Discrete & Computational Geometry

, Volume 55, Issue 4, pp 854–906 | Cite as

Categorified Reeb Graphs

  • Vin de Silva
  • Elizabeth Munch
  • Amit Patel


The Reeb graph is a construction which originated in Morse theory to study a real-valued function defined on a topological space. More recently, it has been used in various applications to study noisy data which creates a desire to define a measure of similarity between these structures. Here, we exploit the fact that the category of Reeb graphs is equivalent to the category of a particular class of cosheaf. Using this equivalency, we can define an ‘interleaving’ distance between Reeb graphs which is stable under the perturbation of a function. Along the way, we obtain a natural construction for smoothing a Reeb graph to reduce its topological complexity. The smoothed Reeb graph can be constructed in polynomial time.


Reeb graph Cosheaf Interleaving distance Stability Smoothing 



The work described in this article is a result of a collaboration made possible by the Institute for Mathematics and its Applications (IMA), University of Minnesota. It was carried out while the authors were at the IMA during the annual program 2013–14, on Scientific and Engineering Applications of Algebraic Topology. We thank the IMA staff for their outstanding support and help throughout the year, and we thank the organizing committee of the thematic year for putting together an excellent program. The IMA was funded at that time by the National Science Foundation. EM and AP have been supported by IMA Postdoctoral Fellowships during this project. VdS thanks his home institution, Pomona College, for granting him a sabbatical leave of absence in 2013–14. The sabbatical was hosted by the IMA, and the second semester of leave was supported by the Simons Foundation (grant #267571). The authors gratefully thank Ulrich Bauer, Justin Curry, Tamal Dey, Jeff Erickson, Robert MacPherson, Dmitriy Morozov, Sara Kališnik Verovšek, João Pita Costa, Primoz Škraba, Blair Sullivan, Mikael Vejdemo-Johansson, Yusu Wang, John Wilmes and Song Yu for many helpful discussions during the course of this work.


