Computing the Gromov-Hausdorff Distance for Metric Trees

Abstract

The Gromov-Hausdorff (GH) distance is a natural way to measure distance between two metric spaces. We prove that it is NP-hard to approximate the Gromov-Hausdorff distance better than a factor of 3 for geodesic metrics on a pair of trees. We complement this result by providing a polynomial time O(min{n, √ rn})approximation algorithm for computing the GH distance between a pair of metric trees, where r is the ratio of the longest edge length in both trees to the shortest edge length. For metric trees with unit length edges, this yields an O( √ n)-approximation algorithm.The Gromov-Hausdorff distance (or GH distance for brevity) [10] is one of the most natural distance measures between metric spaces, and has been used, for example, for matching deformable shapes [3,15], and for analyzing hierarchical clustering trees [5]. Informally, the Gromov-Hausdorff distance measures the additive distortion suffered when mapping one metric space to another using a correspondence between their points. Multiple approaches have been proposed to estimate the Gromov-Hausdorff distance [3,14,15]. Despite much effort, the problem of computing, either exactly or approximately, GH distance has remained elusive. The problem is not known to be NP-hard, and computing the GH distance, even approximately, for graphic metrics 1 is at least as hard as the graph isomorphism problem. Indeed, the metrics for two graphs have GH distance 0 if and only if the two graphs are isomorphic. Motivated by this trivial hardness result, it is natural to ask whether GH distance becomes easier in more restrictive settings such as geodesic metrics over trees, where efficient algorithms are known for checking isomorphism [1].Related work. Most work on associating points between two metric spaces involves embedding a given high dimensional metric space into an infinite host space of lower dimensional metric spaces. However, there is some work on finding a bijection between points in two given finite metric spaces that minimizes typically multiplicative distortion of distances between points and their images, with some limited results on additive distortion. Kenyon et al. [13] give an optimal algorithm for minimizing the multiplicative distortion of a bijection between two equal-sized finite metric spaces, and a parameterized polynomial time algorithm that finds the optimal bijection between an arbitrary unweighted graph metric and a bounded-degree tree metric.Papadimitriou and Safra [17] show that it is NP-hard to approximate the multiplicative distortion of any bijection between two finite 3-dimensional point sets to within any additive constant or to a factor better than 3.Hall and Papadimitriou [11] discuss the additive distortion problem -given two equal-sized point sets S, T ⊂ R d , find the smallest ∆ such that there exists a bijection f :They show that it is NP-hard to approximate by a factor better than 3 in R 3 , and also give a 2-approximation for R 1 and a 5-approximation for the more general problem of embedding an arbitrary metric space onto R 1 . ...

Spatiotemporal Persistent Homology for Dynamic Metric Spaces

“…which is slightly more general than the condition in (1). (Definition 2.9), which exists by the assumption d dyn (γ X , γ Y ) < ε.…”

“…Remark 2.13. From Remark 2.12, we conclude that the computation of d dyn is in general not tractable: On the class of constant DMSs the metric d dyn reduces to the Gromov-Hausdorff distance, which leads to NP-hard problems [1,60,61].…”

“…In order to ensure that we are in a position to utilize the metric d I for comparing rank invariants of DMSs, we extend the domain of the rank invariant of a DMS to the poset R 6 × . Given any (v 1…”

“…, l − 1. Then, invoking R is an (ε/2)-tripod between γ X and γ Y (see (1)), together with assumption (18) and Remark 6.12, (1) If γ X ≡ (X , d X ), then the SLHC dendrogram θ(X , d X ) is encoded along any vertical ray, such as blue or red rays in the figure (Remark 6.19 (i)).…”

“…The rank function rk(M X ) of M X has also been extensively considered [17,19,51,58,59]. We observe that both of these functions (1) can themselves be computed in polynomial time, (2) can be compared to each other via the interleaving distance d Z I for integer-valued functions (see Section 3.2) and (3) are stable to perturbations of (X , d X (·)) under d dyn (Theorems 4. 4 and 4.5).…”