The EDGE Language: Extended General Einsums for Graph Algorithms

TL;DR

The paper introduces EDGE, a unified tensor algebra-based language for expressing graph algorithms, aiming to simplify comparison, implementation, and discovery of algorithm variants.

Contribution

It presents the design of EDGE, an extended Einsum notation supporting complex graph operations, enabling algebraic manipulation and clearer algorithm expression.

Findings

EDGE provides a rigorous mathematical framework for graph algorithms.

The notation allows expressing complex graph operations succinctly.

Potential to facilitate algorithm comparison and variant discovery.

Abstract

In this work, we propose a unified abstraction for graph algorithms: the Extended General Einsums language, or EDGE. The EDGE language expresses graph algorithms in the language of tensor algebra, providing a rigorous, succinct, and expressive mathematical framework. EDGE leverages two ideas: (1) the well-known foundations provided by the graph-matrix duality, where a graph is simply a 2D tensor, and (2) the power and expressivity of Einsum notation in the tensor algebra world. In this work, we describe our design goals for EDGE and walk through the extensions we add to Einsums to support more complex operations common in graph algorithms. Additionally, we provide a few examples of how to express graph algorithms in our proposed notation. We hope that a single, mathematical notation for graph algorithms will (1) allow researchers to more easily compare different algorithms and different implementations of a graph algorithm; (2) enable developers to factor complexity by separating the concerns of what to compute (described with the extended Einsum notation) from the lower level details of how to compute; and (3) enable the discovery of different algorithmic variants of a problem through algebraic manipulations and transformations on a given EDGE expression.