $k$.

We assume a perfect multicast channel, i.e., each active node $k\in\mathbf{Q}$ receives perfectly all the transmitted signals

[TABLE] 3. 3.

Reduce Phase: Using the received signals $X^{{\mathbf{Q}}}$ from the shuffle phase and the local IVAs $\mathcal{C}_{k}$ computed in the map phase, node $k$ has to be able to compute all the IVAs

[TABLE]

where $\psi^{{\mathbf{Q}}}_{k}:\mathbb{F}_{2}^{\sum_{k\in\mathbf{Q}}|X_{k}^{{\mathbf{Q}}}|}\times\mathbb{F}_{2}^{|\mathcal{C}_{k}|\mathsf{V}}\times\mathbf{\Delta}\rightarrow\mathbb{F}_{2}^{\frac{\mathsf{NDV}}{\mathsf{Q}}}$. Finally, with the restored IVAs, it computes each assigned function via the reduce function, namely,

[TABLE]

To measure the storage and communication costs, we introduce the following definitions.

Definition 1** (Storage Load).**

Storage load* $\overline{r}$ is defined as the total number of files stored across the $\mathsf{K}$ nodes normalized by the total number of files $\mathsf{N}$, i.e.,*

[TABLE]

Definition 2** (Communication Load).**

Communication load* $\overline{L}$ is defined as the average total number of bits sent in the shuffle phase, normalized by the total number of bits of all intermediate values, i.e.,*

[TABLE]

where the expectation is taken over the random active set $\mathbf{Q}$.

Definition 3** (Storage-Communication (SC) Tradeoff).**

A pair of real numbers $(r,L)$ is achievable if for any $\epsilon>0$, there exist positive integers $\mathsf{N,D,U,V,W}$, a storage design $\{\mathcal{M}_{k}\}_{k=1}^{\mathsf{K}}$ of storage load less than $r+\epsilon$, a set of uniform assignments of output functions $\left\{\mathbf{\Gamma}^{\mathcal{Q}}\right\}_{\mathcal{Q}\in\mathbf{\Omega}_{\mathsf{K}}^{\mathsf{Q}}}$, and a collection of encoding functions \big{\{}\big{\{}\varphi^{\mathcal{Q}}_{k}\big{\}}_{k\in\mathcal{Q}}\big{\}}_{\mathcal{Q}\in\mathbf{\Omega}_{\mathsf{K}}^{\mathsf{Q}}} with communication load less than $L+\epsilon$, such that all the output functions $\phi_{1},\ldots,\phi_{\mathsf{D}}$ can be computed successfully. For a fixed $\mathsf{Q}\in[\mathsf{K}]$, we define the fundamental storage-communication (SC) tradeoff as

[TABLE]

Note that the non-trivial interval for the values of $r$ is $[\mathsf{K}-\mathsf{Q}+1,\mathsf{K}]$. Indeed, if $r>\mathsf{K}$, then each node can store all the files and compute any function locally. On the other hand, from the assumption (2), we have for any feasible scheme

[TABLE]

Therefore, throughout the paper, we only focus on the interval $r\in[\mathsf{K}-\mathsf{Q}+1,\mathsf{K}]$ for any given $\mathsf{Q}\in[\mathsf{K}]$. Further, for a given storage design $\{\mathcal{M}_{k}\}_{k=1}^{\mathsf{K}}$, by the symmetry assumption of the reduce functions and the fact that each node has all the IVAs of all $\mathsf{D}$ output functions in the files it has stored, the optimal communication load is independent of the reduce function assignment. This is similar to the case without straggling nodes (see [5, Remark 3]).

Definition 4** (File Complexity).**

The smallest number of files $\mathsf{N}$ required to implement a given scheme is called the file complexity of this scheme.

In above problem definition, the various nodes store entire files during the map phase and during the shuffle phase they reconstruct all the IVAs corresponding to their output functions. This system definition does not allow to reduce storage or communication loads by exploiting special structures of the map or reduce functions as proposed for example in [31, 32]. As a consequence, all the coded computing schemes presented in this paper universally apply to arbitrary map and reduce functions and the SC tradeoff in Definition 3 applies only to such universal schemes. In fact, as we will explain, for linear reduce functions the SC tradeoff derived in [32] improves over the one in Definition 3, since it was derived for a system where nodes do not have to store individual files and reconstruct all the required IVAs, but linear combinations of them suffice.

III Placement Delivery Arrays for Straggling

Systems

III-A Definitions

Placement delivery arrays (PDA) introduced in [33] are the main tool of this paper. To adapt to our setup, we use the following definition from [8].

Definition 5** (Placement Delivery Array (PDA)).**

For positive integers $\mathsf{K,F,T}$ and a nonnegative integer $\mathsf{S}$, an $\mathsf{F}\times\mathsf{K}$ array $\mathbf{A}=[a_{j,k}]$, $j\in[\mathsf{F}],k\in[\mathsf{K}]$, composed of $\mathsf{T}$ special symbols $``*"$ and some ordinary symbols $1,\ldots,\mathsf{S}$, each occurring at least once, is called a $(\mathsf{K,F,T,S})$ PDA, if for any two distinct entries $a_{j_{1},k_{1}}$ and $a_{j_{2},k_{2}}$, we have $a_{j_{1},k_{1}}=a_{j_{2},k_{2}}=s$ with $s$ an ordinary symbol only if

1. a)

$j_{1}\neq j_{2}$, $k_{1}\neq k_{2}$, i.e., they lie in distinct rows and distinct columns; and 2. b)

$a_{j_{1},k_{2}}=a_{j_{2},k_{1}}=*$,

i.e., the corresponding $2\times 2$ subarray formed by rows $j_{1},j_{2}$ and columns $k_{1},k_{2}$ must be of the following form

[TABLE]

*A PDA with all “$*$” entries is called trivial. Notice that in this case $\mathsf{S}=0$ and $\mathsf{KF}=\mathsf{T}$. A PDA is called a $g$-regular PDA if each ordinary symbol occurs exactly $g$ times. *

Example 1*.*

The following array is a $3$-regular $(4,6,12,4)$ PDA.

[TABLE]

For our purpose, we introduce the following definitions similarly to the ones in [8].

Definition 6** (PDA for Distributed Computing (Comp-PDA)).**

A Comp-PDA is a PDA with at least one $``*"$-symbol in each row.

Definition 7** (Minimum Storage Number).**

Given a Comp-PDA $\mathbf{A}$, its minimum storage number $\tau$ is defined as the minimum number of $``*"$-symbols in any of the rows of $\mathbf{A}$.

