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Dynamically Maintaining Shortest Path Trees under Batches of Updates

  • Annalisa D’Andrea
  • Mattia D’Emidio
  • Daniele Frigioni
  • Stefano Leucci
  • Guido Proietti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8179)

Abstract

In this paper we focus on dynamic batch algorithms for single source shortest paths in graphs with positive real edge weights. A dynamic algorithm is called batch if it is able to handle graph changes that consist of multiple edge updates at a time, i.e. a batch. We propose a new algorithm to process a decremental batch (containing only delete and weight increase operations), a new algorithm to process an incremental batch (containing only insert and weight decrease operations), and a combination of these algorithms to process arbitrary sequences of incremental and decremental batches. These algorithms are update-sensitive, namely they are efficient w.r.t. to the number of nodes in the shortest paths tree that change the parent and/or the distance from the source as a consequence of the changes.

Keywords

Short Path Priority Queue Short Path Problem Dynamic Algorithm Colored Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abraham, I., Delling, D., Goldberg, A.V., Werneck, R.F.: Hierarchical hub labelings for shortest paths. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 24–35. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Bauer, R., Delling, D., Sanders, P., Schieferdecker, D., Schultes, D., Wagner, D.: Combining hierarchical and goal-directed speed-up techniques for dijkstra’s algorithm. ACM Journal on Experimental Algorithms 15, Article 2.3 (2010)Google Scholar
  3. 3.
    Bauer, R., Wagner, D.: Batch dynamic single-source shortest-path algorithms: An experimental study. In: Vahrenhold, J. (ed.) SEA 2009. LNCS, vol. 5526, pp. 51–62. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Bernstein, A.: Maintaining shortest paths under deletions in weighted directed graphs. In: Proceedings of 45th ACM STOC, pp. 725–734. ACM (2013)Google Scholar
  5. 5.
    Brodal, G.S.: Worst-case efficient priority queues. In: Proceedings seventh ACM-SIAM Symposium on Discrete algorithms, pp. 52–58. SIAM (1996)Google Scholar
  6. 6.
    Bruera, F., Cicerone, S., D’Angelo, G., Di Stefano, G., Frigioni, D.: Dynamic multi-level overlay graphs for shortest paths. Mathematics in Computer Science 1(4), 709–736 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buriol, L.S., Resende, M.G.C., Thorup, M.: Speeding up dynamic shortest-path algorithms. INFORMS Journal on Computing 20(2), 191–204 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chan, E.P.F., Yang, Y.: Shortest path tree computation in dynamic graphs. IEEE Transactions on Computers 4(58), 541–557 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Delling, D., Goldberg, A.V., Pajor, T., Werneck, R.F.: Customizable route planning. In: Pardalos, P.M., Rebennack, S. (eds.) SEA 2011. LNCS, vol. 6630, pp. 376–387. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Frigioni, D., Marchetti-Spaccamela, A., Nanni, U.: Semidynamic algorithms for maintaining single source shortest paths trees. Algorithmica 22(3), 250–274 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Frigioni, D., Marchetti-Spaccamela, A., Nanni, U.: Fully dynamic algorithms for maintaining shortest paths trees. J. of Algorithms 34(2), 251–281 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Frigioni, D., Marchetti-Spaccamela, A., Nanni, U.: Fully dynamic shortest paths in digraphs with arbitrary arc weights. J. of Algorithms 49(1), 86–113 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Geisberger, R., Sanders, P., Schultes, D., Vetter, C.: Exact routing in large road networks using contraction hierarchies. Transportation Sc. 46(3), 388–404 (2012)CrossRefGoogle Scholar
  15. 15.
    Narváez, P., Siu, K.Y., Tzeng, H.Y.: New dynamic algorithms for shortest path tree computation. IEEE/ACM Transactions on Networking 8(6), 734–746 (2000)CrossRefGoogle Scholar
  16. 16.
    Ramalingam, G., Reps, T.W.: An incremental algorithm for a generalization of the shortest paths problem. Journal of Algorithms 21, 267–305 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ramalingam, G., Reps, T.W.: On the computational complexity of dynamic graph problems. Theor. Comput. Sci. 158(1&2), 233–277 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Roditty, L., Zwick, U.: Dynamic approximate all-pairs shortest paths in undirected graphs. SIAM J. on Computing 41(3), 670–683 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Annalisa D’Andrea
    • 1
  • Mattia D’Emidio
    • 1
  • Daniele Frigioni
    • 1
  • Stefano Leucci
    • 1
  • Guido Proietti
    • 1
  1. 1.Department of Information Engineering, Computer Science and MathematicsUniversity of L’AquilaItaly

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