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We consider the problem of private information retrieval from N storage-constrained databases. In this problem, a user wishes to retrieve a single message out of M messages (of size L) without revealing any information about the identity of the message to individual databases. Each database stores µM L symbols, i.e., a µ fraction of the entire library, where 1 N ≤ µ ≤ 1. Our goal is to characterize the optimal tradeoff curve for the storage cost (captured by µ) and the normalized download cost (D/L). We show that the download cost can be reduced by employing a hybrid storage scheme that combines MDS coding ideas with uncoded partial replication ideas. When there is no coding, our scheme reduces to Attia-Kumar-Tandon storage scheme, which was initially introduced by Maddah-Ali-Niesen in the context of the caching problem, and when there is no uncoded partial replication, our scheme reduces to Banawan-Ulukus storage scheme; in general, our scheme outperforms both.

We consider the private information retrieval (PIR) problem from decentralized uncoded caching databases. There are two phases in our problem setting, a caching phase, and a retrieval phase. In the caching phase, a data center containing all the K files, where each file is of size L bits, and several databases with storage size constraint µKL bits exist in the system. Each database independently chooses µKL bits out of the total KL bits from the data center to cache through the same probability distribution in a decentralized manner. In the retrieval phase, a user (retriever) accesses N databases in addition to the data center, and wishes to retrieve a desired file privately. We characterize the optimal normalized download cost to be D L = N +1 n=1 N n−1 µ n−1 (1 − µ) N +1−n 1 + 1 n + · · · + 1 n K−1 . We show that uniform and random caching scheme which is originally proposed for decentralized coded caching by Maddah-Ali and Niesen, along with Sun and Jafar retrieval scheme which is originally proposed for PIR from replicated databases surprisingly result in the lowest normalized download cost. This is the decentralized counterpart of the recent result of Attia, Kumar and Tandon for the centralized case. The converse proof contains several ingredients such as interference lower bound, induction lemma, replacing queries and answering string random variables with the content of distributed databases, the nature of decentralized uncoded caching databases, and bit marginalization of joint caching distributions.

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