Shortest Paths in a Weighted Simplicial Complex

TL;DR

This paper introduces the concept of weighted simplicial complexes and develops algorithms for shortest path computation within them, extending classical graph algorithms like Dijkstra's and Bellman-Ford to more complex topological structures.

Contribution

It defines weighted simplicial complexes and $d$-paths, and adapts fundamental shortest path algorithms to this new setting, bridging graph theory and algebraic topology.

Findings

Developed algorithms for shortest paths in weighted simplicial complexes

Extended classical graph algorithms to higher-dimensional structures

Provided foundational tools for applications in distributed computing and neural networks

Abstract

Simplicial complexes are extensively studied in the field of algebraic topology. They have gained attention in recent time due to their applications in fields like theoretical distributed computing and simplicial neural networks. Graphs are mono-dimensional simplicial complex. Graph theory has application in topics like theoretical computer science, operations research, bioinformatics and social sciences. This makes it natural to try to adapt graph-theoretic results for simplicial complexes, which can model more intricate and detailed structures appearing in real-world systems. Though seemingly obvious, we did not find any previous work that looked into this prospect of simplicial complexes. In this article, we define the concept of weighted simplicial complex and $d$-path in a simplicial complex. Both these concepts have the potential to have numerous real-life applications. We start by adapting the Depth-First Search and Breadth-First Search algorithms for our setup. Next, we provide two novel algorithms to find the shortest paths in a weighted simplicial complex. The core principles of our algorithms align with those of Dijkstra$^\prime$s algorithm and Bellman-Ford algorithm for graphs. Hence, this work lays a building block for the sake of integrating graph-theoretic concepts with abstract simplicial complexes.