Fundamental Limits of Exact-Repair Regenerating Codes

Understanding the fundamental limits of communication systems involves both constructing efficient coding schemes as well as proving mathematically that certain performance is impossible to achieve; the latter is known as the converse problem in information theory. This thesis focused on the converse problems for complex information systems such as self-repair distributed storage and coded caching systems, and our goal was to establish tight converse results for such systems by exploiting problem-specific combinatorial structures. The main part of this thesis dealt with exact-repair regenerating codes, which were first proposed by Dimakis et al. in 2010. In particular, we considered two extensions of the original setting of Dimakis et al., namely 1) multilevel diversity coding with regeneration and 2) secure exact-repair regenerating codes. For the problem of multilevel diversity coding with regeneration, we showed, via the proposed combinatorial approach, that the natural separate encoding strategy can achieve the optimal tradeoff between the normalized storage capacity and repair bandwidth at the minimum-bandwidth rate (MBR) point. This settled a conjecture by Tian and Liu in 2015. For the problem of secure exact-repair regenerating codes, all known results from the literature showed that the achievable tradeoff regions between the normalized storage capacity and repair bandwidth have a single corner point, achieved by a scheme proposed by Shah, Rashmi and Kumar (the SRK point). Since the achievable tradeoff regions of the exact-repair regenerating code problem without any secrecy constraints were known to have multiple corner points in general, these existing results suggested a phase-change-like behavior, i.e., enforcing a secrecy constraint immediately reduces the tradeoff region to one with a single corner point. In our work, we first showed that when the secrecy parameter is sufficiently large, the SRK point is indeed the only corner point of the tradeoff region. However, when the secrecy parameter is small, we showed that the tradeoff region can, in fact, have multiple corner points. In particular, we established a precise characterization of the tradeoff region for a particular problem instance, which has exactly two corner points. Thus, a smooth transition, instead of a phase-change-type of transition, should be expected as the secrecy constraint is gradually strengthened

Fundamental Limits of Caching: Symmetry Structure and Coded Placement Schemes

Caching is a technique to reduce the communication load in peak hours by prefetching contents during off-peak hours. In 2014, Maddah-Ali and Niesen introduced a framework for coded caching, and showed that significant improvement can be obtained compared to uncoded caching. Considerable efforts have been devoted to identify the precise information theoretic fundamental limit of such systems, however the difficulty of this task has also become clear. One of the reasons for this difficulty is that the original coded caching setting allows multiple demand types during delivery, which in fact introduces tension in the coding strategy to accommodate all of them. We seek to develop a better understanding of the fundamental limit of coded caching. In order to characterize the fundamental limit of the tradeoff between the amount of cache memory and the delivery transmission rate of multiuser caching systems, various coding schemes have been proposed in the literature. These schemes can largely be categorized into two classes, namely uncoded prefetching schemes and coded prefetching schemes. While uncoded prefetching schemes in general over order-wise optimal performance, coded prefetching schemes often have better performance at the low cache memory regime. At first sight it seems impossible to connect these two different types of coding schemes, yet finding a unified coding scheme that achieves the optimal memory-rate tradeoff is an important and interesting problem. We take the first step on this direction and provide a connection between the uncoded prefetching scheme proposed by Maddah Ali and Niesen (and its improved version by Yu et al.) and the coded prefetching scheme proposed by Tian and Chen. The intermediate operating points of this general scheme can in fact provide new memory-rate tradeoff points previously not known to be achievable in the literature. This new general coding scheme is then presented and analyzed rigorously, which yields a new inner bound to the memory-rate tradeoff for the caching problem. While studying the general case can be difficult, we found that studying the single demand type systems will provide important insights. Motivated by these findings, we focus on systems where the number of users and the number of files are the same, and the demand type is when all files are being requested. A novel coding scheme is proposed, which provides several optimal memory transmission operating points. Outer bounds for this class of systems are also considered, and their relation with existing bounds is discussed. Outer-bounding the fundamental limits of coded caching problem is difficult, not only because there are tons of information inequalities and problem specific equalities to choose from, but also because of identifying a useful subset (and often a quite small subset) from them and how to combine them to produce an improved outerbound is a hard problem. Information inequalities can be used to derive the fundamental limits of information systems. Many information inequalities and problem-specific constraints are linear equalities or inequalities of joint entropies, and thus outer bounding the fundamental limits can be viewed as and in principle computed through linear programming. However, for many practical engineering problems, the resultant linear program (LP) is very large, rendering such a computational approach almost completely inapplicable in practice. We provide a method to pinpoint this reduction by counting the number of orbits induced by the symmetry on the set of the LP variables and the LP constraints, respectively. We proposed a generic three-layer decomposition of the group structures for this purpose. This general approach can also be applied to various other problems such as extremal pairwise cyclically symmetric entropy inequalities and the regenerating code problem. Decentralized coded caching is applicable in scenarios when the server is uninformed of the number of active users and their identities in a wireless or mobile environment. We propose a decentralized coded prefetching strategy where both prefetching and delivery are coded. The proposed strategy indeed outperforms the existing decentralized uncoded caching strategy in regimes of small cache size when the numbers of files is less than the number of users. Methods to manage the coding overhead are further suggested

