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Publication# Successive Refinement to Caching for Dynamic Requests

Résumé

In the celebrated coded caching problem studied by Maddah-Ali and Niesen, the peak-traffic network load is to be reduced by first caching some information about contents into individual memories of end users during the off-peak hours and then upon user requests broadcasting some other information about the contents, which, combined with cached information, can let each user recover their requested content. Thus, information-theoretic studies of coded caching involve the optimal tradeoff among communication rates for the two phases of cache placement and content delivery, and the optimal construction of codes for cache placement and content delivery. In order to allow better utilization of network resources, this paper introduces a new caching model in which user requests can arise at any point of time during the cache placement phase, and proposes a successive refinement approach as an answer to this dynamic caching problem. For uniformly random file requests, the optimal tradeoff among average-case delivery rates are characterized when the cache rate is above a well-defined threshold. For arbitrary file requests, a successive caching algorithm is developed to simultaneously reduce worst-case delivery rates at every request time, which are uniformly within a constant multiplicative factor of their respective optima.

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This thesis looks at efficient information processing for two network applications: content delivery with caching and collecting summary statistics in wireless sensor networks. Both applications are studied under the same paradigm: function computation over networks, where distributed source nodes cooperatively communicate some functions of individual observations to one or multiple destinations. One approach that always works is to convey all observations and then let the destinations compute the desired functions by themselves. However, if the available communication resources are limited, then revealing less unwanted information becomes critical. Centered on this goal, this thesis develops new coding schemes using information-theoretic tools.
The first part of this thesis focuses on content delivery with caching. Caching is a technique that facilitates reallocation of communication resources in order to avoid network congestion during peak-traffic times. An information-theoretic model, termed sequential coding for computing, is proposed to analyze the potential gains offered by the caching technique. For the single-user case, the proposed framework succeeds in verifying the optimality of some simple caching strategies and in providing guidance towards optimal caching strategies. For the two-user case, five representative subproblems are considered, which draw connections with classic source coding problems including the Gray-Wyner system, successive refinement, and the Kaspi/Heegard-Berger problem. Afterwards, the problem of distributed computing with successive refinement is considered. It is shown that if full data recovery is required in the second stage of successive refinement, then any information acquired in the first stage will be useful later in the second stage.
The second part of this thesis looks at the collection of summary statistics in wireless sensor networks. Summary statistics include arithmetic mean, median, standard deviation, etc, and they belong to the class of symmetric functions. This thesis develops arithmetic computation coding in order to efficiently perform in-network computation for weighted arithmetic sums and symmetric functions. The developed arithmetic computation coding increases the achievable computation rate from $\Theta((\log L)/L)$ to $\Theta(1/\log L)$, where $L$ is the number of sensors. Finally, this thesis demonstrates that interaction among sensors is beneficial for computation of type-threshold functions, e.g., the maximum and the indicator function, and that a non-vanishing computation rate is achievable.

