Blind Updates in Coded Caching

Abstract

We consider the centralized coded caching system where a library of files is available at the server and their subfiles are cached at the clients as prescribed by a placement delivery array (PDA). We are interested in the problem where a specific file in the library is replaced with a new file at the server, the contents of which are correlated with the file being replaced, and this change needs to be communicated to the caches. Upon replacement, the server has access only to the updated file and is unaware of its differences with the original, while each cache has access to specific subfiles of the original file as dictated by the PDA. Scenarios such as (a) updating an unlogged file library, (b) ensuring privacy of updated subfiles against an eavesdropper, and (c) correcting errors in the client caches, can be captured as special cases of this problem framework. We model the correlation between the updated and the original files by assuming that they differ in at the most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> subfiles, and aim to reduce the number of bits broadcast by the server to update the caches. We design a new elegant coded transmission strategy for the server to update the caches blindly, and also identify another simple scheme that is based on MDS codes. We then derive converse bounds on the minimum communication cost <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell ^{*}$ </tex-math></inline-formula> among all linear strategies. For two well-known families of PDAs – Maddah-Ali & Niesen’s caching scheme and a PDA by Tang & Ramamoorthy and Yan <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> – our new scheme has cost <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell ^{*}(1 + o(1))$ </tex-math></inline-formula> as the number of users grows large, when the updates are sufficiently sparse and the caching ratio is an arbitrary constant; whereas the scheme using MDS codes has order-optimal cost when the updates are dense.