Coded Caching Schemes with Reduced Subpacketization from Linear Block Codes

Coded caching is a technique that generalizes conventional caching and promises significant reductions in traffic over caching networks. However, the basic coded caching scheme requires that each file hosted in the server be partitioned into a large number (i.e., the subpacketization level) of non-overlapping subfiles. From a practical perspective, this is problematic as it means that prior schemes are only applicable when the size of the files is extremely large. In this work, we propose coded caching schemes based on combinatorial structures called resolvable designs. These structures can be obtained in a natural manner from linear block codes whose generator matrices possess certain rank properties.We obtain several schemes with subpacketization levels substantially lower than the basic scheme at the cost of an increased rate. Depending on the system parameters, our approach allows us to operate at various points on the subpacketization level vs. rate tradeoff

Coded Caching based on Combinatorial Designs

We consider the standard broadcast setup with a single server broadcasting information to a number of clients, each of which contains local storage (called \textit{cache}) of some size, which can store some parts of the available files at the server. The centralized coded caching framework, consists of a caching phase and a delivery phase, both of which are carefully designed in order to use the cache and the channel together optimally. In prior literature, various combinatorial structures have been used to construct coded caching schemes. In this work, we propose a binary matrix model to construct the coded caching scheme. The ones in such a \textit{caching matrix} indicate uncached subfiles at the users. Identity submatrices of the caching matrix represent transmissions in the delivery phase. Using this model, we then propose several novel constructions for coded caching based on the various types of combinatorial designs. While most of the schemes constructed in this work (based on existing designs) have a high cache requirement (uncached fraction being $\Theta(\frac{1}{\sqrt{K}})$ or $\Theta(\frac{1}{K})$, $K$ being the number of users), they provide a rate that is either constant or decreasing ($O(\frac{1}{K})$) with increasing $K$, and moreover require competitively small levels of subpacketization (being $O(K^i), 1\leq i\leq 3$), which is an extremely important parameter in practical applications of coded caching. We mark this work as another attempt to exploit the well-developed theory of combinatorial designs for the problem of constructing caching schemes, utilizing the binary caching model we develop.Comment: 10 pages, Appeared in Proceedings of IEEE ISIT 201

Algebraic approaches for coded caching and distributed computing

This dissertation examines the power of algebraic methods in two areas of modern interest: caching for large scale content distribution and straggler mitigation within distributed computation. Caching is a popular technique for facilitating large scale content delivery over the Internet. Traditionally, caching operates by storing popular content closer to the end users. Recent work within the domain of information theory demonstrates that allowing coding in the cache and coded transmission from the server (referred to as coded caching) to the end users can allow for significant reductions in the number of bits transmitted from the server to the end users. The first part of this dissertation examines problems within coded caching. The original formulation of the coded caching problem assumes that the server and the end users are connected via a single shared link. In Chapter 2, we consider a more general topology where there is a layer of relay nodes between the server and the users. We propose novel schemes for a class of such networks that satisfy a so-called resolvability property and demonstrate that the performance of our scheme is strictly better than previously proposed schemes. Moreover, the original coded caching scheme requires that each file hosted in the server be partitioned into a large number (i.e., the subpacketization level) of non-overlapping subfiles. From a practical perspective, this is problematic as it means that prior schemes are only applicable when the size of the files is extremely large. In Chapter 3, we propose a novel coded caching scheme that enjoys a significantly lower subpacketization level than prior schemes, while only suffering a marginal increase in the transmission rate. We demonstrate that several schemes with subpacketization levels that are exponentially smaller than the basic scheme can be obtained. The second half of this dissertation deals with large scale distributed matrix computations. Distributed matrix multiplication is an important problem, especially in domains such as deep learning of neural networks. It is well recognized that the computation times on distributed clusters are often dominated by the slowest workers (called stragglers). Recently, techniques from coding theory have found applications in straggler mitigation in the specific context of matrix-matrix and matrix-vector multiplication. The computation can be completed as long as a certain number of workers (called the recovery threshold) complete their assigned tasks. In Chapter 4, we consider matrix multiplication under the assumption that the absolute values of the matrix entries are sufficiently small. Under this condition, we present a method with a significantly smaller recovery threshold than prior work. Besides, the prior work suffers from serious numerical issues owing to the condition number of the corresponding real Vandermonde-structured recovery matrices; this condition number grows exponentially in the number of workers. In Chapter 5, we present a novel approach that leverages the properties of circulant permutation matrices and rotation matrices for coded matrix computation. In addition to having an optimal recovery threshold, we demonstrate an upper bound on the worst case condition number of our recovery matrices grows polynomially in the number of workers

Optimal Placement Delivery Arrays from tt-Designs with Application to Hierarchical Coded Caching

Coded caching scheme originally proposed by Maddah-Ali and Niesen (MN) achieves an optimal transmission rate $R$ under uncoded placement but requires a subpacketization level $F$ which increases exponentially with the number of users $K$ where the number of files $N \geq K$. Placement delivery array (PDA) was proposed as a tool to design coded caching schemes with reduced subpacketization level by Yan \textit{et al.} in \cite{YCT}. This paper proposes two novel classes of PDA constructions from combinatorial $t$-designs that achieve an improved transmission rate for a given low subpacketization level, cache size and number of users compared to existing coded caching schemes from $t$-designs. A $(K, F, Z, S)$ PDA composed of a specific symbol $\star$ and $S$ non-negative integers corresponds to a coded caching scheme with subpacketization level $F$, $K$ users each caching $Z$ packets and the demands of all the users are met with a rate $R=\frac{S}{F}$. For a given $K$, $F$ and $Z$, a lower bound on $S$ such that a $(K, F, Z, S)$ PDA exists is given by Cheng \textit{et al.} in \cite{MJXQ} and by Wei in \cite{Wei} . Our first class of proposed PDA achieves the lower bound on $S$. The second class of PDA also achieves the lower bound in some cases. From these two classes of PDAs, we then construct hierarchical placement delivery arrays (HPDA), proposed by Kong \textit{et al.} in \cite{KYWM}, which characterizes a hierarchical two-layer coded caching system. These constructions give low subpacketization level schemes.Comment: Title has been changed. Some changes have been incorporated in the results. 11 pages, 5 figures and 3 table