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Fast fully dynamic labelling for distance queries

Abstract

Finding the shortest-path distance between an arbitrary pair of vertices is a fundamental problem in graph theory. A tremendous amount of research has explored this problem, most of which is limited to static graphs. Due to the dynamic nature of real-world networks, such as social networks or web graphs in which a link between two entities may fail or become alive at any time, there is a pressing need to address this problem for dynamic networks. Existing work can only accommodate distance queries over moderately large dynamic networks due to high space cost and long pre-processing time required for constructing distance labelling, and even on such moderately large dynamic networks, distance labelling can hardly be updated efficiently. In this article, we propose a fully dynamic labelling method to efficiently update distance labelling so as to answer distance queries over large dynamic graphs. At its core, our proposed method incorporates two building blocks: (i) incremental algorithm for handling incremental update operations, i.e. edge insertions, and (ii) decremental algorithm for handling decremental update operations, i.e. edge deletions. These building blocks are built in a highly scalable framework of distance query answering. We theoretically prove the correctness of our fully dynamic labelling method and its preservation of the minimality of labelling. We have also evaluated on 13 real-world large complex networks to empirically verify the efficiency, scalability and robustness of our method.

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References

  1. 1.

    Abraham, I., Delling, D., Goldberg, A.V., Werneck, R.F.: Hierarchical hub labelings for shortest paths. In: Proceedings of the 20th Annual European Conference on Algorithms, pp. 24–35 (2012)

  2. 2.

    Akiba, T., Iwata, Y., Yoshida, Y.: Fast exact shortest-path distance queries on large networks by pruned landmark labeling. In: Proceedings of the 2013 ACM SIGMOD International Conference on Management of Data, pp. 349–360 (2013)

  3. 3.

    Akiba, T., Iwata, Y., Yoshida, Y.: Dynamic and historical shortest-path distance queries on large evolving networks by pruned landmark labeling. In: Proceedings of the 23rd International Conference on World Wide Web, pp. 237–248 (2014)

  4. 4.

    Akiba, T., Sommer, C., Kawarabayashi, K.: Shortest-path queries for complex networks: exploiting low tree-width outside the core. In: Proceedings of the 15th International Conference on Extending Database Technology, pp. 144–155 (2012)

  5. 5.

    Backstrom, L., Huttenlocher, D., Kleinberg, J., Lan, X.: Group formation in large social networks: Membership, growth, and evolution. In: Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD, pp. 44–54 (2006). https://doi.org/10.1145/1150402.1150412

  6. 6.

    Bernstein, A.: Fully dynamic (2 + epsilon) approximate all-pairs shortest paths with fast query and close to linear update time. In: Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 693–702 (2009)

  7. 7.

    Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U.: Complex networks: structure and dynamics. Phys. Rep. 424(4–5), 175–308 (2006). https://doi.org/10.1016/j.physrep.2005.10.009

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Boldi, P., Rosa, M., Santini, M., Vigna, S.: Layered label propagation: a multiresolution coordinate-free ordering for compressing social networks. In: Proceedings of the 20th International Conference on World Wide Web, WWW, pp. 587–596 (2011). https://doi.org/10.1145/1963405.1963488

  9. 9.

    Boldi, P., Santini, M., Vigna, S.: A large time-aware graph. SIGIR Forum 42(2), 33–38 (2008)

    Article  Google Scholar 

  10. 10.

    Boldi, P., Vigna, S.: The webgraph framework i: compression techniques. In: Proceedings of the 13th International Conference on World Wide Web, WWW, pp. 595–602 (2004). https://doi.org/10.1145/988672.988752

  11. 11.

    Callaway, D.S., Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Network robustness and fragility: percolation on random graphs. Phys. Rev. Lett. 85, 5468–5471 (2000). https://doi.org/10.1103/PhysRevLett.85.5468

    Article  Google Scholar 

  12. 12.

    Chang, L., Yu, J.X., Qin, L., Cheng, H., Qiao, M.: The exact distance to destination in undirected world. VLDB J. 21(6), 869–888 (2012). https://doi.org/10.1007/s00778-012-0274-x

    Article  Google Scholar 

  13. 13.

    Cheng, J., Yu, J.X.: On-line exact shortest distance query processing. In: Proceedings of the 12th International Conference on Extending Database Technology: Advances in Database Technology, EDBT, pp. 481–492 (2009). https://doi.org/10.1145/1516360.1516417

  14. 14.

    Cohen, E., Halperin, E., Kaplan, H., Zwick, U.: Reachability and distance queries via 2-hop labels. In: Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 937–946 (2002)

  15. 15.

    Delling, D., Goldberg, A.V., Pajor, T., Werneck, R.F.: Robust distance queries on massive networks. In: European Symposium on Algorithms, pp. 321–333. Berlin, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44777-2_27

  16. 16.

    Demetrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths. J. ACM 51(6), 968–992 (2004). https://doi.org/10.1145/1039488.1039492

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    D’Andrea, A., D’Emidio, M., Frigioni, D., Leucci, S., Proietti, G.: Dynamically maintaining shortest path trees under batches of updates. In: Revised Selected Papers of the 20th International Colloquium on Structural Information and Communication Complexity - Volume 8179, SIROCCO, pp. 286–297 (2013)

  18. 18.

    D’Andrea, A., D’Emidio, M., Frigioni, D., Leucci, S., Proietti, G.: Experimental evaluation of dynamic shortest path tree algorithms on homogeneous batches. In: Proceedings of the 13th International Symposium on Experimental Algorithms - Volume 8504, pp. 283–294 (2014). https://doi.org/10.1007/978-3-319-07959-2_24

  19. 19.

    D’angelo, G., D’emidio, M., Frigioni, D.: Fully dynamic 2-hop cover labeling. J. Exp. Algorithmics 24(1) (2019). https://doi.org/10.1145/3299901

  20. 20.

    D’Emidio, M.: Faster algorithms for mining shortest-path distances from massive time-evolving graphs. Algorithms 13(8), 191 (2020)

  21. 21.

    Farhan, M., Wang, Q.: Efficient maintenance of distance labelling for incremental updates in large dynamic graphs. In: 24th International Conference on Extending Database Technology EDBT (2021)

  22. 22.

    Farhan, M., Wang, Q., Lin, Y., McKay, B.D.: A highly scalable labelling approach for exact distance queries in complex networks. In: 22nd International Conference on Extending Database Technology EDBT, pp. 13–24 (2019)

  23. 23.

    Fu, A.W.C., Wu, H., Cheng, J., Wong, R.C.W.: Is-label: an independent-set based labeling scheme for point-to-point distance querying. Proc. VLDB Endow. 6(6), 457–468 (2013)

    Article  Google Scholar 

  24. 24.

    Goldberg, A.V., Harrelson, C.: Computing the shortest path: a search meets graph theory. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 156–165 (2005)

  25. 25.

    Gutenberg, M.P., Wulff-Nilsen, C.: Fully-dynamic all-pairs shortest paths: Improved worst-case time and space bounds. In: Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 2562–2574 (2020)

  26. 26.

    Hayashi, T., Akiba, T., Kawarabayashi, K.: Fully dynamic shortest-path distance query acceleration on massive networks. In: Proceedings of the 25th ACM International on Conference on Information and Knowledge Management, pp. 1533–1542 (2016)

  27. 27.

    Jiang, M., Fu, A.W.C., Wong, R.C.W., Xu, Y.: Hop doubling label indexing for point-to-point distance querying on scale-free networks. Proc. VLDB Endow. 7(12), 1203–1214 (2014). https://doi.org/10.14778/2732977.2732993

    Article  Google Scholar 

  28. 28.

    Jin, R., Ruan, N., Xiang, Y., Lee, V.: A highway-centric labeling approach for answering distance queries on large sparse graphs. In: Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data, pp. 445–456 (2012)

  29. 29.

    Kumar, R., Novak, J., Tomkins, A.: Structure and evolution of online social networks. In: Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD, pp. 611–617 (2006). https://doi.org/10.1145/1150402.1150476

  30. 30.

    Kunegis, J.: Konect: The koblenz network collection. In: Proceedings of the 22nd International Conference on World Wide Web, WWW, pp. 1343–1350 (2013). https://doi.org/10.1145/2487788.2488173

  31. 31.

    Leskovec, J., Kleinberg, J., Faloutsos, C.: Graphs over time: densification laws, shrinking diameters and possible explanations. KDD (2005). https://doi.org/10.1145/1081870.1081893

    Article  Google Scholar 

  32. 32.

    Leskovec, J., Lang, K.J., Dasgupta, A., Mahoney, M.W.: Community structure in large networks: natural cluster sizes and the absence of large well-defined clusters. Internet Math. 6(1), 29–123 (2009)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Leskovec, J., Sosič, R.: Snap: A general-purpose network analysis and graph-mining library. ACM Trans. Intell. Syst. Technol. 8(1), 2898361 (2016). https://doi.org/10.1145/2898361

    Article  Google Scholar 

  34. 34.

    Li, W., Qiao, M., Qin, L., Zhang, Y., Chang, L., Lin, X.: Scaling distance labeling on small-world networks. In: Proceedings of the 2019 International Conference on Management of Data, pp. 1060–1077 (2019)

  35. 35.

    Li, Y., U, L.H., Yiu, M.L., Kou, N.M.: An experimental study on hub labeling based shortest path algorithms. Proc. VLDB Endow. 11(4), 445–457 (2017)

  36. 36.

    Mislove, A., Marcon, M., Gummadi, K.P., Druschel, P., Bhattacharjee, B.: Measurement and analysis of online social networks. In: IMC, pp. 29–42 (2007)

  37. 37.

