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The Capacity of Private Information Retrieval from Decentralized Uncoded Caching Databases

27 November 2018
Yi-Peng Wei
Batuhan Arasli
Karim A. Banawan
S. Ulukus
ArXiv (abs)PDFHTML
Abstract

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 KKK files, where each file is of size LLL bits, and several databases with storage size constraint μKL\mu K LμKL bits exist in the system. Each database independently chooses μKL\mu K LμKL bits out of the total KLKLKL bits from the data center to cache through the same probability distribution in a decentralized manner. In the retrieval phase, a user (retriever) accesses NNN databases in addition to the data center, and wishes to retrieve a desired file privately. We characterize the optimal normalized download cost to be DL=∑n=1N+1(Nn−1)μn−1(1−μ)N+1−n(1+1n+⋯+1nK−1)\frac{D}{L} = \sum_{n=1}^{N+1} \binom{N}{n-1} \mu^{n-1} (1-\mu)^{N+1-n} \left( 1+ \frac{1}{n} + \dots+ \frac{1}{n^{K-1}} \right)LD​=∑n=1N+1​(n−1N​)μn−1(1−μ)N+1−n(1+n1​+⋯+nK−11​). 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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