On lossy Compression of Directed Graphs

TL;DR

This paper extends the method of types to directed graphs for lossy compression, providing bounds on the rate-distortion trade-off using a new distortion measure that preserves local graph structure.

Contribution

It introduces a natural distortion measure for directed graphs and extends the method of types to analyze lossy compression in this context.

Findings

Derived lower and upper bounds on rate-distortion for directed graphs.

Extended the method of types to a two-dimensional setting.

Proposed a distortion measure that preserves local graph structure.

Abstract

The method of types presented by Csiszar and Korner is a central tool used to develop and analyze the basic properties and constraints on sequences of data over finite alphabets. A central problem considered using these tools is that of data compression, and specifically lossy data compression. In this work we consider this very problem, however, instead of sequences of data we consider directed graphs. We show that given a more natural distortion measure, fitting the data structure of a directed graph, the method of types cannot be applied. The suggested distortion measure aims to preserves the local structure of a directed graph. We build on the recent work of Barvinok and extend the method of types to the two dimensional setting of directed graphs. We see that the extension is quite natural in many ways. Given this extension we provide a lower and upper bound on the rate-distortion problem of lossy compression given the suggested distortion measure.