On the Generic Capacity of $K$-User Symmetric Linear Computation Broadcast

TL;DR

This paper determines the generic capacity of symmetric linear computation broadcast for large user numbers, showing it is achievable and significantly better than baseline schemes, with implications for index coding complexity.

Contribution

It establishes the asymptotic generic capacity of symmetric LCBC for large user counts, providing a tight characterization and demonstrating substantial gains over traditional methods.

Findings

The generic capacity is $C_g=1/\Delta_g$, achievable and asymptotically optimal.

Significant capacity gains over baseline schemes, scaling with number of users and data dimensions.

Capacity characterized within a factor of 2 for arbitrary user counts.

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 retrieve 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 $\mathbb{F}_{p^n}$ where $p^n$ 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 is the capacity of LCBC. The capacity is known for up to $K=3$ 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. Instead of the general setting (all instances), by focusing on the generic setting (almost all instances) this work shows that the generic capacity of the symmetric LCBC (where every user has $m'$ dimensions of side-information and $m$ dimensions of demand) for large number of users ($K>d$ suffices) is $C_g=1/\Delta_g$, where $\Delta_g=\min\left\{\max\{0,d-m'\}, \frac{dm}{m+m'}\right\}$, is the broadcast cost that is both achievable and unbeatable asymptotically almost surely for large $n$, among all LCBC instances with the given parameters $p,K,d,m,m'$. Relative to baseline schemes of random coding or separate transmissions, $C_g$ shows an extremal gain by a factor of $K$ as a function of number of users, and by a factor of $\approx d/4$ 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 $2$.