A Search Algorithm for Simplicial Complexes

TL;DR

This paper introduces the Basic S* algorithm, which computes shortest paths in simplicial complexes derived from metric graphs, providing more accurate geodesic approximations in configuration spaces without requiring global embedding information.

Contribution

The paper presents a novel shortest path algorithm for simplicial complexes that better approximates geodesic paths in configuration spaces using only local graph data.

Findings

The Basic S* algorithm closely approximates geodesic paths in the original metric space.

Its computational complexity is comparable to Dijkstra's algorithm.

It improves path accuracy in motion planning applications.

Abstract

We present the `Basic S*' algorithm for computing shortest path through a metric simplicial complex. In particular, given a metric graph, $G$, which is constructed as a discrete representation of an underlying configuration space (a larger "continuous" space/manifold typically of dimension greater than one), we consider the Rips complex, $\mathcal{R}(G)$, associated with it. Such a complex, and hence shortest paths in it, represent the underlying metric space more closely than what the graph does. While discrete graph representations of continuous spaces is convenient for motion planning in configuration spaces of robotic systems, the metric induced in them by the ambient configuration space is significantly different from the metric of the configuration space itself. We remedy this problem using the simplicial complex representation. Our algorithm requires only an abstract graph, $G=(V,E)$, and a cost/length function, $d:E\rightarrow \mathbb{R}_+$, as inputs, and no global information such as an embedding or a global coordinate chart is required. The complexity of the Basic S* algorithm is comparable to that of Dijkstra's search, but, as the results presented in this paper demonstrate, the shortest paths obtained using the proposed algorithm represent/approximate the geodesic paths in the original metric space significantly more closely.