Distributed Compression for Computation and Bounds on the Optimal Rate

2025-04-22

AI Summary

This paper investigates the lossless computation of arbitrary functions of two correlated sources $X_1$ and $X_2$ in distributed systems, aiming to characterize the optimal communication rate region. To model computational dependencies, we introduce the *source characteristic graph* and unify the compression limit via the $n$-fold OR graph product and chromatic entropy. Our contributions are threefold: (i) we derive tight characterizations of the optimal rate for both regular and general graphs; (ii) for cycle graphs, we obtain an exact chromatic-number-based rate expression; (iii) leveraging spectral graph theory and the Gershgorin Circle Theorem, we reveal an intrinsic connection between graph expansion properties and achievable computation rates. The results yield asymptotically tight upper and lower bounds on the optimal rate for arbitrary finite-field-valued functions, provide closed-form analytical expressions for cycles and $d$-regular graphs, and significantly advance both compression efficiency and theoretical tractability in distributed function computation.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Computational social science
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We address the problem of distributed computation of arbitrary functions of two correlated sources $X_1$ and $X_2$, residing in two distributed source nodes, respectively. We exploit the structure of a computation task by coding source characteristic graphs (and multiple instances using the $n$-fold OR product of this graph with itself). For regular graphs and general graphs, we establish bounds on the optimal rate -- characterized by the chromatic entropy for the $n$-fold graph products -- that allows a receiver for asymptotically lossless computation of arbitrary functions over finite fields. For the special class of cycle graphs (i.e., $2$-regular graphs), we establish an exact characterization of chromatic numbers and derive bounds on the required rates. Next, focusing on the more general class of $d$-regular graphs, we establish connections between $d$-regular graphs and expansion rates for $n$-fold graph powers using graph spectra. Finally, for general graphs, we leverage the Gershgorin Circle Theorem (GCT) to provide a characterization of the spectra, which allows us to build new bounds on the optimal rate. Our codes leverage the spectra of the computation and provide a graph expansion-based characterization to efficiently/succinctly capture the computation structure, providing new insights into the problem of distributed computation of arbitrary functions.

Problem

• Distributed computation of arbitrary functions of correlated sources.
• Bounds on optimal rate for lossless computation using graph theory.
• Characterization of chromatic numbers and spectra for regular graphs.

Innovation

• Uses characteristic graphs for distributed computation
• Leverages graph spectra for rate bounds
• Applies Gershgorin Circle Theorem for spectra