Path Optimization Sheaves

TL;DR

This paper reinterprets pathfinding algorithms, especially Dijkstra's, using cellular sheaves on graphs, providing a new mathematical framework that could unify and extend understanding of such algorithms.

Contribution

It introduces a sheaf-theoretic framework for pathfinding algorithms, including a general sheaf construction and a specific one modeling Dijkstra's decision process.

Findings

Dijkstra's algorithm can be viewed as extending local sections to global sections in a sheaf.

Two sheaves are constructed: a general one and a Dijkstra-specific one.

The framework suggests potential for unifying various pathfinding algorithms.

Abstract

Motivated by efforts to incorporate sheaves into networking, we seek to reinterpret pathfinding algorithms in terms of cellular sheaves, using Dijkstra's algorithm as an example. We construct sheaves on a graph with distinguished source and sink vertices, in which paths are represented by sections. The first sheaf is a very general construction that can be applied to other algorithms, while the second is created specifically to capture the decision making of Dijkstra's algorithm. In both cases, Dijkstra's algorithm can be described as a systematic process of extending local sections to global sections. We discuss the relationship between the two sheaves and summarize how other pathfinding algorithms can be interpreted in a similar way. While the sheaves presented here address paths and pathfinding algorithms, we suggest that future work could explore connections to other concepts from graph theory and other networking algorithms. This work was supported by the NASA Internship Project and SCaN Internship Project during the summer of 2020.