We establish tight bi-Lipschitz bounds certifying quasi-universality (universality up to a constant factor) for various distances between Reeb graphs: the interleaving distance, the functional distortion distance, and the functional contortion distance. The definition of the latter distance is a novel contribution, and for the special case of contour trees we also prove strict universality of this distance. Furthermore, we prove that for the special case of merge trees the functional contortion distance coincides with the interleaving distance, yielding universality of all four distances in this case.

翻译：我们建立了紧的双Lipschitz界，证明了Reeb图之间多种距离的拟普适性（在常数因子意义下的普适性）：交织距离、函数失真距离和函数扭曲距离。其中，最后一种距离的定义是一项新的贡献，并且对于轮廓树的特例，我们证明了该距离的严格普适性。此外，我们证明了对合并树的特例，函数扭曲距离与交织距离一致，从而在此情况下实现了所有四种距离的普适性。