Definition 8** (Symbol Frequencies).**

For a given nontrivial $(\mathsf{K},\mathsf{F},\mathsf{T},\mathsf{S})$ Comp-PDA, let $\mathsf{S}_{t}$ denote the number of ordinary symbols that occur exactly $t$ times, for $t\in[\mathsf{K}]$. The symbol frequencies $\theta_{1},\theta_{2},\ldots,\theta_{\mathsf{K}}$ of the Comp-PDA are then defined as

[TABLE]

They indicate the fractions of ordinary entries of the Comp-PDA that occur exactly $1,2,\ldots,\mathsf{K}$ times, respectively. For completeness, we also define $\theta_{t}\triangleq 0$ for $t>K$.

III-B Constructing a Coded Computing Scheme from a Comp-PDA: A Toy Example

In this subsection we illustrate the connection between Comp-PDAs and coded computing schemes with stragglers at hand of a toy example. Section V ahead describes a general procedure to obtain a coded computing scheme with stragglers from any Comp-PDA, and it presents a performance analysis for the obtained scheme.

Consider the $(4,6,12,4)$ Comp-PDA $\mathbf{A}$ in Example 1, and assume a $(\mathsf{K},\mathsf{Q})=(4,3)$ straggling system with $\mathsf{N}=6$ files and $\mathsf{D}=3$ output functions. The scheme is illustrated in Fig. 1 for the case that node $3$ is straggling. In this Fig. 1, the line “files” in each of the four boxes indicates the files stored at the nodes. The remaining lines in the boxes illustrate the computed IVAs, where red circles, green triangles, and blue squares depict IVAs pertaining to output functions $\phi_{1}$, $\phi_{2}$, and $\phi_{3}$, respectively. More specifically, a red circle with the number $i$ in the middle stands for IVA $v_{i,1}$, and so on. The lines below the boxes of the active nodes 1, 2, and 4 indicate the IVAs that the nodes have to learn to be able to compute their output functions. In this example it is assumed that node 1 computes $\phi_{1}$, node 2 computes $\phi_{2}$, and node $4$ computes $\phi_{3}$. The signals on the left/right side of the boxes indicate the signals sent by the nodes. Here, splitting of IVAs indicates that the IVA is decomposed into a substring consisting of the first half of the bits and a substring consisting of the second half of the bits, and the plus symbol stands for a bit-wise XOR-operation on the substrings.

We now explain the distributed coding scheme associated with the PDA ${\mathbf{A}}$ more formally. We start by associating column $k$ of ${\mathbf{A}}$ with node $k$ in the system, ($k\in[4]$), and row $j$ of ${\mathbf{A}}$ with file $w_{j}$ in the system, ($j\in[6]$). In the map phase, node $k$ stores file $w_{j}$ if the row-$k$ and column-$j$ entry of ${\mathbf{A}}$ is a $``*"$-symbol. For example, node $1$, which is associated with the first column of the Comp-PDA, stores files $w_{1},w_{2}$ and $w_{3}$. Each node then computes all the IVAs corresponding to the files it has stored.

In our example, we assume that node $3$ is the only straggler. Nodes 1, 2, and 4 thus form the active set and as such continue with the shuffle and reduce procedures. Accordingly, we extract from the PDA ${\mathbf{A}}$ the subarray ${\mathbf{A}}^{\{1,2,4\}}$ consisting of columns $1,2$ and $4$ (the columns corresponding to the active set):

[TABLE]

Notice that ${\mathbf{A}}^{\{1,2,4\}}$ is also a Comp-PDA (in particular it has at least one “$*$” symbol in each row) and the node corresponding to a given column has stored all the files indicated by the $``*"$-symbols in this column. The same statement applies also to the subarrays associated with any other possible active set of size $3$. After the shuffling phase, we are thus in the same situation as described in [8, 44] when a coded computing scheme without stragglers is to be constructed from a Comp-PDA, and as a consequence, the same shuffle and reduce procedures can be applied. We described these procedures here in detail for completeness.

The shuffle phase is as follows. For each $s\in\{1,2,3,4\}$ occuring $g$ times ($g=2$ or $3$), pick out the $g\times g$ array containing $s$. For example, symbol $s=2$ is associated with the following 3-by-3 subarray:

[TABLE]

Each occurence of the symbol $``2"$ in this subarray stands for an IVA desired by the node in the corresponding column and computed at the other nodes in this subarray. The row of the symbol indicates to which file this IVA pertains, and the $``*"$ symbols in this row indicate that the IVA can indeed be computed by all nodes in the active set except for the one corresponding to the column of the $``2"$ symbol. In the above example, the three $``2"$ symbols from top to down represent the IVAs $v_{3,1}$, $v_{2,3}$, and $v_{1,5}$, respectively. These IVAs are shuffled in a coded manner. To this end, they are first split into $g-1=2$ equally-large sub-IVAs, and each of these sub-IVAs is labeled by one of the nodes where the IVA has been computed (i.e,. by the columns with $``*"$ symbols). In our example, we split $v_{3,1}=(v_{3,1}^{1},v_{3,1}^{2})$, $v_{2,3}=(v_{2,3}^{1},v_{2,3}^{4})$ and $v_{1,5}=(v_{1,5}^{2},v_{1,5}^{4})$. The signal sent by a given node $i$ is then simply the componentwise XOR of the sub-IVAs with superscript $i$. So, nodes $1$, $2$, $4$ send $v_{2,3}^{1}\oplus v_{3,1}^{1}$, $v_{3,1}^{2}\oplus v_{1,5}^{2}$ and $v_{1,5}^{4}\oplus v_{2,3}^{4}$, respectively. The same procedure is applied for all other ordinary symbols $1$, $3$, and $4$ in subarray ${\mathbf{A}}^{\{1,2,4\}}$. The following table lists all the signals sent at the 4 nodes, where the first line lists their associated ordinary symbols:

[TABLE]

We now explain how the nodes extract their missing IVAs from the shuffled signals. Since node $1$ has computed $v_{1,1},v_{1,2}$ and $v_{1,3}$ in the map phase, it still needs to decode $v_{1,4},v_{1,5},v_{1,6}$. It directly obtains the IVAs $v_{1,4}$ and $v_{1,6}$ from the uncoded signals sent by nodes $2$ and $4$ respectively. Moreover, it reconstructs the two sub-IVAs $v_{1,5}^{2}$ and $v_{1,5}^{4}$, by XORing the signals $v_{3,1}^{2}\oplus v_{1,5}^{2}$ and $v_{1,5}^{4}\oplus v_{2,3}^{4}$ shuffled by nodes $2$ and $4$ with its locally stored sub-IVAs $v_{3,1}^{2}$ and $v_{2,3}^{4}$. Nodes $2$ and $4$ reconstruct their missing IVAs in a similar way.

