Cache-Aided Private Information Retrieval with Partially Known Uncoded Prefetching: Fundamental Limits
This work addresses efficient and private data retrieval in distributed systems with caching, offering incremental theoretical insights into PIR with partially known side information.
The paper tackles the problem of private information retrieval (PIR) with a cache holding uncoded fractions of messages, deriving fundamental limits on download cost as a function of caching ratio, number of messages, and databases. It provides lower and upper bounds that match in extreme cases, fully characterizes the tradeoff for 3 messages, and shows a maximum gap of 5/32 for general parameters.
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 $r$ of the symbols from each of the $K$ stored messages in the databases. This model operates in a two-phase scheme, namely, the prefetching phase where the user acquires side information and the retrieval phase where the user privately downloads the desired message. In the prefetching phase, the user receives $\frac{r}{N}$ uncoded fraction of each message from the $n$th database. This side information is known only to the $n$th database and unknown to the remaining databases, i.e., the user possesses \emph{partially known} side information. We investigate the optimal normalized download cost $D^*(r)$ in the retrieval phase as a function of $K$, $N$, $r$. We develop lower and upper bounds for the optimal download cost. The bounds match in general for the cases of very low caching ratio ($r \leq \frac{1}{N^{K-1}}$) and very high caching ratio ($r \geq \frac{K-2}{N^2-3N+KN}$). 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 $\frac{5}{32}$.