Fundamental Limits of Caching: Symmetry Structure and Coded Placement Schemes

Caching is a technique to reduce the communication load in peak hours by prefetching contents during off-peak hours. In 2014, Maddah-Ali and Niesen introduced a framework for coded caching, and showed that significant improvement can be obtained compared to uncoded caching. Considerable efforts have been devoted to identify the precise information theoretic fundamental limit of such systems, however the difficulty of this task has also become clear. One of the reasons for this difficulty is that the original coded caching setting allows multiple demand types during delivery, which in fact introduces tension in the coding strategy to accommodate all of them. We seek to develop a better understanding of the fundamental limit of coded caching. In order to characterize the fundamental limit of the tradeoff between the amount of cache memory and the delivery transmission rate of multiuser caching systems, various coding schemes have been proposed in the literature. These schemes can largely be categorized into two classes, namely uncoded prefetching schemes and coded prefetching schemes. While uncoded prefetching schemes in general over order-wise optimal performance, coded prefetching schemes often have better performance at the low cache memory regime. At first sight it seems impossible to connect these two different types of coding schemes, yet finding a unified coding scheme that achieves the optimal memory-rate tradeoff is an important and interesting problem. We take the first step on this direction and provide a connection between the uncoded prefetching scheme proposed by Maddah Ali and Niesen (and its improved version by Yu et al.) and the coded prefetching scheme proposed by Tian and Chen. The intermediate operating points of this general scheme can in fact provide new memory-rate tradeoff points previously not known to be achievable in the literature. This new general coding scheme is then presented and analyzed rigorously, which yields a new inner bound to the memory-rate tradeoff for the caching problem. While studying the general case can be difficult, we found that studying the single demand type systems will provide important insights. Motivated by these findings, we focus on systems where the number of users and the number of files are the same, and the demand type is when all files are being requested. A novel coding scheme is proposed, which provides several optimal memory transmission operating points. Outer bounds for this class of systems are also considered, and their relation with existing bounds is discussed. Outer-bounding the fundamental limits of coded caching problem is difficult, not only because there are tons of information inequalities and problem specific equalities to choose from, but also because of identifying a useful subset (and often a quite small subset) from them and how to combine them to produce an improved outerbound is a hard problem. Information inequalities can be used to derive the fundamental limits of information systems. Many information inequalities and problem-specific constraints are linear equalities or inequalities of joint entropies, and thus outer bounding the fundamental limits can be viewed as and in principle computed through linear programming. However, for many practical engineering problems, the resultant linear program (LP) is very large, rendering such a computational approach almost completely inapplicable in practice. We provide a method to pinpoint this reduction by counting the number of orbits induced by the symmetry on the set of the LP variables and the LP constraints, respectively. We proposed a generic three-layer decomposition of the group structures for this purpose. This general approach can also be applied to various other problems such as extremal pairwise cyclically symmetric entropy inequalities and the regenerating code problem. Decentralized coded caching is applicable in scenarios when the server is uninformed of the number of active users and their identities in a wireless or mobile environment. We propose a decentralized coded prefetching strategy where both prefetching and delivery are coded. The proposed strategy indeed outperforms the existing decentralized uncoded caching strategy in regimes of small cache size when the numbers of files is less than the number of users. Methods to manage the coding overhead are further suggested