The total number of stored files at the nodes is $3\times 4=12$, thus the storage load is $r=\frac{3\times K}{N}=2$. The total length of the transmitted signals is $7.5\mathsf{V}$, which remains unchanged also when any of the other nodes straggles. The communication load is thus $L=\frac{7.5\mathsf{V}}{6\times 3\times\mathsf{V}}=\frac{5}{12}$.

IV Main Results

In this section, we present our main results. Details and proofs are deferred to Sections V–VII.

IV-A *Coded Computing Schemes for Straggling Systems from Comp-PDAs *

In Section V, we propose a coded computing scheme for a $(\mathsf{K},\mathsf{Q})$ straggling system based on any Comp-PDA with $\mathsf{K}$ columns and minimum storage number $\tau\geq\mathsf{K}-\mathsf{Q}+1$. Theorem 1 is proved by analyzing the coded computing scheme, which is deferred to Section V-B.

Theorem 1**.**

From any given $\mathsf{(K,F,T,S)}$ Comp-PDA $\mathbf{A}$ with symbol frequencies $\{\theta_{t}\}_{t=1}^{\mathsf{K}}$ and minimum storage number $\tau\in[\mathsf{K-Q+1}:\mathsf{K}]$, one can construct a coded computing scheme for a $(\mathsf{K},\mathsf{Q})$ straggling system achieving the SC pair

[TABLE]

with file complexity $\mathsf{F}$.

Theorem 1 characterizes the performance of the coded computing scheme obtained from a Comp-PDA as described in Section V in terms of the Comp-PDA parameters. In the following, we will simply say that a Comp-PDA achieves this performance.

Notice that the file complexity of any Comp-PDA based scheme coincides with the number of rows $\mathsf{F}$ of the Comp-PDA. We shall therefore call the parameter $\mathsf{F}$ of a Comp-PDA its file complexity.

As we show in the following, Theorem 1 can be simplified for regular Comp-PDAs.

Corollary 1**.**

From any given $g$-regular $\mathsf{(K,F,T,S)}$ Comp-PDA $\mathbf{A}$, with $g\in[\mathsf{K}]$ and minimum storage number $\tau\in[\mathsf{K-Q+1}:\mathsf{K}]$, one can construct a coded computing scheme for a $(\mathsf{K,Q})$ straggling system achieving thre SC pair

[TABLE]

with file complexity $\mathsf{F}$.

Proof:

From Theorem 1, we only need to evaluate $L_{\mathbf{A}}$ when $\mathbf{A}$ is a $g$-$\mathsf{(K,F,T,S)}$ Comp-PDA. In this case, all the $\mathsf{S}$ symbols occur $g$ times, i.e.,

[TABLE]

Then the conclusion directly follows from Theorem 1. ∎

Corollary 1 is of particular interest since there are several explicit regular PDA constructions for coded caching in the literature, such as [33, 42, 43], which are also Comp-PDAs. In particular, the following PDAs obtained from the coded caching scheme proposed by Maddah-Ali and Niesen [34] are important.

Definition 9** (Maddah-Ali Niesen PDA (MAN-PDA)).**

Fix any integer $i\in[\mathsf{K}]$, and let $\{\mathcal{T}_{j}\}_{j=1}^{C_{\mathsf{K}}^{i}}$ denote all subsets of $[\mathsf{K}]$ of size $i$. Also, choose an arbitrary bijective function $\kappa$ from the collection of all subsets of $[\mathsf{K}]$ with cardinality $i+1$ to the set \big{[}C_{\mathsf{K}}^{i+1}\big{]}. Then, define the array $\mathbf{P}_{i}=[p_{j,k}]$ as

[TABLE]

We observe that for any $i\in[\mathsf{K}-1]$, the array $\mathbf{P}_{i}$ is an $(i+1)$-regular \big{(}\mathsf{K},C_{\mathsf{K}}^{i},\mathsf{K}{C_{\mathsf{K}-1}^{i-1}},{C_{\mathsf{K}}^{i+1}}\big{)} Comp-PDA (see [33] for details). For $i=\mathsf{K}$, the Comp-PDA $\mathbf{P}_{i}$ consists only of “$*$”-entries and is thus a trivial PDA. By Corollary 1, we directly obtain the following result.

Corollary 2**.**

For a $(\mathsf{K},\mathsf{Q})$ straggling system, and each $r$ in the discrete set $[\mathsf{K}-\mathsf{Q}+1:\mathsf{K}]$, the MAN-PDA $\mathbf{P}_{r}$ achieves the SC pair $(r,L_{\mathbf{P}_{r}})$, where

[TABLE]

The coded computing scheme associated to $\mathbf{P}_{r}$ is equivalent to our proposed coded computing for straggling systems (CCS) in [1]. Here, we present it as a special case of the more general Comp-PDA framework. As we shall see, the Comp-PDA framework allows us to design new coded computing schemes with lower file complexity.

IV-B The Fundamental Storage-Communication Tradeoff

We are ready to present our result on the fundamental SC tradeoff, which is proved in Section VI.

Theorem 2**.**

For a $(\mathsf{K,Q})$ straggling system, with a given integer storage load $r$ in the discrete set $[\mathsf{K}-\mathsf{Q}+1:\mathsf{K}]$, the fundamental SC tradeoff is

[TABLE]

which is achievable with a scheme of file complexity $C_{\mathsf{K}}^{r}$. For a general $r$ in the interval $[\mathsf{K}-\mathsf{Q}+1,\mathsf{K}]$, the fundamental SC tradeoff $L_{\mathsf{K},\mathsf{Q}}^{*}(r)$ is given by the lower convex envelope formed by the above points in (38).

Fig. 2 shows the fundamental SC tradeoff curves for $\mathsf{K}=10$ and different values of $\mathsf{Q}$. When $\mathsf{Q}=1$, the curve reduces to a single point $(\mathsf{K},0)$, while when $\mathsf{Q}=\mathsf{K}$, the curve corresponds to the fundamental tradeoff without straggling nodes (cf. [5, Fig. 1]). In this latter case without stragglers, the fundamental SC tradeoff curve is achieved by the CDC scheme in [5]. For a general value of $\mathsf{Q}$ and integer storage $r\in[\mathsf{K}-\mathsf{Q}+1:\mathsf{K}]$, the fundamental SC tradeoff pair $(r,L_{\mathsf{K},\mathsf{Q}}^{*}(r))$ is achieved by the MAN-PDA $\mathbf{P}_{r}$, see Corollary 2. This implies that for a fixed integer storage load $r\in[1:\mathsf{K}]$, the SC pairs $\{(r,L_{\mathsf{K},\mathsf{Q}}^{*}(r))\}_{\mathsf{Q}=\mathsf{K}-r+1}^{K}$ are all achieved by the same PDA $\mathbf{P}_{r}$, irrespective of the size of the active set $\mathsf{Q}$. As we show in Section V-A, the map procedures of the coded computing scheme corresponding to a given Comp-PDA at a given node $k$ only depends on the $``*"$-symbols in the $k$-th column of the PDA. Therefore, all the points on the fundamental SC tradeoff curve with same integer storage load $r$ can be attained with the same map procedures described by the MAN-PDA $\mathbf{P}_{r}$. (See also Remark 3 in Section V-A.)

