A Search Algorithm for Simplicial Complexes
This work addresses a specific problem in robotics motion planning, where accurate path representation in high-dimensional spaces is crucial, though it appears incremental by building on existing simplicial complex and graph search techniques.
The paper tackles the problem of approximating geodesic paths in continuous configuration spaces using discrete graph representations, which often distort the underlying metric, by introducing the Basic S* algorithm that computes shortest paths through simplicial complexes, resulting in significantly closer approximations to the true geodesics compared to traditional graph-based methods.
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.