  1. 1.
    Acar, U.A., Blelloch, G.E., Harper, R., Vittes, J.L., Woo, S.L.M.: Dynamizing static algorithms, with applications to dynamic trees and history independence. In: ACM-SIAM Symposium on Discrete Algorithms, SODA ’04, pp. 531–540 (2004)Google Scholar
  2. 2.
    Agarwal, P.K., Edelsbrunner, H., Harer, J., Wang, Y.: Extreme elevation on a 2-manifold. In: Proceedings of the Twentieth Annual Symposium on Computational Geometry, SoCG ’04, pp. 357–365, New York, NY. ACM (2004)Google Scholar
  3. 3.
    Alstrup, S., Holm, J., De Lichtenberg, K., Thorup, M.: Maintaining information in fully dynamic trees with top trees. ACM Trans. Algorithms 1(2), 243–264 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bauer, U., Ge, X., Wang, Y.: Measuring distance between Reeb graphs. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SoCG ’14 (2014)Google Scholar
  5. 5.
    Bauer, U., Munch, E., Wang, Y.: Strong equivalence of the interleaving and functional distortion metrics for Reeb graphs. In: Arge, Lars, Pach, János (eds.) 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), vol. 34, pp. 461–475. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2015)Google Scholar
  6. 6.
    Biasotti, S., Falcidieno, B., Spagnuolo, M.: Extended Reeb graphs for surface understanding and description. In: Borgefors, G., Nyström, I., Baja, G.S. (eds.) Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 1952, pp. 185–197. Springer, Berlin (2000)CrossRefGoogle Scholar
  7. 7.
    Biasotti, S., Giorgi, D., Spagnuolo, M., Falcidieno, B.: Reeb graphs for shape analysis and applications. Theor. Comput. Sci. Comput. Algebr. Geom. Appl. 392(13), 5–22 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bubenik, P., Scott, J.A.: Categorification of persistent homology. Discrete Comput. Geom. 51(3), 600–627 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bubenik, P., de Silva, V., Scott, J.: Metrics for generalized persistence modules. Found. Comput. Math. 15(6), 1501–1531 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Burghelea, D., Dey, T.K.: Topological persistence for circle-valued maps. Discrete Comput. Geom. 50(1), 69–98 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Carlsson, G., de Silva, V., Morozov, D.: Zigzag persistent homology and real-valued functions. In: Proceedings 25th ACM Symposium on Computational Geometry (SoCG), pp. 247–256 (2009)Google Scholar
  12. 12.
    Carr, H., Snoeyink, J., Axen, U.: Computing contour trees in all dimensions. Comput. Geom. 24(2), 75–94 (2003). Special Issue on the Fourth CGC Workshop on Computational Geometry.
  13. 13.
    Chazal, F., Sun, J.: Gromov–Hausdorff approximation of filament structure using Reeb-type graph. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SOCG’14, pp. 491:491–491:500. ACM, New York, NY (2014)Google Scholar
  14. 14.
    Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.Y.: Proximity of persistence modules and their diagrams. In: Proceedings of the 25th annual symposium on computational geometry, SoCG ’09, pp. 237–246. ACM, New York, NY (2009)Google Scholar
  15. 15.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Extending persistence using Poincaré and Lefschetz duality. Found. Comput. Math. 9(1), 79–103 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cole-McLaughlin, K., Edelsbrunner, H., Harer, J., Natarajan, V., Pascucci, V.: Loops in Reeb graphs of 2-manifolds. In: Proceedings of the Nineteenth Annual Symposium on Computational Geometry, SoCG ’03, pp. 344–350. ACM, New York, NY (2003)Google Scholar
  18. 18.
    Coste, M.: An Introduction to o-minimal Geometry. Istituti Editoriali e Poligrafici Internazionali, Pisa (2000)Google Scholar
  19. 19.
    Curry, J: Sheaves, Cosheaves and Applications. PhD thesis, University of Pennsylvania (2014)Google Scholar
  20. 20.
    Dey, T.K., Fan, F., Wang, Y.: An efficient computation of handle and tunnel loops via Reeb graphs. ACM Trans. Graph. 32(4), 32:1–32:10 (2013)CrossRefzbMATHGoogle Scholar
  21. 21.
    Di Fabio, B., Landi, C.: The edit distance for Reeb graphs of surfaces. arXiv:1411.1544 (2014)
  22. 22.
    Doraiswamy, H., Natarajan, V.: Output-sensitive construction of Reeb graphs. IEEE Trans. Vis. Comput. Graph. 18(1), 146–159 (2012)CrossRefzbMATHGoogle Scholar
  23. 23.
    Edelsbrunner, H., Harer, J., Patel, A.: Reeb spaces of piecewise linear mappings. In: Proceedings of the Twenty-fourth Annual Symposium on Computational Geometry, SoCG ’08, pp. 242–250. ACM, New York, NY (2008)Google Scholar
  24. 24.
    Eppstein, D., Galil, Z., Italiano, G.F., Nissenzweig, A.: Sparsification: a technique for speeding up dynamic graph algorithms. J. ACM 44(5), 669–696 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Escolano, F., Hancock, E.R., Biasotti, S.: Complexity fusion for indexing reeb digraphs. In: Wilson, R., Hancock, E., Bors, A., Smith, W. (eds.) Computer Analysis of Images and Patterns. Lecture Notes in Computer Science, vol. 8047, pp. 120–127. Springer, Berlin (2013)CrossRefGoogle Scholar
  26. 26.
    Fox, R.H.: Covering spaces with singularities. In: Algebraic Geometry and Topology: A Symposium in Honor of S. Lefschetz, pp. 243–257 (1957)Google Scholar
  27. 27.
    Funk, J.: The display locale of a cosheaf. Cah. Topologie et Géom. Différ. Catégoriques 36(1), 53–93 (1995)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Ge, X., Safa, I.I., Belkin, M., Wang, Y.: Data skeletonization via Reeb graphs. In: Shawe-Taylor, J., Zemel, R.S., Bartlett, P., Pereira, F.C.N., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems, vol. 24, pp. 837–845 (2011)Google Scholar
