Abstract
The Gromov-Hausdorff distance is a natural way to measure distance between two metric spaces. We give the first proof of hardness and first non-trivial approximation algorithm for computing the Gromov-Hausdorff distance for geodesic metrics in trees. Specifically, we prove it is \(\mathrm {NP}\)-hard to approximate the Gromov-Hausdorff distance better than a factor of 3. We complement this result by providing a polynomial time \(O(\min \{n, \sqrt{rn}\})\)-approximation algorithm 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 \(O(\sqrt{n})\)-approximation algorithm.
Work on this paper by P. K. Agarwal, K. Fox and A. Nath was supported by NSF under grants CCF-09-40671, CCF-10-12254, CCF-11-61359, and IIS-14-08846, and by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. A. Sidiropoulos was supported by NSF under grants CAREER-1453472 and CCF-1423230. Y. Wang was supported by NSF under grant CCF–1319406.
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Notes
- 1.
A graphic metric measures the shortest path distance between vertices of a graph with unit length edges.
- 2.
In fact, our hardness result can be easily extended to the GH distance between discrete tree metrics and the interleaving distance between merge trees.
References
Agarwal, P.K., Fox, K., Nath, A., Sidiropoulos, A., Wang, Y.: Computing the Gromov-Hausdorff distance for metric trees (2015). CoRR, abs/1509.05751
Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)
Bauer, U., Ge, X., Wang, Y.: Measuring distance between Reeb graphs. In: 30th Annual Symposium on Computational Geometry, p. 464 (2014)
Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Efficient computation of isometry-invariant distances between surfaces. SIAM J. Sci. Comput. 28(5), 1812–1836 (2006)
Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. American Mathematical Society, Providence (2001)
Carlsson, G., Mémoli, F.: Characterization, stability and convergence of hierarchical clustering methods. J. Mach. Learn. Res. 11, 1425–1470 (2010)
Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Disc. Comput. Geom. 37(1), 103–120 (2007)
de Silva, V., Munch, E., Patel, A.: Categorification of Reeb graphs, Preprint (2014)
Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser Basel, Basel (2007)
Hall, A., Papadimitriou, C.: Approximating the distortion. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 111–122. Springer, Heidelberg (2005)
Hopcroft, J.E., Karp, R.M.: An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comp. 2(4), 225–231 (1973)
Kenyon, C., Rabani, Y., Sinclair, A.: Low distortion maps between point sets. SIAM J. Comp. 39(4), 1617–1636 (2009)
Memoli, F.: On the use of Gromov-Hausdorff distances for shape comparison. In: Eurographics Symposium on Point-Based Graphics (2007)
Mémoli, F., Sapiro, G.: A theoretical and computational framework for isometry invariant recognition of point cloud data. Found. Comput. Math. 5(3), 313–347 (2005)
Morozov, D., Beketayev, K., Weber, G.H.: Interleaving distance between merge trees. In: Workshop on Topological Methods in Data Analysis and Visualization: Theory, Algorithms and Applications (2013)
Papadimitriou, C., Safra, S.: The complexity of low-distortion embeddings between point sets. In: 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 112–118 (2005)
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Agarwal, P.K., Fox, K., Nath, A., Sidiropoulos, A., Wang, Y. (2015). Computing the Gromov-Hausdorff Distance for Metric Trees. In: Elbassioni, K., Makino, K. (eds) Algorithms and Computation. ISAAC 2015. Lecture Notes in Computer Science(), vol 9472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48971-0_45
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