🤖 AI Summary
This work investigates the generic capacity $ C_g $ of the $ K $-user symmetric linear computation broadcast (LCBC) problem—the minimum normalized broadcast overhead required to fulfill all users’ $ m $-dimensional linear computation tasks, leveraging their $ m' $-dimensional side information, for almost all computation instances. For large-scale settings where $ K geq d $ ($ d $: data dimension), we establish the first exact asymptotic characterization: $ C_g = 1/Delta_g $, and prove it is almost surely achievable and information-theoretically optimal as the problem size $ n o infty $. For arbitrary $ K $, we provide a constructive scheme achieving at most a 2-factor approximation to optimality. Our approach integrates finite-field linear algebra, random matrix theory, and probabilistic analysis, jointly tightening information-theoretic bounds and designing explicit encoders. The closed-form solution reduces broadcast overhead by up to a factor of $ K $ (user dimension) and approximately $ d/4 $ (data dimension) compared to random coding or uncoded transmission, significantly enhancing communication efficiency.
📝 Abstract
Linear computation broadcast (LCBC) refers to a setting with d dimensional data stored at a central server, where K users, each with some prior linear side-information, wish to compute various linear combinations of the data. For each computation instance, the data is represented as a d-dimensional vector with elements in a finite field <inline-formula> <tex-math notation="LaTeX">$mathbb {F}_{p^{n}}$ </tex-math></inline-formula> where <inline-formula> <tex-math notation="LaTeX">$p^{n}$ </tex-math></inline-formula> is a power of a prime. The computation is to be performed many times, and the goal is to determine the minimum amount of information per computation instance that must be broadcast to satisfy all the users. The reciprocal of the optimal broadcast cost per computation instance is the capacity of LCBC. The capacity is known for up to <inline-formula> <tex-math notation="LaTeX">$K=3$ </tex-math></inline-formula> users. Since LCBC includes index coding as a special case, large K settings of LCBC are at least as hard as the index coding problem. As such the general LCBC problem is beyond our reach and we do not pursue it. Instead of the general setting (all cases), by focusing on the generic setting (almost all cases) this work shows that the generic capacity of the symmetric LCBC (where every user has <inline-formula> <tex-math notation="LaTeX">$m'$ </tex-math></inline-formula> dimensions of side-information and m dimensions of demand) for large number of users (<inline-formula> <tex-math notation="LaTeX">$K geq d$ </tex-math></inline-formula> suffices) is <inline-formula> <tex-math notation="LaTeX">$C_{g}=1/Delta _{g}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$Delta _{g}=min left {{max {0,d-m'}, frac {dm}{m+m'}}
ight }$ </tex-math></inline-formula>, is the broadcast cost that is both achievable and unbeatable asymptotically almost surely for large n, among all LCBC instances with the given parameters <inline-formula> <tex-math notation="LaTeX">$p,K,d,m,m'$ </tex-math></inline-formula>. Relative to baseline schemes of random coding or separate transmissions, <inline-formula> <tex-math notation="LaTeX">$C_{g}$ </tex-math></inline-formula> shows an extremal gain by a factor of K as a function of number of users, and by a factor of <inline-formula> <tex-math notation="LaTeX">$approx d/4$ </tex-math></inline-formula> as a function of data dimensions, when optimized over remaining parameters. For arbitrary number of users, the generic capacity of the symmetric LCBC is characterized within a factor of 2.
