Placement Delivery Arrays for Coded Caching with Shared and Private Caches
K. K. Krishnan Namboodiri, Elizabath Peter, B. Sundar Rajan
TL;DR
The paper tackles coded caching in a network with Λ helper caches and per-user private caches, where the server knows the user-to-cache associations. It introduces Shared and Private Placement Delivery Arrays (SP-PDAs) to jointly describe placement in helper/private caches and the server’s transmissions, and develops SP-PDAs from two PDAs to exploit structure and permutations. A key theoretical contribution is a construction that yields a SP-PDA with parameters F, Z, Z^(h), and S, achieving a rate R = S/F while controlling memory allocations via M_h/N = Z^(h)/F and M_p/N = (Z−Z^(h))/F. The work demonstrates that column permutations of component PDAs can significantly affect performance, provides selection criteria to minimize S, and shows that SP-PDAs subsume existing PNR3 schemes while enabling much smaller subpacketization with only modest rate loss, thus offering practical gains for caching in networks with shared and private caches.
Abstract
We consider a coded caching network consisting of a server with a library of $N$ files connected to $K$ users, where each user is equipped with a dedicated cache of size $M_p$ units. In addition to that, the network consists of $Λ\leq K$ helper caches, each with a size $M_h$ units. Each helper cache can serve an arbitrary number of users; however, each user can access only a single helper cache. Also, we assume that the server knows the user-to-helper cache association, defined as the sets of users connected to each helper cache, during the cache placement phase. We propose a solution for the aforementioned coded caching problem by introducing a combinatorial structure called a Shared and Private Placement Delivery Array (SP-PDA). These SP-PDAs describe the helper cache placement, private cache placement, and the server transmissions in a single array. Further, we propose a novel construction of SP-PDAs using two Placement Delivery Arrays (PDAs). Interestingly, we observe that the permutations of the columns of the two chosen PDAs result in SP-PDAs with different performances. Moreover, we characterize the conditions for selecting the best column permutations of the chosen PDAs. Furthermore, the coded caching schemes resulting from SP-PDAs subsume two existing coded caching schemes as special cases. Additionally, SP-PDAs enable the construction of coded caching schemes with much smaller subpacketization numbers -subpacketization number is defined as the number of subfiles to which a file is divided- compared to the existing schemes, without paying much in terms of rate (the size of the transmission in the delivery phase).