Weighted Graph Coloring for Quantized Computing

TL;DR

This paper introduces a novel weighted graph coloring approach for distributed lossless computation of functions, leveraging edge weights to improve compression efficiency beyond traditional methods.

Contribution

It develops the concept of edge-weighted characteristic graphs and demonstrates how fractional coloring and edge weights can significantly enhance distributed source coding.

Findings

Achieves over 30% compression gains compared to traditional coloring schemes.

Provides a fundamental limit characterization for the proposed compression setup.

Validates the approach through an illustrative example.

Abstract

We consider the problem of distributed lossless computation of a function of two sources by one common user. To do so, we first build a bipartite graph, where two disjoint parts denote the individual source outcomes. We then project the bipartite graph onto each source to obtain an edge-weighted characteristic graph (EWCG), where edge weights capture the function's structure, by how much the source outcomes are to be distinguished, generalizing the classical notion of characteristic graphs. Via exploiting the notions of characteristic graphs, the fractional coloring of such graphs, and edge weights, the sources separately build multi-fold graphs that capture vector-valued source sequences, determine vertex colorings for such graphs, encode these colorings, and send them to the user that performs minimum-entropy decoding on its received information to recover the desired function in an asymptotically lossless manner. For the proposed EWCG compression setup, we characterize the fundamental limits of distributed compression, verify the communication complexity through an example, contrast it with traditional coloring schemes, and demonstrate that we can attain compression gains higher than $\% 30$ over traditional coloring.