Fundamental Limits of Cache-aided Shared-link Broadcast Networks and Combination Networks

Abstract : In this thesis, we investigated the coded caching problem by building the connection between coded caching with uncoded placement and index coding, and leveraging the index coding results to characterize the fundamental limits of coded caching problem. We mainly analysed the caching problem in shared-link broadcast model and in combination networks. In the first part of this thesis, for cache-aided shared-link broadcast networks, we considered the constraint that content is placed uncoded within the caches. When the cache contents are uncoded and the user demands are revealed, the caching problem can be connected to an index coding problem. We derived fundamental limits for the caching problem by using tools for the index coding problem. A novel index coding achievable scheme was first derived based on distributed source coding. This inner bound was proved to be strictly better than the widely used “composite (index) coding” inner bound by leveraging the ignored correlation among composites and the non-unique decoding. For the centralized caching problem, an outer bound under the constraint of uncoded cache placement is proposed based on the “acyclic index coding outer bound”. This outer bound is proved to be achieved by the cMAN scheme when the number of files is not less than the number of users, and by the proposed novel index coding achievable scheme otherwise. For the decentralized caching problem, this thesis proposes an outer bound under the constraint that each user stores bits uniformly and independently at random. This outer bound is achieved by dMAN when the number of files is not less than the number of users, and by our proposed novel index coding inner bound otherwise. In the second part of this thesis, we considered the centralized caching problem in two-hop relay networks, where the server communicates with cache-aided users through some intermediate relays. Because of the hardness of analysis on the general networks, we mainly considered a well-known symmetric relay networks, combination networks, including H relays and binom{H}{r} users where each user is connected to a different r-subset of relays. We aimed to minimize the max link-load for the worst cases. We derived outer and inner bounds in this thesis. For the outer bound, the straightforward way is that each time we consider a cut of x relays and the total load transmitted to these x relays could be outer bounded by the outer bound for the shared-link model including binom{x}{r} users. We used this strategy to extend the outer bounds for the shared-link model and the acyclic index coding outer bound to combination networks. In this thesis, we also tightened the extended acyclic index coding outer bound in combination networks by further leveraging the network topology and joint entropy of the various random variables. For the achievable schemes, there are two approaches, separation and non-separation. In the separation approach, we use cMAN cache placement and multicast message generation independent of the network topology. We then deliver cMAN multicast messages based on the network topology. In the non-separation approach, we design the placement and/or the multicast messages on the network topology. We proposed four delivery schemes on separation approach. On non-separation approach, firstly for any uncoded cache placement, we proposed a delivery scheme by generating multicast messages on network topology. Moreover, we also extended our results to more general models, such as combination networks with cache-aided relays and users, and caching systems in more general relay networks. Optimality results were given under some constraints and numerical evaluations showed that our proposed schemes outperform the state-of-the-art.
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Kai Wan. Fundamental Limits of Cache-aided Shared-link Broadcast Networks and Combination Networks. Information Theory [cs.IT]. Université Paris-Saclay, 2018. English. ⟨NNT : 2018SACLS217⟩. ⟨tel-01842269⟩

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