The Capacity of Private Information Retrieval From Heterogeneous Uncoded Caching Databases

We consider private information retrieval (PIR) of a single file out of <italic>K</italic> files from <italic>N</italic> non-colluding databases with <italic>heterogeneous storage constraints</italic> <inline-formula> <tex-math notation="LaTeX">${{m}}=({m}_{1}, \cdots,{m}_{{N}})$ </tex-math></inline-formula>. The aim of this work is to jointly design the content placement phase and the information retrieval phase in order to minimize the download cost in the PIR phase. We characterize the optimal PIR download cost as a linear program. By analyzing the structure of the optimal solution of this linear program, we show that, surprisingly, the optimal download cost in our heterogeneous case matches its homogeneous counterpart where all databases have the same average storage constraint <inline-formula> <tex-math notation="LaTeX">$\mu =\frac {1}{{N}} \sum _{{n}=1}^{N} {m}_{{n}}$ </tex-math></inline-formula>. Thus, we show that there is no loss in the PIR capacity due to heterogeneity of storage spaces of the databases. We provide the optimum content placement explicitly for <italic>N</italic> = 3.