As a consequence, the fundamental SC-tradeoff points that have integer storage load $r\in[1,\mathsf{K}]$ remain achievable (and optimal) also in a related setup where the size of the active set $\mathsf{Q}$ is unknown during the map procedure. By simple time and memory-sharing arguments, this conclusion extends to all points on the fundamental SC tradeoff curve with arbitrary real-valued storage loads $r\in[1,\mathsf{K}]$. Related is also the scenario where instead of fixing the size of the active set $\mathsf{Q}$, the system imposes a hard time-limit for the map phase and proceeds to the shuffle and reduce phases with the (random) number of nodes that have terminated within due time. For given storage load $r$, the MAN-PDA based coded computing scheme promises that when $\mathsf{Q}\geq\lceil\mathsf{K}+1-r\rceil$ nodes have terminated during the map phase, all IVAs are computed at least once and thus the system can proceed to data shuffling, and achieves the minimum required communication load $L_{\mathsf{K},\mathsf{Q}}^{*}(r)$. When only $\mathsf{Q}<\lceil\mathsf{K}+1-r\rceil$ nodes have terminated, some IVAs are not computed, and hence the system can not proceed.

It is further worth pointing out that all our PDA based coded computing schemes are universal and achieve the same performance for any choice of map and reduce functions. No structure is assumed on these functions. Similarly, our information-theoretic converse applies only to such universal coded computing schemes. If the map or reduce functions have certain properties, for example, linearity, it is possible to achieve better SC tradeoffs by storing combinations of files instead of each file separately [31, 32]. Fig. 3 compares Theorem 2 to the results in [31, 32]. It can be observed that the MAN-PDA based scheme outperforms the scheme in [31] but is inferior to the improved version in [32]. As already mentioned, the scheme in [32] however works only for linear map functions, and not for arbitrary functions as our schemes. Another advantage of our schemes is that they work over the binary field, and are thus easier to implement than the MDS-based schemes in [31, 32]. which require a large enough field size.

IV-C Optimality and Reduction of File Complexity

From Theorem 1 and Corollary 2, the coded computing scheme based on the MAN-PDA $\mathbf{P}_{r}$, for $r\in[\mathsf{K-Q+1}:\mathsf{K}]$, has file complexity {\color[rgb]{0,0,0}\mathsf{F}=C_{\mathsf{K}}^{r}} and achieves the fundamental SC tradeoff. The following theorem indicates that, this is the smallest file complexity to achieve the same tradeoff point in most cases. The proof is deferred to Section VII.

Theorem 3**.**

For a $(\mathsf{K,Q})$ straggling system, if a Comp-PDA based scheme achieves the fundamental SC tradeoff \big{(}r,L_{\mathsf{K,Q}}^{*}(r)\big{)} for some $r\in[\mathsf{K-Q}+1:\mathsf{K}]$, if $\mathsf{Q}\notin\{2,\mathsf{K}\}$ or $r\neq\mathsf{K}-\mathsf{Q}+1$, then the file complexity $\mathsf{F}\geq C_{\mathsf{K}}^{r}$.

Remark 1*.*

It is easy to verify that, in the case $\mathsf{Q}\in\{2,\mathsf{K}\}$ and $r=\mathsf{K}-\mathsf{Q}+1$, the fundamental SC tradeoff can be achieved with $\mathsf{F}=1$ with the Comp-PDAs $[*,*,\ldots,*,1]$ and $[*,1,2,\ldots,\mathsf{K}-1]$, respectively.

We next present Comp-PDAs with lower file complexity $\mathsf{F}$ that achieve SC tradeoffs close to the optimal ones. We consider two existing PDA constructions originally proposed for coded caching in [33, Theorems 4 and 5]. Let $q\in[2:\mathsf{K}-1]$ be such that $q\,|\,\mathsf{K}$, and $m=\frac{\mathsf{K}}{q}$. There exists

1. P$1$)

an $m$-regular \big{(}mq,q^{m-1},mq^{m-1},(q-1)q^{m-1}\big{)} Comp-PDA with minimum storage number $m$; 2. P$2$)

an $m(q-1)$-regular \big{(}mq,(q-1)q^{m-1},m(q-1)^{2}q^{m-1},q^{m-1}\big{)} Comp-PDA with minimum storage number $m(q-1)$.

Corollary 3**.**

For any integer $r\in[\mathsf{K-Q}+1:\mathsf{K}-1]$, such that either $r\,|\,\mathsf{K}$ or {\color[rgb]{0,0,0}(\mathsf{K}-r)}\,|\,\mathsf{K}, the communication load

[TABLE]

can be achieved with file complexity $\mathsf{F}=\frac{r}{\mathsf{K}}\cdot\left(\frac{\mathsf{K}}{\min\{r,\mathsf{K}-r\}}\right)^{\min\{r,\mathsf{K}-r\}}$.

Proof:

If $r\,|\,\mathsf{K}$, then specialize the Comp-PDA in P1) to parameter $q=\frac{\mathsf{K}}{r}$. This results in a $r$-regular $\left(\mathsf{K},q^{r-1},\mathsf{K}q^{r-2},(q-1)q^{r-1}\right)$ Comp-PDA with minimum storage number $r$, and the proof is then immediate from Corollary 1. If $\mathsf{K}-r|\mathsf{K}$, then specialize the Comp-PDA in P2) to parameter $q=\frac{\mathsf{K}}{\mathsf{K}-r}$. This results in a $r$-regular \big{(}\mathsf{K},(q-1)q^{\mathsf{K}-r-1}, $\mathsf{K}(q-1)^{2}q^{\mathsf{K}-r-2},$ q^{\mathsf{K}-r-1}\big{)} Comp-PDA, and the proof again follows from Corollary 1. ∎

In the following proposition, we quantify how close the above SC tradeoff point is to the optimal, and by how much we can reduce the file complexity.

