On the Fundamental Limits of Coded Caching With Correlated Files of Combinatorial Overlaps

This paper studies the fundamental limits of the shared-link coded caching problem with correlated files, where a server with a library of
${\mathsf N}$
files communicates with
${\mathsf K}$
users who can locally cache
${\mathsf M}$
files. Given an integer
${\mathsf r}\in [{\mathsf N}]$
, correlation is modelled as follows: each
${\mathsf r}$
-subset of files contains a unique common block. The tradeoff between the cache size and the average transmitted load over the uniform demand distribution is studied. First, a converse bound under the constraint of uncoded cache placement (i.e., each user directly stores a subset of the library bits) is derived. Then, a caching scheme for the case where every user demands a distinct file (possible for
${\mathsf N}\geq {\mathsf K}$
) is shown to be optimal under the constraint of uncoded cache placement. This caching scheme is further proved to be decodable and optimal under the constraint of uncoded cache placement when (i)
${\mathsf K} {\mathsf r} {\mathsf M}\leq 2 {\mathsf N}$
or
${\mathsf K} {\mathsf r} {\mathsf M}\geq ({\mathsf K}-1) {\mathsf N}$
or
${\mathsf r}\in \{1,2, {\mathsf N}-1, {\mathsf N}\}$
, and (ii) when the number of distinct demanded files is no larger than four. Finally, a new delivery scheme based on interference alignment which jointly serves the users’ demands is shown to be order optimal to within a factor of 2 under the constraint of uncoded cache placement. As an extension, the above exact and order optimal results can be extended to the worst-case load. As by-products, an extension of the proposed scheme for
${\mathsf M}= {\mathsf N}/ {\mathsf K}$
is shown to reduce the load of state-of-the-art schemes for the coded caching problem where the users can request multiple files; the proposed scheme for distinct demands can be extended to the coded distributed computing problem with a central server, which achieves the optimal transmission load over the binary field.