  29. 29.
    Grothendieck, A.: A General Theory of Fibre Spaces With Structure Sheaf. University of Kansas, Lawrence, KS (1955)Google Scholar
  30. 30.
    Harvey, W., Wang, Y.: Topological landscape ensembles for visualization of scalar-valued functions. In: Proceedings of the 12th Eurographics/IEEE—VGTC Conference on Visualization, EuroVis’10, pp. 993–1002, Aire-la-Ville, Switzerland. Eurographics Association (2010)Google Scholar
  31. 31.
    Harvey, W., Wang, Y., Wenger, R.: A randomized \(O(m \log m)\) time algorithm for computing Reeb graphs of arbitrary simplicial complexes. In: Proceedings of the 2010 Annual Symposium on Computational Geometry, SoCG ’10, pp. 267–276. ACM, New York, NY (2010)Google Scholar
  32. 32.
    Hilaga, M., Shinagawa, Y., Kohmura, T., Kunii, T.L.: Topology matching for fully automatic similarity estimation of 3D shapes. In: Proceedings of the 28th Annual Conference on Computer graphics and Interactive Techniques, SIGGRAPH ’01, pp. 203–212. ACM, New York, NY (2001)Google Scholar
  33. 33.
    Johnstone, P.T.: Stone Spaces. Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press, Cambridge (1986)Google Scholar
  34. 34.
    Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Springer, Berlin (1990)CrossRefzbMATHGoogle Scholar
  35. 35.
    Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, Berlin (1998)Google Scholar
  36. 36.
    Mather, J.: Notes on topological stability. Bull. AMS 49(4) (2012)Google Scholar
  37. 37.
    Mendelson, B.: Introduction to Topology. Allyn and Bacon Inc, Newton, MA (1975)zbMATHGoogle Scholar
  38. 38.
    Morozov, D., Beketayev, K., Weber, G.: Interleaving distance between merge trees. In: Proceedings of TopoInVis (2013)Google Scholar
  39. 39.
    Nicolau, M., Levine, A.J., Carlsson, G.: Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proc. Natl. Acad. Sci. 108(17), 7265–7270 (2011)CrossRefGoogle Scholar
  40. 40.
    Parsa, S.: A deterministic \(O(m \log m)\) time algorithm for the Reeb graph. In: Proceedings of the 28th annual ACM symposium on Computational geometry, SoCG ’12. ACM (2012)Google Scholar
  41. 41.
    Pascucci, V., Cole-McLaughlin, K.: Parallel computation of the topology of level sets. Algorithmica 38(1), 249–268 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Pascucci, V., Scorzelli, G., Bremer, P.-T., Mascarenhas, A.: Robust on-line computation of Reeb graphs: simplicity and speed. ACM Trans. Graph. 26(3) (2007)Google Scholar
  43. 43.
    Reeb, G.: Sur les points singuliers d’une forme de Pfaff complèment intégrable ou d’une fonction numérique. C. R. L’Acad. Sci. 222, 847–849 (1946)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Shinagawa, Y., Kunii, T.L., Kergosien, Y.L.: Surface coding based on Morse theory. IEEE Comput. Graph. Appl. 11(5), 66–78 (1991)CrossRefGoogle Scholar
  45. 45.
    Singh, G., Mémoli, F., Carlsson, G.: Topological methods for the analysis of high dimensional data sets and 3D object recognition. In: Eurographics Symposium on Point-Based Graphics (2007)Google Scholar
  46. 46.
    Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26(3), 362–391 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Sleator, D.D., Tarjan, R.E.: Self-adjusting binary search trees. J. ACM 32(3), 652–686 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Szymczak, A.: A categorical approach to contour, split and join trees with application to airway segmentation. In: Pascucci, V., Tricoche, H., Hagen, H., Tierny, J. (eds.) Topological Methods in Data Analysis and Visualization, pp. 205–216. Springer, Berlin (2011)CrossRefGoogle Scholar
  49. 49.
    Tarjan, R.E., Werneck, R.F.: Self-adjusting top trees. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’05, pp. 813–822. Society for Industrial and Applied Mathematics, Philadelphia, PA (2005)Google Scholar
  50. 50.
    Treumann, D.: Exit paths and constructible stacks. Compos. Math. 145, 1504–1532 (2009). 11MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Tucker, A.W.: Branched and folded coverings. Bull. Am. Math. Soc 42(12), 859–862 (1936). 12MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    van den Dries, L.: Tame Topology and O-minimal Structures. London Mathematical Society Lecture Note Series, vol. 248. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  53. 53.
    van Kreveld, M., van Oostrum, R., Bajaj, C., Pascucci, V., Schikore, D.: Contour trees and small seed sets for isosurface traversal. In: Proceedings of the Thirteenth Annual Symposium on Computational Geometry, SoCG ’97, pp. 212–220. ACM, New York, NY (1997)Google Scholar
  54. 54.
    Weber, G.H., Bremer, P.-T., Pascucci, V.: Topological landscapes: a terrain metaphor for scientific data. IEEE Trans. Vis. Comput. Graph. 13(6), 1416–1423 (2007)CrossRefGoogle Scholar
  55. 55.
    Wood, Z., Hoppe, H., Desbrun, M., Schröder, P.: Removing excess topology from isosurfaces. ACM Trans. Graph. 23(2), 190–208 (2004)CrossRefGoogle Scholar
  56. 56.
    Woolf, J.: The fundamental category of a stratified space. J. Homotopy Relat. Struct. 4(1), 359–387 (2009)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Yao, Y., Sun, J., Huang, X., Bowman, G.R., Singh, G., Lesnick, M., Guibas, L.J., Pande, V.S., Carlsson, G.: Topological methods for exploring low-density states in biomolecular folding pathways. J. Chem. Phys. 130(14), 144115 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.ClaremontUSA
  2. 2.AlbanyUSA
  3. 3.PrincetonUSA

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