Proposition 1*.*

Consider a $(\mathsf{K},\mathsf{Q})$ straggling system and an integer $r\in[\mathsf{K}-\mathsf{Q}+1:\mathsf{K}]$ such that \frac{r}{\mathsf{K}}=c\in\big{\{}\frac{1}{q},\frac{q-1}{q}\big{\}} for some integer $q\in[2:\mathsf{K}-1]$. There exist $\alpha\in[0,2]$ and $\beta\in[0,\sqrt{2\pi}e^{2}]$, such that the SC tradeoff $L_{\mathsf{K},\mathsf{Q}}(r)$ and the file complexity $\mathsf{F}$ achieved by constructions P1) or P2) above satisfy

[TABLE]

and

[TABLE]

where $A_{q}\triangleq\frac{\sqrt{q-1}}{cq}$ and $B_{q}\triangleq\left(\frac{q}{q-1}\right)^{\frac{q-1}{q}}$.

The proof is given in Appendix B. From the above proposition, for a fixed integer $q$, whenever $\mathsf{\frac{r}{K}}\in\{\frac{1}{q},\frac{q-1}{q}\}$ and $\mathsf{K},r$ scale proportionally to infinity, the communication load is close to optimal, while the file complexity can be reduced by a factor that increases exponentially in $\mathsf{K}$.

Remark 2*.*

In this work, we only consider two particular PDAs. There has been extensive research in coded caching schemes with low subpacketization level using various approaches. Most of them have an equivalent PDA representation. For examples, PDAs can be constructed from hyper-graphs [42], bipartite graphs [43], linear block codes [45], Ruzsa-Szemer$\acute{\mbox{e}}$di graphs [46]. The result in Theorem 1 makes it possible to apply all these results straightforwardly to straggling systems.

V Coded Computing Schemes for Straggling Systems from Comp-PDAs (Proof of Theorem 1)

In this section, we prove Theorem 1 by describing how to construct a coded computing scheme from a given Comp-PDA and analyzing its performance.

V-A Constructing a Coded Computing Scheme for a Straggling System from a Comp-PDA

In [8], we described how to obtain a coded computing scheme without stragglers from any given Comp-PDA. A similar procedure is possible in the presence of stragglers if the minimum storage number $\tau\geq\mathsf{K}-\mathsf{Q}+1$. In fact, assume a given Comp-PDA $\mathbf{A}$. The storage design in the map phase corresponding to $\mathbf{A}$ is the same as without straggling nodes. As part of the map phase, each node computes all the IVAs that it can compute from its stored files. For the reduce phase of the straggling system, we restrict to the subarray $\mathbf{A}^{\mathbf{Q}}$ of $\mathbf{A}$ formed by the columns of $\mathbf{A}$ with indices in the active set ${\mathbf{Q}}$. Notice that $\mathbf{A}^{\mathbf{Q}}$ is again a Comp-PDA, because the minimum storage number of $\mathbf{A}$ is at least $\mathsf{K}-\mathsf{Q}+1$ and after eliminating $\mathsf{K}-\mathsf{Q}$ columns from $\mathbf{A}$ each row still contains at least one $``*^{\prime\prime}$ symbol. Shuffle and reduce phases are performed as in a non-straggling setup, see [8], but where the Comp-PDA $\mathbf{A}$ is replaced by the new Comp-PDA $\mathbf{A}^{\mathbf{Q}}$. For completeness, we explain the map, shuffle, and reduce phases in detail.

Fix a $(\mathsf{K,F,T,S})$ Comp-PDA $\mathbf{A}=[a_{i,j}]$ with minimum storage number $\tau\geq\mathsf{K-Q}+1$. Partition the $\mathsf{N}$ files into $\mathsf{F}$ batches $\mathcal{W}_{1},\mathcal{W}_{2},\ldots,\mathcal{W}_{\mathsf{F}}$, each containing

[TABLE]

files and so that $\mathcal{W}_{1},\mathcal{W}_{2},\ldots,\mathcal{W}_{\mathsf{F}}$ form a partition for $\mathcal{W}$. It is implicitly assumed here that $\eta$ is an integer number.

V-A1 Map Phase

Each node $k$ stores the files in

[TABLE]

and computes the IVAs in (1). The map phase terminates whenever any $\mathsf{Q}$ nodes accomplish their computations. Throughout this section, let $\mathbf{Q}=\mathcal{Q}$ be the realization of the active set. Then, $\mathbf{A}^{\mathcal{Q}}$ denotes the subarray of $\mathbf{A}$ composed of the columns in $\mathcal{Q}$. Also, let $g_{s}^{\mathcal{Q}}$ denote the number of occurrences of the symbol $s$ in $\mathbf{A}^{\mathcal{Q}}$, i.e.,

[TABLE]

and $\mathcal{I}^{\mathcal{Q}}$ be the set of symbols occuring only once in $\mathbf{A}^{\mathcal{Q}}$:

[TABLE]

The symbols in $\mathcal{I}^{\mathcal{Q}}$ are partitioned into $\mathsf{Q}$ subsets $\{\mathcal{I}^{\mathcal{Q}}_{k}:k\in\mathcal{Q}\}$ as follows. For each $s\in\mathcal{I}^{\mathcal{Q}}$, let $(i,j)$ be the unique pair in $[\mathsf{F}]\times\mathcal{Q}$ such that $a_{i,j}=s$. Since the number of $``*"$ symbols in the $i$-th row of $\mathbf{A}$ is equal or larger than $\mathsf{K}-\mathsf{Q}+1$ by the assumption $\tau\geq\mathsf{K}-\mathsf{Q}+1$, there exists at least one $k\in\mathcal{Q}$ such that $a_{i,k}=*$. Arbitrarily choose such a $k$ and assign $s$ into $\mathcal{I}^{\mathcal{Q}}_{k}$.

Let $\mathcal{A}_{k}^{\mathcal{Q}}$ denote the set of ordinary symbols in column $k$ occurring at least twice:

[TABLE]

Pick any uniform assignment of reduce functions $\mathbf{D}^{\mathcal{Q}}=\{\mathcal{D}^{\mathcal{Q}}_{k}\}_{k\in\mathcal{Q}}$. Let $\mathcal{U}_{i,j}^{\mathcal{Q}}$ denote the set of IVAs for node $j$ computed from the files in $\mathcal{W}_{i}$, i.e.,

[TABLE]

V-A2 Shuffle Phase

Node $k$ multicasts the signal

[TABLE]

where the signals $X_{k,s}^{\mathcal{Q}}$ are created as described in the following, depending on whether $s\in\mathcal{I}_{k}$ or $s\in\mathcal{A}_{k}^{\mathcal{Q}}$. For all $s\in\mathcal{I}_{k}$, set

[TABLE]

where $(i,j)$ is the unique index in $[\mathsf{F}]\times\mathcal{Q}$ such that $a_{i,j}=s$.

