Combination Networks With End-User-Caches: Novel Achievable and Converse Bounds Under Uncoded Cache Placement

Caching is an efficient way to reduce network traffic congestion during peak hours by storing some content at the users’ local caches. For the shared-link network with end-user-caches, Maddah-Ali and Niesen proposed a two-phase coded caching strategy. In practice, users may communicate with the server through intermediate relays. This paper studies the tradeoff between the memory size <inline-formula> <tex-math notation="LaTeX">${\mathsf M}$ </tex-math></inline-formula> and the network load <inline-formula> <tex-math notation="LaTeX">${\mathsf R}$ </tex-math></inline-formula> for the networks where a server with <inline-formula> <tex-math notation="LaTeX">${\mathsf N}$ </tex-math></inline-formula> files is connected to <inline-formula> <tex-math notation="LaTeX">${\mathsf H}$ </tex-math></inline-formula> relays (without caches), which in turn are connected to <inline-formula> <tex-math notation="LaTeX">${\mathsf K}$ </tex-math></inline-formula> users equipped with caches of <inline-formula> <tex-math notation="LaTeX">${\mathsf M}$ </tex-math></inline-formula> files. When each user is connected to a different subset of <inline-formula> <tex-math notation="LaTeX">${\mathsf r}$ </tex-math></inline-formula> relays, i.e., <inline-formula> <tex-math notation="LaTeX">${\mathsf K}= \binom {\mathsf H}{\mathsf r}$ </tex-math></inline-formula>, the system is referred to as a <italic>combination network with end-user-caches</italic>. In this work, converse bounds are derived for the practically motivated case of <italic>uncoded</italic> cache contents, that is, bits of the various files are directly pushed into the user caches without any coding. In this case, once the cache contents and the users’ demands are known, the problem reduces to a general index coding problem. This paper shows that relying on a well-known “acyclic index coding converse bound” results in converse bounds that are not tight for combination networks with end-user-caches. A novel converse bound that leverages the network topology is proposed, which is the tightest converse bound known to date. As a result of independent interest, an inequality that generalizes the well-known sub-modularity of entropy is derived. Several novel caching schemes are proposed, based on the Maddah-Ali and Niesen cache placement. These schemes leverage the structure of the combination network or/and perform interference elimination at the end-users. The proposed schemes are proved: (i) to be (order) optimal for some <inline-formula> <tex-math notation="LaTeX">$({\mathsf N}, {\mathsf M}, {\mathsf H}, {\mathsf r})$ </tex-math></inline-formula> parameters regimes under the constraint of uncoded cache placement, and (ii) to outperform the state-of-the-art schemes in numerical evaluations.

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