Cache-Aided Private Information Retrieval With Partially Known Uncoded Prefetching: Fundamental Limits

We consider the problem of private information retrieval from <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> non-colluding and replicated databases, when the user is equipped with a cache that holds an uncoded fraction <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> of the symbols from each of the <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">${r}/{N}$ </tex-math></inline-formula> uncoded fraction of each message from the <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>th database. This side information is known only to the <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>th database and unknown to the remaining databases, i.e., the user possesses <italic>partially known</italic> side information. We investigate the optimal normalized download cost <inline-formula> <tex-math notation="LaTeX">$D^{*}(r)$ </tex-math></inline-formula> in the retrieval phase as a function of <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>. We develop lower and upper bounds for the optimal download cost. The bounds match in general for the cases of very low caching ratio and very high caching ratio. We fully characterize the optimal download cost caching ratio tradeoff for <inline-formula> <tex-math notation="LaTeX">$K=3$ </tex-math></inline-formula>. For general <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> values, we show that the largest additive gap between the achievability and the converse bounds is 5/32.

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