To describe the signal $X_{k,s}^{\mathcal{Q}}$ for $s\in\mathcal{A}_{k}^{\mathcal{Q}}$, we first describe a partition of the IVA $\mathcal{U}_{i,j}^{\mathcal{Q}}$ for each pair $(i,j)\in[\mathsf{F}]\times\mathcal{Q}$ such that $a_{i,j}\in\mathcal{A}_{j}^{\mathcal{Q}}$. Let $s^{\prime}=a_{i,j}$, then $g_{s^{\prime}}^{\mathcal{Q}}\geq 2$. Let $(l_{1},j_{1}),(l_{2},j_{2}),\ldots,(l_{g_{s}^{\mathcal{Q}}-1},j_{g_{s}^{\mathcal{Q}}-1})\in[\mathsf{F}]\times\mathcal{Q}$ indicate all the other $g_{s^{\prime}}^{\mathcal{Q}}-1$ occurrences of the ordinary symbol $s^{\prime}$ in $\mathbf{A}^{\mathcal{Q}}$, i.e.,

[TABLE]

Partition the set of IVAs $\mathcal{U}_{i,j}^{\mathcal{Q}}$ into $g_{s^{\prime}}^{\mathcal{Q}}-1$ subsets of equal size and denote these subsets by $\mathcal{U}_{i,j}^{\mathcal{Q},j_{1}},\mathcal{U}_{i,j}^{\mathcal{Q},j_{2}},\ldots,\mathcal{U}_{i,j}^{\mathcal{Q},j_{g_{s}^{\mathcal{Q}}-1}}$:

[TABLE]

For all $s\in\mathcal{A}_{k}^{\mathcal{Q}}$, set

[TABLE]

V-A3 Reduce Phase

Node $k$ computes all IVAs in

[TABLE]

In the map phase, node $k$ has already computed all IVAs in \big{\{}\mathcal{U}_{i,k}^{\mathcal{Q}}:a_{i,k}=*\big{\}}. It thus remains to compute all IVAs in

[TABLE]

Fix an arbitrary $i\in[\mathsf{F}]$ such that $a_{i,k}\neq*$, and set $s=a_{i,k}$. If $s\in\mathcal{A}_{k}^{\mathcal{Q}}$, each subblock $\mathcal{U}_{i,k}^{\mathcal{Q},j}$ in (43) can be restored by node $k$ from the signal $X_{j,s}^{\mathcal{Q}}$ sent by node $j$ (see (44)):

[TABLE]

where $(l_{t},j_{t})$ ($t\in[g_{s}^{\mathcal{Q}}-2]$) indicate the other $g_{s}^{\mathcal{Q}}-2$ occurrences of the symbol $s$ in ${\mathbf{A}}^{\mathbf{Q}}$, i.e., $a_{l_{t},j_{t}}=s$. Notice that the sub-IVAs on the right-hand side of (45) have been computed by node $k$ during the map phase, because by the PDA properties, $a_{l_{t},j_{t}}=a_{i,k}=s$ and $j_{t}\neq k$ imply that $l_{t}\neq i$ and $a_{l_{t},k}=*$. Therefore, $\mathcal{U}_{i,k}^{\mathcal{Q},j}$ can be decoded from (45).

If $s\notin\mathcal{A}_{k}^{\mathcal{Q}}$, then $s\in\mathcal{I}^{\mathcal{Q}}$ by (41). There exists thus an index $j\in\mathcal{Q}\backslash\{k\}$ such that $s\in\mathcal{I}_{j}$ and therefore, by (42), the subset $\mathcal{U}_{i,k}^{\mathcal{Q}}$ can be recovered from the signal $X_{j,s}^{\mathcal{Q}}$ sent by node $j$.

Remark 3*.*

It is worth pointing out that the storage design $\{\mathcal{M}_{k}\}_{k=1}^{\mathsf{K}}$ only depends on the positions of the $``*"$ symbols in $\mathbf{A}$, but not on the parameter $\mathsf{Q}$ (See (40)). This indicates that, in practice the map phase can be carried out even without knowing how many nodes will be participating in the shuffle and reduce phases.

V-B Performance Analysis

We have analyzed the performances of storage and communication loads in the no-stragglers setup in [8]. For the scheme in the preceding subsection, the analysis of storage load follows the same lines as in [8]. When computing the communication load defined in (3), we have to average over all realizations of the active set $\mathbf{Q}$.

V-B1 Storage Load

Since the Comp-PDA $\mathbf{A}$ contains $\mathsf{T}$ $``*"$ symbols, and each $``*"$ symbol indicates that a batch of $\eta=\frac{\mathsf{N}}{\mathsf{F}}$ files is stored at a given node, see (40), the storage load of the proposed scheme is:

[TABLE]

V-B2 Communication Load

We first analyze the length of the signals sent for a given realization of the active set $\mathbf{Q}=\mathcal{Q}$. For any $s\in[\mathsf{S}]$, let $g_{s}$ be the occurrence of $s$ in $\mathbf{A}$, and $g_{s}^{\mathcal{Q}}$ be the occurrence of $s$ in the columns in $\mathcal{Q}$. By (42) and (44), the length of the signals associated to symbol $s$ is

[TABLE]

when $\mathcal{Q}$ is the active set. The total length of all the signals is thus

[TABLE]

We now compute the communication load as defined in (3) where we have to average over all realizations of the active set $\mathbf{Q}$:

[TABLE]

where $(a)$ holds by (50); $(b)$ holds since for each $s\in[\mathsf{S}]$, $\sum_{l=0}^{\mathsf{Q}}\mathbbm{1}(g_{s}^{\mathcal{Q}}=l)=1$ and $\sum_{g=1}^{\mathsf{K}}\mathbbm{1}(g_{s}=g)=1$; $(c)$ follows from (49); and $(d)$ holds since for each symbol occurring $g$ times, it has occurrence $l$ in exact $C_{g}^{l}\cdot C_{\mathsf{K}-g}^{\mathsf{Q}-l}$ subsets of $\mathcal{K}$ with cardinality $\mathsf{Q}$; in $(e)$, we defined $\mathsf{S}_{g}$ to be the number of ordinary symbols occurring $g$ times for each $g\in[\mathsf{K}]$; in $(f)$, we used the equality $C_{g}^{l}=\frac{g}{l}\cdot C_{g-1}^{l-1}$; in $(g)$, we eliminated the indices of zero terms in the summation of (V-B2); and $(h)$ follows from the definition of symbol frequencies.

V-C File Complexity of the Proposed Schemes

The analysis of file complexity is similar to the no-straggler setup in [8]. The files are partitioned into $\mathsf{F}$ batches so that each batch contains $\eta=\frac{\mathsf{N}}{\mathsf{F}}>0$ files. It is assumed that $\eta$ is a positive integer. The smallest number of files $\mathsf{N}$ where this assumption can be met is $\mathsf{F}$. Therefore, the file complexity of the scheme is $\mathsf{F}$.

