Gromov--Hausdorff Distance for Directed Spaces

The Gromov--Hausdorff distance measures the similarity between two metric spaces by isometrically embedding them into an ambient metric space. In this work, we introduce an analogue of this distance for metric spaces endowed with directed structures. The directed Gromov--Hausdorff distance measures the distance between two new (extended) metric spaces, where the new metric, on the same underlying space, is induced from the length of the zigzag paths. This distance is then computed by isometrically embedding the directed spaces, endowed with the zigzag metric, into an ambient directed space with respect to such zigzag distance. Analogously to the standard Gromov--Hausdorff distance, we propose alternative definitions based on the distortion of d-maps and d-correspondences. Unlike the classical case, these directed distances are not equivalent.