The Capacity of Private Information Retrieval from Uncoded Storage Constrained Databases
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Private information retrieval (PIR) allows a user to retrieve a desired message from a set of databases without revealing the identity of the desired message. The replicated databases scenario was considered by Sun and Jafar, 2016, where N databases can store the same K messages completely. A PIR scheme was developed to achieve the optimal download cost given by (1+ 1/N+ 1/N^2+ ... + 1/N^K-1). In this work, we consider the problem of PIR for uncoded storage constrained databases. Each database has a storage capacity of μ KL bits, where L is the size of each message in bits, and μ∈ [1/N, 1] is the normalized storage. On one extreme, μ=1 is the replicated databases case previously settled. On the other hand, when μ= 1/N, then in order to retrieve a message privately, the user has to download all the messages from the databases achieving a download cost of 1/K. We aim to characterize the optimal download cost versus storage trade-off for any storage capacity in the range μ∈ [1/N, 1]. In the storage constrained PIR problem, there are two key challenges: a) construction of communication efficient schemes through storage content design at each database that allow download efficient PIR; and b) characterizing the optimal download cost via information-theoretic lower bounds. The novel aspect of this work is to characterize the optimum download cost of PIR with storage constrained databases for any value of storage. In particular, for any (N,K), we show that the optimal trade-off between storage, μ, and the download cost, D(μ), is given by the lower convex hull of the N pairs (t/N, (1+ 1/t+ 1/t^2+ ... + 1/t^K-1)) for t=1,2,..., N.
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