VI The Fundamental Storage-Communication Tradeoff (Proof of Theorem 2)

By Corollary 2, the SC pair \big{(}r,L_{\mathsf{K},\mathsf{Q}}^{*}(r)\big{)}, $r\in[\mathsf{K}-\mathsf{Q}+1:\mathsf{K}]$ can be achieved by the MAN-PDA $\mathbf{P}_{r}$. For a general non-integer $r\in[\mathsf{K}-\mathsf{Q}+1,\mathsf{K}]$, the lower convex envelope of these points can be achieved by memory- and time- sharing. It remains to prove the converse in Theorem 2.

Let $Z_{\mathsf{K}}^{\mathsf{Q}}(x)$ be a piecewise linear function connecting the points \big{(}u,Z_{\mathsf{K}}^{\mathsf{Q}}(u)\big{)} sequentially over the interval $[\mathsf{K}-\mathsf{Q}+1,\mathsf{K}]$ with

[TABLE]

We shall need the following lemma, proved in Appendix A.

Lemma 1**.**

The sequence $Z_{\mathsf{K}}^{\mathsf{Q}}(u)$ is strictly convex and decreasing for $u\in[\mathsf{K}-\mathsf{Q}+1:\mathsf{K}]$. And the function $Z_{\mathsf{K}}^{\mathsf{Q}}(x)$ is convex and decreasing over $[\mathsf{K}-\mathsf{Q}+1,\mathsf{K}]$.

Let $\mathcal{M}\triangleq\{\mathcal{M}_{k}\}_{k=1}^{\mathsf{K}}$ be a storage design and $(r,L)$ be a SC pair achieved based on $\{\mathcal{M}_{k}\}_{k=1}^{\mathsf{K}}$. For each $u\in[\mathsf{K-Q}+1:\mathsf{K}]$, define

[TABLE]

i.e., $a_{\mathcal{M},u}$ is the number of files stored $u$ times across all the nodes. Then by definition, $a_{\mathcal{M},u}$ satisfies

[TABLE]

For any $\mathcal{Q}\in\mathbf{\Omega}_{\mathsf{K}}^{\mathsf{Q}}$ and any $l\in[\mathsf{Q}]$, define

[TABLE]

i.e., $b_{\mathcal{M},l}^{\mathcal{Q}}$ is the number of files stored exactly $l$ times in the nodes of set $\mathcal{Q}$. Since any file that is stored $u$ times across all the nodes has $l$ occurrences in exactly $C_{u}^{l}\cdot C_{\mathsf{K}-u}^{\mathsf{Q}-l}$ subsets $\mathcal{Q}$ of size $\mathfrak{\mathsf{Q}}$, i.e.,

[TABLE]

Summing over $n\in[\mathsf{N}]$, we obtain

[TABLE]

We can now apply the result in [5, Lemma 1] of the system without stragglers, to lower bound the communication load for any realization of the active set $\mathbf{Q}=\mathcal{Q}$:

[TABLE]

The average communication load over the random realization of the active set $\mathbf{Q}$ is then obtained as:

[TABLE]

where $(a)$ follows from (55); $(b)$ holds because the inner summation in (VI) only includes summation indices $u\in[\mathsf{K}-\mathsf{Q}:\mathsf{K}]$ and it includes the summation index $u\in\{\mathsf{K}-\mathsf{Q}+1,\ldots,\mathsf{K}\}$ if, and only if, the outer summation index $l$ satisfies $l\leq u$ and $l\geq u+\mathsf{Q}-\mathsf{K}$; $(c)$ follows from (52); $(d)$ follows from Lemma 1; $(e)$ follows from (54); and $(f)$ follows from the fact $\overline{r}\leq r+\epsilon$. Since $\epsilon$ can be arbitrarily close to zero, we conclude

[TABLE]

In particular, when $r\in[\mathsf{K}-\mathsf{Q}+1:\mathsf{K}]$, by (52),

[TABLE]

where $(a)$ holds since for $l=\mathsf{Q}$, the term in the summation is zero. This establishes the desired converse proof.

VII Optimality of File Complexity (Proof of Theorem 3)

In order to prove Theorem 3, we need to first derive several lemmas.

VII-A Preliminaries

Lemma 2**.**

If a coded computing scheme achieves the fundamental SC tradeoff pair $\left(r,L_{\mathsf{K},\mathsf{Q}}^{*}(r)\right)$ for any integer $r\in[\mathsf{K}-\mathsf{Q}+1:\mathsf{K}]$, then each file is stored exactly $r$ times across the nodes.

Proof:

According to Lemma 1, the sequence \big{\{}Z_{\mathsf{K}}^{\mathsf{Q}}(u)\big{\}}_{u=\mathsf{K}-\mathsf{Q}+1}^{\mathsf{K}} is strictly convex. Thus for the integer $r=\sum_{u=\mathsf{K}-\mathsf{Q}+1}^{\mathsf{K}}\frac{ua_{\mathcal{M},u}}{\mathsf{N}}$, the equality in (VI) holds if, and only if,

[TABLE]

Therefore, by definition of $a_{\mathcal{M},u}$ in (53), this indicates that each file is stored exactly $r$ times across the system. ∎

Lemma 3**.**

In a $g$-regular $\mathsf{(K,F,T,S)}$ PDA, s.t., $\mathsf{K}\geq g\geq 2$, if there are exactly $g-1$ $``*"$s in each row, then $\mathsf{F}\geq C_{\mathsf{K}}^{g-1}$.

Proof.

With Definition 5 (the definition of PDAs), the conclusion follows directly from [33, Lemma 2]. ∎

For each $u\in[\mathsf{K}]$, define

[TABLE]

Lemma 4**.**

When $\mathsf{Q}\geq 3$, the subsequence $\{U_{\mathsf{K}}^{\mathsf{Q}}(u)\}_{u=2}^{\mathsf{K}}$ strictly decreases with $u\in[2:\mathsf{K}]$.