Security in mobile communications is a topic of increasing relevance in everyday life. We all use mobile devices for everyday communications, maybe even for exchanging confidential information with the work place. This requires security systems that are reliable and stable even though mobility and number of users are increasing. An established solution in today's systems is to use symmetric cryptographic systems by relying on pre-established secrets. There are a few drawbacks. The first problem is that all security information is stored in one central storage place. This implies that users are required to trust this storage place. They need to trust that data is properly handled and well protected. The mechanism of storing pre-established long-term secrets at a central storage also makes it vulnerable to adversaries. As soon as someone gains access to it, he automatically has access to all secret information necessary to capture communications. A second problem is that security mechanisms implemented in today's mobile communication systems were designed for a moderate number of users. Since then, the number of users has grown significantly and continues to do so. It would thus be desirable to have a system that is flexible in terms of the number of users it can handle. The third and maybe the most crucial problem is the fact that today's security systems rely on computational cryptography. This means that the security mechanisms in cellular communication systems were designed with limited computational power of the adversary in mind. However, the computational power of computers is growing and systems might be breakable in the near future. A complementary notion of security is information-theoretic security. Systems built in this spirit are unbreakable even with unlimited computational power. These systems exploit the knowledge of statistical properties of the environment. This means mainly that security systems can be proved to be secure with respect to assumptions on the correlated sources and with respect to the nature of the knowledge the adversary can gain. The main idea of information-theoretic security is that two communication partners have access to correlated random sources. An adversary being potentially present in the system, has only degraded access to the source. The common knowledge can be used in order to generate sequences that are very similar among each other. Because the adversary has access to the random source, he is able to generate his own version of the sequence. It is fundamental that the sequences of the communicationpartners are stronger correlated among each other than with respect to the adversary. This advantage in knowledge can be interpreted as a secret among the communication partners to the adversary. In communicating over an authenticated but public channel, this secret can be extracted from the original sequence by both parties. In order for both keys to be equal, the original sequences need to coincide. This is not necessarily the case, but can be achieved by discussion over the public channel. In the first part of this thesis, we propose a security system that relies on information-theoretic security. We will demonstrate how this system can be implemented in cellular communication systems. The movement of the users is random and we want to use it as a random source. We assume that users are moving along a random itinerary while visiting a random sequence of cells. Users not only visit cells in a random order, they also stay for a random time in each cells. It turns out that it is the sequence of durations that contributes most to the randomness of the sequence. We call the sequence of durations timing. This sequence is known to users and to the infrastructure, but only parts of it are known to potential adversaries. By recording the sequence of cells along with the time of residence, both the user and the infrastructure will be able to acquire very similar sequences. After the collection process, a procedure for key agreement is applied. Due to the underlying process, the key will not be uniformly distributed. As this is a requirement for perfect secret keys, a final part of the protocol is privacy amplification, where the perfectly secret bits will be extracted. This system is lightweight as it is based on readily available information and only a small amount of post-processing is required. In the second part of this thesis, we analyze the performance of the system. For this, we demonstrate how the process of bit collection can be analytically modeled. Our analysis leads to two conclusions. It gives us intuition about how to choose parameters such that the scheme performs optimally. Our scheme can be applied to settings that fulfill certain conditions. Conclusions from analysis allow to specify such conditions. We will show the implication of this conclusion in sketching some alternative applications of our scheme.

In any communication system, there exist two dimensions through which the information at the source becomes distorted before reaching the destination: the noisy channel and time. Messages transmitted through a noisy channel are susceptible to modification in their content, due to the action of the noise of the channel. Claude E. Shannon, in his seminal paper of 1948 "A Mathematical Theory of Communication",
introduces the bit as a unit of measure of information, and he lays down the theoretical foundations needed to understand the problem of sending bits reliably through a noisy channel. The distortion measure, which he used to quantify reliability, is the error probability. In his paper, Shannon shows that any channel is characterized by a number that he calls capacity: It represents the highest transmission rate that
can be used to communicate information with, through this same channel, while guaranteeing a negligible error probability. Whereas, even if the messages are sent through a perfect channel, the time
they take to reach their destination causes the receiver to acquire a distorted view of the status of the source that generated these messages. For instance, take the case of a monitor interested in the status of a distant process. A sender observes this process and, to keep the monitor up-to-date, sends updates to it. However, if, at any time t, the last received update at the monitor was generated at time u(t),
then the information at the receiver reflects the status of the process at time u(t), not at time t. Hence, the monitor has a distorted version of reality. In fact, it has an obsolete version with an age of t-u(t). The concept of age as a distortion measure in communication systems was first used in 2011 by Kaul et al., in order to assess the performance of a given vehicular network. The aim of the authors was to come up with a transmission scheme that would minimize an age-related metric: the average age. Since then, a growing body of works has used this metric to evaluate the performance of multiple communication
systems. The drive behind this interest lies in the importance that status-update applications are gaining in today's life (in vehicular networks, warehouse and environment surveillance, news feed,etc.). In this thesis, we choose age as a distortion measure and derive expressions for the average age and the average peak-age (another age-related metric) for different communication systems. Therefore, we divide this dissertation into two parts: In the first part, we assume that the the updates are transmitted through a noiseless channel that has a random service time. In the second part, we consider a special category of noisy channels, namely the erasure channel. In the first part of this thesis, in order to compute the age-related metrics, we employ queue-theoretic concepts. We study and compare the performance of various transmission schemes under different settings.We show that the optimal transmission scheme when the monitor is interested in a single source loses its optimality when another source of higher priority shares the system. In the second part of this thesis, we introduce, in our age calculations, the distortion caused by the erasure channel on the transmitted updates. In order to combat the erasures of the channel, we first consider two flavors of the hybrid automatic repeat request (HARQ). Finally, we focus on the optimal average age that could be achieved over an erasure channel.