Finite-Length Analysis of Caching-Aided Coded Multicasting

We study a noiseless broadcast link serving K users whose requests arise from a library of N files. Every user is equipped with a cache of size M files each. It has been shown that by splitting all the files into packets and placing individual packets in a random independent manner across all the caches prior to any transmission, at most N/M file transmissions are required for any set of demands from the library. The achievable delivery scheme involves linearly combining packets of different files following a greedy clique cover solution to the underlying index coding problem. This remarkable multiplicative gain of random placement and coded delivery has been established in the asymptotic regime when the number of packets per file F scales to infinity. The asymptotic coding gain obtained is roughly t = K M/N. In this paper, we initiate the finite-length analysis of random caching schemes when the number of packets F is a function of the system parameters M, N, and K. Specifically, we show that the existing random placement and clique cover delivery schemes that achieve optimality in the asymptotic regime can have at most a multiplicative gain of 2 even if the number of packets is exponential in the asymptotic gain t = K(M/N). Furthermore, for any clique cover-based coded delivery and a large class of random placement schemes that include the existing ones, we show that the number of packets required to get a multiplicative gain of (4/3)g is at least O((g/K)(N/M)(g-1)). We design a new random placement and an efficient clique cover-based delivery scheme that achieves this lower bound approximately. We also provide tight concentration results that show that the average (over the random placement involved) number of transmissions concentrates very well requiring only a polynomial number of packets in the rest of the system parameters.