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Fast Deterministic Algorithms for Computing All Eccentricities in (Hyperbolic) Helly Graphs

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Algorithms and Data Structures (WADS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12808))

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Abstract

A graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. The class of Helly graphs is the discrete analogue of the class of hyperconvex metric spaces. It is also known that every graph isometrically embeds into a Helly graph, making the latter an important class of graphs in Metric Graph Theory. We study diameter, radius and all eccentricity computations within the Helly graphs. Under plausible complexity assumptions, neither the diameter nor the radius can be computed in truly subquadratic time on general graphs. In contrast to these negative results, it was recently shown that the radius and the diameter of an n-vertex m-edge Helly graph G can be computed with high probability in \(\tilde{\mathcal O}(m\sqrt{n})\) time (i.e., subquadratic in \(n+m\)). In this paper, we improve that result by presenting a deterministic \({\mathcal O}(m\sqrt{n})\) time algorithm which computes not only the radius and the diameter but also all vertex eccentricities in a Helly graph. Furthermore, we give a parameterized linear-time algorithm for this problem on Helly graphs, with the parameter being the Gromov hyperbolicity \(\delta \). More specifically, we show that the radius and a central vertex of an m-edge \(\delta \)-hyperbolic Helly graph G can be computed in \(\mathcal O(\delta m)\) time and that all vertex eccentricities in G can be computed in \(\mathcal O(\delta ^2 m)\) time. To show this more general result, we heavily use our new structural properties obtained for Helly graphs.

This work was supported by project PN 19 37 04 01 “New solutions for complex problems in current ICT research fields based on modelling and optimization”, funded by the Romanian Core Program of the Ministry of Research and Innovation (MCI), 2019–2022.

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References

  1. Abboud, A., Vassilevska Williams, V., Wang, J.: Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In: SODA, pp. 377–391. SIAM (2016)

    Google Scholar 

  2. Abu-Ata, M., Dragan, F.F.: Metric tree-like structures in real-world networks: an empirical study. Networks 67(1), 49–68 (2016)

    Article  MathSciNet  Google Scholar 

  3. Bandelt, H.-J., Chepoi, V.: Metric graph theory and geometry: a survey. Contemp. Math. 453, 49–86 (2008)

    Article  MathSciNet  Google Scholar 

  4. Bandelt, H.-J., Pesch, E.: Dismantling absolute retracts of reflexive graphs. Eur. J. Comb. 10(3), 211–220 (1989)

    Article  MathSciNet  Google Scholar 

  5. Borassi, M., Coudert, D., Crescenzi, P., Marino, A.: On computing the hyperbolicity of real-world graphs. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 215–226. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48350-3_19

    Chapter  Google Scholar 

  6. Borassi, M., Crescenzi, P., Habib, M.: Into the square: on the complexity of some quadratic-time solvable problems. Electron. Notes TCS 322, 51–67 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Brandstädt, A., Chepoi, V., Dragan, F.F.: The algorithmic use of hypertree structure and maximum neighbourhood orderings. DAM 82(1–3), 43–77 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Brandstädt, A., Dragan, F.F., Chepoi, V., Voloshin, V.: Dually chordal graphs. SIDMA 11(3), 437–455 (1998)

    Article  MathSciNet  Google Scholar 

  9. Bringmann, K., Husfeldt, T., Magnusson, M.: Multivariate analysis of orthogonal range searching and graph distances parameterized by treewidth. In: IPEC (2018)

    Google Scholar 

  10. Brinkmann, G., Koolen, J., Moulton, V.: On the hyperbolicity of chordal graphs. Ann. Comb. 5(1), 61–69 (2001)

    Article  MathSciNet  Google Scholar 

  11. Cabello, S.: Subquadratic algorithms for the diameter and the sum of pairwise distances in planar graphs. ACM TALG 15(2), 21 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Chalopin, J., Chepoi, V., Dragan, F.F., Ducoffe, G., Mohammed, A., Vaxès, Y.: Fast approximation and exact computation of negative curvature parameters of graphs. In: SoCG 2018, pp. 22:1–22:15 (2018)

    Google Scholar 

  13. Chalopin, J., Chepoi, V., Genevois, A., Hirai, H., Osajda, D.: Helly groups (2020)

    Google Scholar 

  14. Chalopin, J., Chepoi, V., Hirai, H., Osajda, D.: Weakly modular graphs and nonpositive curvature. Mem. Amer. Math. Soc. 159 (2020)

    Google Scholar 

  15. Chepoi, V., Dragan, F.F.: Disjoint sets problem (1992)

    Google Scholar 

  16. Chepoi, V., Dragan, F.: A linear-time algorithm for finding a central vertex of a chordal graph. In: van Leeuwen, J. (ed.) ESA 1994. LNCS, vol. 855, pp. 159–170. Springer, Heidelberg (1994). https://doi.org/10.1007/BFb0049406