Proof:

For each $u\in[2:\mathsf{K}-1]$, by (58),

[TABLE]

where in $(a)$, we used (67); in $(b)$, we separate the two summations in (VII-A) and eliminate the indices of zero terms in the separated summations; and in $(c)$, we used the variable change $l^{\prime}=l+1$. Moreover, if $u\geq 2$ and $\mathsf{Q}\geq 3$, from (VII-A), $U_{\mathsf{K}}^{\mathsf{Q}}(u+1)-U_{\mathsf{K}}^{\mathsf{Q}}(u)\leq-\frac{C_{u-1}^{l}C_{\mathsf{K}-u-1}^{\mathsf{Q}-2}}{2}<0$, i.e., $U_{\mathsf{K}}^{\mathsf{Q}}(u)$ is strictly decreasing when $u\geq 2$. ∎

VII-B Proof of Theorem 3

Define the set

[TABLE]

and partition it into three subsets

[TABLE]

Notice that, if $(\mathsf{Q},r)\in\mathcal{E}_{1}$, the bound $\mathsf{F}\geq C_{\mathsf{K}}^{\mathsf{K}}=1$ is trivial. The case $(\mathsf{Q},r)\in\mathcal{E}_{2}$ is excluded. Therefore, in the rest of the proof, we assume $(\mathsf{Q},r)\in\mathcal{E}_{3}$, i.e., $\mathsf{Q}\in[3:\mathsf{K}]$ and $r\in[\max\{\mathsf{K}-\mathsf{Q}+1,2\}:\mathsf{K}-1]$.

Let $\mathbf{A}$ be a $\mathsf{(K,F,T,S)}$ Comp-PDA that achieves the optimal tradeoff point $(r,L_{\mathsf{K},\mathsf{Q}}^{*}(r))$. Recall that each row in a Comp-PDA is associated to a file batch, and a $``*^{\prime\prime}$ symbole in that row and column $k$ indicates that the file batch is stored at node $k$. According to Lemma 2, each file is stored exactly $r$ times across the nodes, i.e., there are exactly $r$ $``*^{\prime\prime}$ symbols in each row of $\mathbf{A}$.

Let $\theta_{g^{\prime}}$ be the fraction of ordinary entries occurring $g^{\prime}$ times in the Comp-PDA $\mathbf{A}$, for all $g^{\prime}\in[\mathsf{K}]$. Since there are $r$ $``*^{\prime\prime}$ symbols in each row, from the PDA properties a) and b) in Definition 5, any ordinary symbol cannot appear more than $r+1$ times, i.e.,

[TABLE]

Therefore,

[TABLE]

From (34), (58), and (61), the communication load of $\mathbf{A}$ has the form

[TABLE]

where $(a)$ follows since by Lemma 4 the sequence $\{U_{\mathsf{K}}^{\mathsf{Q}}(u)\}_{u=2}^{\mathsf{K}}$ is decreasing and because $U_{\mathsf{K}}^{\mathsf{Q}}(1)=U_{\mathsf{K}}^{\mathsf{Q}}(2)=C_{\mathsf{K}-1}^{\mathsf{Q}-1}$; $(b)$ follows from (62); and $(c)$ follows from Theorem 2 and (58). By our assumption $L_{\mathbf{A}}=L_{\mathsf{K},\mathsf{Q}}^{*}(r)$, the equality in (VII-B) must hold. Since $r+1\geq 3$ and the sequence $\{U_{\mathsf{K}}^{\mathsf{Q}}(u)\}_{u=2}^{\mathsf{K}}$ strictly decreases, equality in (VII-B) implies that

[TABLE]

Combining (61) and (64), we conclude that $\mathbf{A}$ is a $(r+1)$-regular PDA, and each row has exactly $r$ $``*^{\prime\prime}$ symbols. Applying Lemma 3, we complete the proof.

VIII Conclusion

In this work, we have explained how to convert any Comp-PDA with at least $\mathsf{K}-\mathsf{Q}+1$ $``*^{\prime\prime}$ symbols in each row into a coded computing scheme for a MapReduce system with $\mathsf{Q}$ non-straggling nodes. We have further characterized the optimal storage-communication (SC) tradeoff for this system. The Comp-PDA framework allows us to design universal coded computing schemes with small file complexities compared to the ones (the MAN-PDAs) achieving the fundamental SC tradeoff.

In our setup, for a given integer storage load $r$, the size of active set $\mathsf{Q}$ has to be no less than $\mathsf{K}-r+1$, since we exclude outage events (See Footnote 2). With a given Comp-PDA, the key to obtaining a coded computing scheme for a given active set is that the subarray formed by the columns corresponding to the active set is still a Comp-PDA. In fact, for the constructions in P1) and P2), it allows to construct coded computing schemes for some (but not all) active sets if the active set size $\mathsf{Q}$ satisfies $\lceil\frac{\mathsf{K}}{r}\rceil\leq\mathsf{Q}\leq\mathsf{K}-\mathsf{r}$.

Appendix A Proof of Lemma 1

We shall prove the first statement of the lemma that the sequence \big{\{}Z_{\mathsf{K}}^{\mathsf{Q}}(u)\big{\}}_{u=\mathsf{K}-\mathsf{Q}+1}^{\mathsf{K}} is strictly convex and decreasing, i.e.,

[TABLE]

The second statement of the lemma on the piecewise linear function is an immediate consequence of the first one.

By (52),

[TABLE]

where in $(a)$, we used the identity $\sum_{l=s+\mathsf{Q}-\mathsf{K}}^{\min\{u,\mathsf{Q}\}}C_{u}^{l}\cdot C_{\mathsf{K}-u}^{\mathsf{Q}-l}=C_{\mathsf{K}}^{\mathsf{Q}}$. Then,

[TABLE]

where in $(a)$, we applied the identity

[TABLE]

in $(b)$, we separated the two summations of (A) into four summations and eliminated indices of zero terms in the separated summations; and in $(c)$, we used the change of variable $l^{\prime}=l-1$ in the first summation. Finally, from (A), for $u\in[\mathsf{K}-\mathsf{Q}+2:\mathsf{K}-1]$, we have

[TABLE]

where in $(a)$ we applied the identity (67); in $(b)$, we separated the two summations in (A) and eliminated the indices of zero terms in the separated summations; and in $(c)$, we used the change of variable $l^{\prime}=l+1$.

Appendix B Proof of Proposition 1

By (38), (39) and (58), we have

[TABLE]

Combining these equalities with (VII-A), we obtain

[TABLE]

where in $(a)$, we used the identity $C_{r-1}^{l-1}=\frac{l}{r}\cdot C_{r}^{l}$. Therefore, with (38) and (69),

[TABLE]

where in $(a)$, we used the fact $\frac{l}{l-1}\leq 2$ for any $l\geq 2$.

To prove the second part, we first note that, by Corollary 3, the number of batches required by constructions P1) and P2) is

[TABLE]

On the other hand, to achieve the fundamental SC tradeoff, the number of required batches is

[TABLE]

where $(a)$ follows by applying Stirling’s approximation $\sqrt{2\pi}n^{n+\frac{1}{2}}e^{-n}\leq n!\leq e\sqrt{2\pi}n^{n+\frac{1}{2}}e^{-n}$ to both the numerator and the denominator. Taking the ratio $\frac{\mathsf{F}}{\mathsf{F}^{*}}$ using (70) and (71), we complete the proof of the second part.