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 extended metric spaces, where the new metric, defined on the same underlying space, is induced by the length of zigzag paths. This distance is then computed by isometrically embedding the directed metric spaces into an ambient directed space equipped with the zigzag distance. Analogously to the standard Gromov--Hausdorff distance, we also propose alternative formulations based on the distortion of d-maps and d-correspondences. However, unlike the classical case, these directed distances are not equivalent.