    Chapter  Google Scholar 

  17. Chepoi, V., Dragan, F.F., Estellon, B., Habib, M., Vaxès, Y.: Diameters, centers, and approximating trees of \(\delta \)-hyperbolic geodesic spaces and graphs. In: SoCG 2008, pp. 59–68. ACM (2008)

    Google Scholar 

  18. Chepoi, V., Dragan, F.F.F., Habib, M., Vaxès, Y., Alrasheed, H.: Fast approximation of eccentricities and distances in hyperbolic graphs. J. Graph Algorithms Appl. 23(2), 393–433 (2019)

    Article  MathSciNet  Google Scholar 

  19. Corneil, D., Dragan, F.F., Habib, M., Paul, C.: Diameter determination on restricted graph families. DAM 113(2–3), 143–166 (2001)

    MathSciNet  MATH  Google Scholar 

  20. Coudert, D., Ducoffe, G.: Recognition of c\({}_{\text{4 }}\)-free and 1/2-hyperbolic graphs. SIDMA 28(3), 1601–1617 (2014)

    Article  MathSciNet  Google Scholar 

  21. Coudert, D., Ducoffe, G., Popa, A.: Fully polynomial FPT algorithms for some classes of bounded clique-width graphs. ACM TALG 15(3) (2019)

    Google Scholar 

  22. Dourado, M., Protti, F., Szwarcfiter, J.: Complexity aspects of the Helly property: graphs and hypergraphs. EJC 1000, 17-12 (2009)

    Google Scholar 

  23. Dragan, F.F.: Centers of Graphs and the Helly Property. Ph.D. thesis, Moldava State University, Chişinău (1989). (in Russian)

    Google Scholar 

  24. Dragan, F.F.: Conditions for coincidence of local and global minima for eccentricity function on graphs and the Helly property. Stud. Appl. Math. Inf. Sci. 49–56 (1990). (in Russian)

    Google Scholar 

  25. Dragan, F.F.: HT-graphs: centers, connected r-domination and Steiner trees. Comput. Sci. J. Moldova (Kishinev) 1(2), 64–83 (1993)

    MathSciNet  Google Scholar 

  26. Dragan, F.F.: Dominating cliques in distance-hereditary graphs. In: Schmidt, E.M., Skyum, S. (eds.) SWAT 1994. LNCS, vol. 824, pp. 370–381. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-58218-5_34

    Chapter  Google Scholar 

  27. Dragan, F.F., Ducoffe, G., Guarnera, H.M.: Fast deterministic algorithms for computing all eccentricities in (hyperbolic) Helly graphs. CoRR, arXiv:2102.08349 (2021)

  28. Dragan, F.F., Guarnera, H.M.: Obstructions to a small hyperbolicity in Helly graphs. Discret. Math. 342(2), 326–338 (2019)

    Article  MathSciNet  Google Scholar 

  29. Dragan, F.F., Guarnera, H.M.: Eccentricity function in distance-hereditary graphs. Theor. Comput. Sci. 833, 26–40 (2020)

    Article  MathSciNet  Google Scholar 

  30. Dragan, F.F., Guarnera, H.M.: Eccentricity terrain of \(\delta \)-hyperbolic graphs. J. Comput. Syst. Sci. 112, 50–65 (2020)

    Article  MathSciNet  Google Scholar 

  31. Dragan, F.F., Habib, M., Viennot, L.: Revisiting radius, diameter, and all eccentricity computation in graphs through certificates. CoRR, arXiv:1803.04660 (2018)

  32. Dragan, F.F., Mohammed, A.: Slimness of graphs. DMTCS 21(3) (2019)

    Google Scholar 

  33. Dragan, F.F., Nicolai, F.: LexBFS-orderings of distance-hereditary graphs with application to the diametral pair problem. DAM 98(3), 191–207 (2000)

    MathSciNet  MATH  Google Scholar 

  34. Dragan, F.F., Nicolai, F., Brandstädt, A.: LexBFS-orderings and powers of graphs. In: d’Amore, F., Franciosa, P.G., Marchetti-Spaccamela, A. (eds.) WG 1996. LNCS, vol. 1197, pp. 166–180. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-62559-3_15

    Chapter  Google Scholar 

  35. Dragan, F.F., Prisakaru, K., Chepoi, V.: The location problem on graphs and the Helly problem. Diskret. Mat. 4(4), 67–73 (1992)