    Myers, S.A., Leskovec, J.: The bursty dynamics of the twitter information network. In: Proceedings of the 23rd International Conference on World Wide Web, pp. 913–924 (2014)

  38. 38.

    Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118 (2001). https://doi.org/10.1103/PhysRevE.64.026118

    Article  Google Scholar 

  39. 39.

    Ouyang, D., Yuan, L., Qin, L., Chang, L., Zhang, Y., Lin, X.: Efficient shortest path index maintenance on dynamic road networks with theoretical guarantees. Proc. VLDB Endow. 13(5), 602–615 (2020)

    Article  Google Scholar 

  40. 40.

    Potamias, M., Bonchi, F., Castillo, C., Gionis, A.: Fast shortest path distance estimation in large networks. In: Proceedings of the 18th ACM Conference on Information and Knowledge Management, pp. 867–876 (2009)

  41. 41.

    Qin, Y., Sheng, Q.Z., Falkner, N.J., Yao, L., Parkinson, S.: Efficient computation of distance labeling for decremental updates in large dynamic graphs. World Wide Web 20(5), 915–937 (2017). https://doi.org/10.1007/s11280-016-0421-1

    Article  Google Scholar 

  42. 42.

    Robertson, N., Seymour, P.D.: Graph minors. iii. planar tree-width. J. Comb. Theory Ser. B 36(1), 49–64 (1984). https://doi.org/10.1016/0095-8956(84)90013-3

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Roditty, L., Zwick, U.: On dynamic shortest paths problems. In: Albers, S., Radzik, T. (eds.) European Symposium on Algorithms, pp. 580–591. Springer, Berlin (2004)

    MATH  Google Scholar 

  44. 44.

    Rossi, R.A., Ahmed, N.K.: The network data repository with interactive graph analytics and visualization. In: Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, AAAI, pp. 4292–4293 (2015)

  45. 45.

    Tarjan, R.E.: Data Structures and Network Algorithms. Society for Industrial and Applied Mathematics, 3600 University City Science Center Philadelphia, PA, United States (1983). https://doi.org/10.1137/1.9781611970265

  46. 46.

    Tretyakov, K., Armas-Cervantes, A., García-Bañuelos, L., Vilo, J., Dumas, M.: Fast fully dynamic landmark-based estimation of shortest path distances in very large graphs. In: Proceedings of the 20th ACM International Conference on Information and Knowledge Management, CIKM, pp. 1785–1794 (2011). https://doi.org/10.1145/2063576.2063834

  47. 47.

    Ukkonen, A., Castillo, C., Donato, D., Gionis, A.: Searching the wikipedia with contextual information. In: Proceedings of the 17th ACM Conference on Information and Knowledge Management, CIKM, pp. 1351–1352 (2008). https://doi.org/10.1145/1458082.1458274

  48. 48.

    Vieira, M.V., Fonseca, B.M., Damazio, R., Golgher, P.B., Reis, D.d.C., Ribeiro-Neto, B.: Efficient search ranking in social networks. In: Proceedings of the Sixteenth ACM Conference on Conference on Information and Knowledge Management, CIKM, pp. 563–572 (2007). https://doi.org/10.1145/1321440.1321520

  49. 49.

    Wang, Y., Wang, Q., Koehler, H., Lin, Y.: Query-by-sketch: Scaling shortest path graph queries on very large networks. In: Proceedings of the 2021 International Conference on Management of Data, pp. 1946–1958 (2021)

  50. 50.

    Wei, F.: Tedi: efficient shortest path query answering on graphs. In: Proceedings of the 2010 ACM SIGMOD International Conference on Management of Data, pp. 99–110 (2010)

  51. 51.

    Xu, B., Huang, Y., Kwak, H., Contractor, N.: Structures of broken ties: exploring unfollow behavior on twitter. In: Proceedings of the 2013 Conference on Computer Supported Cooperative Work, CSCW, pp. 871–876 (2013). https://doi.org/10.1145/2441776.2441875

  52. 52.

    Yahia, S.A., Benedikt, M., Lakshmanan, L.V.S., Stoyanovich, J.: Efficient network aware search in collaborative tagging sites. Proc. VLDB Endow. 1(1), 710–721 (2008). https://doi.org/10.14778/1453856.1453934

    Article  Google Scholar 

  53. 53.

    Yang, J., Leskovec, J.: Defining and evaluating network communities based on ground-truth. Knowl. Inf. Syst. 42(1), 181–213 (2015). https://doi.org/10.1007/s10115-013-0693-z

    Article  Google Scholar 

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Correspondence to Muhammad Farhan.

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Farhan, M., Wang, Q., Lin, Y. et al. Fast fully dynamic labelling for distance queries. The VLDB Journal (2021). https://doi.org/10.1007/s00778-021-00707-z

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Keywords

  • Graph algorithms
  • Dynamic graphs
  • Distance labelling
  • Query processing