Towards the Optimal Rate Memory Tradeoff in Caching With Coded Placement

Abstract

The idea of coded caching for content distribution networks was introduced by Maddah-Ali and Niesen, who considered the canonical <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(N, K)$ </tex-math></inline-formula> cache network in which a server with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> files satisfies the demands of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> users (each equipped with an independent cache of size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> ). The optimal rate memory tradeoff for demands where all files are requested by at least one user has been characterized only for small caches where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M\leq \frac {1}{K}$ </tex-math></inline-formula> and large caches where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M\geq N-\frac {N}{K}$ </tex-math></inline-formula> . For the case <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N \leq K \leq 2N-1$ </tex-math></inline-formula> , we derive new lower bounds for small and large caches and propose a new coded caching scheme for large caches. Along with the scheme proposed by Gómez-Vilardebó, this leads to a characterization of the optimal rate memory tradeoff for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M\leq \frac {1}{K}+\frac {1}{K(N-1)}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M\geq N-\frac {N}{K}-\frac {N-1}{K(K-1)}$ </tex-math></inline-formula> . For the case <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2N-1\leq K$ </tex-math></inline-formula> , we derive a new lower bound for large caches, which proves the optimality of the scheme proposed by Yu et al. and leads to a characterization of the optimal rate memory tradeoff for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M\geq N-\frac {2N}{K}$ </tex-math></inline-formula> . We also derive a new lower bound for small caches, which improves upon previously known lower bounds.