    MathSciNet  Google Scholar 

  36. Dress, A.: Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces. Adv. Math. 53(3), 321–402 (1984)

    Article  MathSciNet  Google Scholar 

  37. Ducoffe, G.: Around the diameter of AT-free graphs. CoRR, arXiv:2010.15814 (2020)

  38. Ducoffe, G.: Distance problems within Helly graphs and k-Helly graphs. CoRR, arXiv:2011.00001 (2020)

  39. Ducoffe, G.: Optimal diameter computation within bounded clique-width graphs. CoRR, arXiv:2011.08448 (2020)

  40. Ducoffe, G.: Beyond Helly graphs: the diameter problem on absolute retracts. CoRR, arXiv:2101.03574 (2021)

  41. Ducoffe, G., Dragan, F.F.: A story of diameter, radius, and (almost) Helly property. Networks, to appear

    Google Scholar 

  42. Ducoffe, G., Habib, M., Viennot, L.: Diameter computation on \(H\)-minor free graphs and graphs of bounded (distance) VC-dimension. In: SODA, pp. 1905–1922. SIAM (2020)

    Google Scholar 

  43. Farley, A., Proskurowski, A.: Computation of the center and diameter of outerplanar graphs. DAM 2(3), 185–191 (1980)

    MathSciNet  MATH  Google Scholar 

  44. Fournier, H., Ismail, A., Vigneron, A.: Computing the Gromov hyperbolicity of a discrete metric space. IPL 115(6–8), 576–579 (2015)

    Article  MathSciNet  Google Scholar 

  45. Gawrychowski, P., Kaplan, H., Mozes, S., Sharir, M., Weimann, O.: Voronoi diagrams on planar graphs, and computing the diameter in deterministic \(\tilde{O}(n^{5/3})\) time. In: SODA, pp. 495–514. SIAM (2018)

    Google Scholar 

  46. Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory. MSRI, vol. 8, pp. 75–263. Springer, New York (1987). https://doi.org/10.1007/978-1-4613-9586-7_3

    Chapter  Google Scholar 

  47. Isbell, J.: Six theorems about injective metric spaces. Commentarii Mathematici Helvetici 39(1), 65–76 (1964)

    Article  MathSciNet  Google Scholar 

  48. Kennedy, W.S., Saniee, I., Narayan, O.: On the hyperbolicity of large-scale networks and its estimation. In: BigData 2016, pp. 3344–3351. IEEE (2016)

    Google Scholar 

  49. Koolen, J.H., Moulton, V.: Hyperbolic bridged graphs. Eur. J. Comb. 23(6), 683–699 (2002)

    Article  MathSciNet  Google Scholar 

  50. Koschützki, D., Lehmann, K.A., Peeters, L., Richter, S., Tenfelde-Podehl, D., Zlotowski, O.: Centrality indices. In: Brandes, U., Erlebach, T. (eds.) Network Analysis. LNCS, vol. 3418, pp. 16–61. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-31955-9_3

    Chapter  Google Scholar 

  51. Lin, M., Szwarcfiter, J.: Faster recognition of clique-Helly and hereditary clique-Helly graphs. Inf. Process. Lett. 103(1), 40–43 (2007)

    Article  MathSciNet  Google Scholar 

  52. Narayan, O., Saniee, I.: Large-scale curvature of networks. Phys. Rev. E, 84(6), 066108 (2011)

    Google Scholar 

  53. Olariu, S.: A simple linear-time algorithm for computing the center of an interval graph. Int. J. Comput. Math. 34(3–4), 121–128 (1990)

    Article  Google Scholar 

  54. Roditty, L., Vassilevska Williams, V.: Fast approximation algorithms for the diameter and radius of sparse graphs. In: STOC, pp. 515–524. ACM (2013)

    Google Scholar 

  55. Wu, Y., Zhang, C.: Hyperbolicity and chordality of a graph. Electr. J. Comb. 18(1), Paper #P43 (2011)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Computer Science Department, Kent State University, Kent, USA

    Feodor F. Dragan

  2. National Institute for Research and Development in Informatics and University of Bucharest, Bucureşti, Romania

    Guillaume Ducoffe

  3. Department of Mathematical and Computational Sciences, The College of Wooster, Wooster, USA

    Heather M. Guarnera

Authors
  1. Feodor F. Dragan
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  2. Guillaume Ducoffe
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  3. Heather M. Guarnera
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Corresponding author

Correspondence to Feodor F. Dragan .

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Editors and Affiliations

  1. University of Waterloo, Waterloo, ON, Canada

    Anna Lubiw

  2. University of Alberta, Edmonton, AB, Canada

    Mohammad Salavatipour

  3. Dalhousie University, Halifax, Canada

    Meng He

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Dragan, F.F., Ducoffe, G., Guarnera, H.M. (2021). Fast Deterministic Algorithms for Computing All Eccentricities in (Hyperbolic) Helly Graphs. In: Lubiw, A., Salavatipour, M., He, M. (eds) Algorithms and Data Structures. WADS 2021. Lecture Notes in Computer Science(), vol 12808. Springer, Cham. https://doi.org/10.1007/978-3-030-83508-8_22

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  • DOI: https://doi.org/10.1007/978-3-030-83508-8_22

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