Non-Linear Function Computation Broadcast

TL;DR

The paper studies the K-user computation broadcast problem where a master with datasets $X_{[N]} \in \mathbb{F}_q$ broadcasts to $K$ users with side information $\mathcal{S}_i$ who demand functions $f_i(X_{[N]})$, aiming asymptotically lossless recovery. It develops a graph-based coding framework based on Körner's characteristic graphs and union graphs to encode dependencies among datasets, demands, and side information, and derives both achievability and converse bounds for the broadcast rate. The authors demonstrate general non-linear and linear demand results, with concrete 3-user examples showing rate reductions to as low as $1.5$ bits for Boolean non-linear demands and $1.42$ bits for linear demands, outperforming prior approaches. Overall, the work extends computation broadcast to general demands over finite fields, providing new rate bounds and practical coding schemes that leverage structure in data and side information to reduce communication costs in distributed computation settings.

Abstract

This work addresses the $K$-user computation broadcast problem consisting of a master node, that holds all datasets and users for a general class of function demands, including linear and non-linear functions, over finite fields. The master node sends a broadcast message to enable each of $K$ distributed users to compute its demanded function in an asymptotically lossless manner with user's side information. We derive bounds on the optimal $K$-user computation broadcast rate that allows the users to compute their demanded functions by capturing the structures of the computations and available side information. Our achievability scheme involves the design of a novel graph-based coding model to build a broadcast message to meet each user's demand, by leveraging the structural dependencies among the datasets, the user demands, and the side information of each user, drawing on K{ö}rner's characteristic graph framework. The converse uses the structures of the demands and the side information available at $K$ users to yield a tight lower bound on the broadcast rate. With the help of examples, we demonstrate our scheme achieves a better communication rate than the existing state of the art.

Figures (2)

• Figure 1: Computation broadcast model for $K$ users, each with side information $\mathcal{S}_i$ and demand $f_i$, for $i\in [K]$sun2019capacity.
• Figure 2: Individual characteristic graphs and broadcast graph for Example \ref{['ex-nonlinear-comp']}: (Left) $G_{f_1}$, (Middle left) $G_{f_2}$, and (Middle right) $G_{f_3}$, based on users' demands $f_1$, $f_2$, and $f_3$, and the side information $\mathcal{S}_i$ for each $i\in[3]$, respectively, and (Right) broadcast graph $G_{\cup_{i=1}^{3}f_{i}}$.

Theorems & Definitions (9)

• Definition 1: Source characteristic graphs OrlRoc2001
• Definition 2: Union graphkorner1986fredman
• Example 1: -user Boolean computation broadcast
• Example 2: -user linear computation broadcast yao2024capacity
• Theorem 1
• proof
• Proposition 1
• proof
• Proposition 2