Private information retrieval (PIR) allows a user to retrieve a desired message out of K possible messages from N databases without revealing the identity of the desired message. There has been significant recent progress on understanding fundamental information-theoretic limits of PIR, and in particular the download cost of PIR for several variations. Majority of existing works however, assume the presence of replicated databases, each storing all the K messages. In this work, we consider the problem of PIR from storage constrained databases. Each database has a storage capacity of μKL bits, where K is the number of messages, L is the size of each message in bits, and μ ∈ [1/N, 1] is the normalized storage. In the storage constrained PIR problem, there are two key design questions: a) how to store content across each database under storage constraints; and b) construction of schemes that allow efficient PIR through storage constrained databases. The main contribution of this work is a general achievable scheme for PIR from storage constrained databases for any value of storage. In particular, for any (N, K), with normalized storage μ = t/N, where the parameter t can take integer values t ∈ {1, 2, ..., N}, we show that our proposed PIR scheme achieves a download cost of (1 + 1/t + 1/2 + ⋯ + 1/t K-1 ). The extreme case when μ = 1 (i.e., t = N) corresponds to the setting of replicated databases with full storage. For this extremal setting, our scheme recovers the information-theoretically optimal download cost characterized by Sun and Jafar as (1 + 1/N + ⋯ +1/N K-1 ). For the other extreme, when μ = 1/N (i.e., t = 1), the proposed scheme achieves a download cost of K. The most interesting aspect of the result is that for intermediate values of storage, i.e., 1/N <; μ <; 1, the proposed scheme can strictly outperform memory-sharing between extreme values of storage.
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The Capacity of Private Information Retrieval from Decentralized Uncoded Caching Databases
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 μ K L bits exist in the system. Each database independently chooses μ K L bits out of the total K L 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 * = ∑ 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 results 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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- Award ID(s):
- 1713977
- NSF-PAR ID:
- 10201781
- Date Published:
- Journal Name:
- Information
- Volume:
- 10
- Issue:
- 12
- ISSN:
- 2078-2489
- Page Range / eLocation ID:
- 372
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/info10120372
