On the Placement Delivery Array Design for Centralized Coded Caching Scheme

Caching is a promising solution to satisfy the ever-increasing demands for the multi-media traffics. In caching networks, coded caching is a recently proposed technique that achieves significant performance gains over the uncoded caching schemes. However, to implement the coded caching schemes, each file has to be split into <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> packets, which usually increases exponentially with the number of users <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>. Thus, designing caching schemes that decrease the order of <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> is meaningful for practical implementations. In this paper, by reviewing the Ali-Niesen caching scheme, the placement delivery array (PDA) design problem is first formulated to characterize the placement issue and the delivery issue with a single array. Moreover, we show that, through designing appropriate PDA, new centralized coded caching schemes can be discovered. Second, it is shown that the Ali-Niesen scheme corresponds to a special class of PDA, which realizes the best coding gain with the least <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula>. Third, we present a new construction of PDA for the centralized coded caching system, wherein the cache size <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> at each user (identical cache size is assumed at all users) and the number of files <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> satisfies <inline-formula> <tex-math notation="LaTeX">$M/N=1/q$ </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">${(q-1)}/{q}$ </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> is an integer, such that <inline-formula> <tex-math notation="LaTeX">$q\geq 2$ </tex-math></inline-formula>). The new construction can decrease the required <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> from the order <inline-formula> <tex-math notation="LaTeX">$O(e^{K\cdot (({M}/{N})\ln ({N}/{M}) +(1-({M}/{N}))\ln ~(N/(N-M))})$ </tex-math></inline-formula> of Ali-Niesen scheme to <inline-formula> <tex-math notation="LaTeX">$O(e^{K\cdot ({M}/{N})\ln \!\,({N}/\!{M})})$ </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">$O(e^{K\cdot (1-({M}/\!{N}))\ln \!\,(N/(N-M))})$ </tex-math></inline-formula>, respectively, while the coding gain loss is only 1.