Fundamental Limits of Cache-Aided Private Information Retrieval with Unknown and Uncoded Prefetching

We consider the problem of private information retrieval (PIR) from $N$ non-colluding and replicated databases when the user is equipped with a cache that holds an uncoded fraction $r$ from each of the $K$ stored messages in the databases. We assume that the databases are unaware of the cache content. We investigate $D^*(r)$ the optimal download cost normalized with the message size as a function of $K$, $N$, $r$. For a fixed $K$, $N$, we develop an inner bound (converse bound) for the $D^*(r)$ curve. The inner bound is a piece-wise linear function in $r$ that consists of $K$ line segments. For the achievability, we develop explicit schemes that exploit the cached bits as side information to achieve $K-1$ non-degenerate corner points. These corner points differ in the number of cached bits that are used to generate one side information equation. We obtain an outer bound (achievability) for any caching ratio by memory-sharing between these corner points. Thus, the outer bound is also a piece-wise linear function in $r$ that consists of $K$ line segments. The inner and the outer bounds match in general for the cases of very low caching ratio ($r \leq \frac{1}{1+N+N^2+\cdots+N^{K-1}}$) and very high caching ratio ($r \geq \frac{K-2}{(N+1)K+N^2-2N-2}$). As a corollary, we fully characterize the optimal download cost caching ratio tradeoff for $K=3$. For general $K$, $N$, and $r$, we show that the largest gap between the achievability and the converse bounds is $\frac{1}{6}$. Our results show that the download cost can be reduced beyond memory-sharing if the databases are unaware of the cached content.