Skip to main content
SpringerLink
Log in

Approximate Shortest Paths Guided by a Small Index

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Distance oracles and graph spanners are excerpts of a graph that allow to compute approximate shortest paths. Here, we consider the situation where it is possible to access the original graph in addition to the graph excerpt while computing paths. This allows for asymptotically much smaller excerpts than distance oracles or spanners. The quality of an algorithm in this setting is measured by the size of the excerpt (in bits), by how much of the original graph is accessed (in number of edges), and the stretch of the computed path (as the ratio between the length of the path and the distance between its end points). Because these three objectives are conflicting goals, we are interested in a good trade-off. We measure the number of accesses to the graph relative to the number of edges in the computed path.

We present a parametrized construction that, for constant stretches, achieves excerpt sizes and number of accessed edges that are both sublinear in the number of graph vertices. We also show that within these limits, a stretch smaller than 5 cannot be guaranteed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alon, N., Spencer, J.: The Probabilistic Method. Wiley, New York (1992)

    MATH  Google Scholar 

  2. Althöfer, I., Das, G., Dobkin, D.P., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete Comput. Geom. 9(1), 81–100 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baswana, S., Sen, S.: Approximate distance oracles for unweighted graphs in \(\tilde{O}(n^{2})\) time. In: J.I. Munro (ed.) Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 271–280 (2004)

  4. Bollobás, B.: Extremal Graph Theory. Academic Press, San Diego (1978)

    MATH  Google Scholar 

  5. Chvátal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4, 233–235 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cohen, E., Halperin, E., Kaplan, H., Zwick, U.: Reachability and distance queries via 2-hop labels. SIAM J. Comput. 32(5), 1338–1355 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Demetrescu, C., Finocchi, I., Ribichini, A.: Trading off space for passes in graph streaming problems. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 714–723 (2006)

  8. Demetrescu, C., Goldberg, A., Johnson, D. (eds.): 9th DIMACS Challenge on Shortest Paths (to appear). Available at http://www.dis.uniroma1.it/~challenge9

  9. Dijkstra, E.W.: A note on two problems in connection with graphs. Numer. Math. 1, 269–271 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  10. Feigenbaum, J., Kannan, S., McGregor, A., Suri, S., Zhang, J.: Graph distances in the streaming model: the value of space. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 745–754. SIAM, Philadelphia (2005)

    Google Scholar 

  11. Garey, M.R., Johnson, D.S.: Computer and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)

    Google Scholar 

  12. Goldberg, A., Harrelson, C.: Computing the shortest path: A* search meets graph theory. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, vol. 16 (2005)

  13. Katriel, I., Meyer, U.: Elementary graph algorithms in external memory. In: Meyer, U., Sanders, P., Sibeyn, J.F. (eds.) Algorithms for Memory Hierarchies. Lecture Notes in Computer Science, vol. 2625, pp. 62–84. Springer, Berlin (2002)

    Chapter  Google Scholar 

  14. Meyer, U., Zeh, N.: I/O-efficient undirected shortest paths. In: Battista, G.D., Zwick, U. (eds.) Proceedings of the 11th Annual European Symposium on Algorithms. Lecture Notes in Computer Science, vol. 2832, pp. 434–445. Springer, Berlin (2003)

    Google Scholar 

  15. Peleg, D., Schäffer, A.A.: Graph spanners. J. Graph Theory 13(1), 99–116 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  16. Prüfer, H.: Neuer Beweis eines Satzes über Permutationen. Arch. Math. Phys. 27, 742–744 (1918)

    Google Scholar 

  17. Regev, H.: The weight of the greedy graph spanner. Technical Report CS95-22, Weizmann Institute of Science, July 1995

  18. Sanders, P., Schultes, D.: Highway hierarchies hasten exact shortest path queries. In: Brodal, G.S., Leonardi, S. (eds.) Proceedings of the 13th Annual European Symposium on Algorithms. Lecture Notes in Computer Science, vol. 3669, pp. 568–579. Springer, Berlin (2005)

    Google Scholar 

  19. Stirling, J.: Methodus Differentialis. London (1730)

  20. Thorup, M., Zwick, U.: Approximate distance oracles. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp. 183–192. Assoc. Comput. Mach. Press, New York (2001)

    Chapter  Google Scholar 

  21. Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52(1), 1–24 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Zwick, U.: Exact and approximate distances in graphs—a survey. In: auf der Heide, F.M. (ed.) Proceedings of the 9th Annual European Symposium on Algorithms. Lecture Notes in Computer Science, vol. 2161, pp. 33–48. Springer, Berlin (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Theoretical Computer Science, ETH Zurich, Switzerland ETH Zentrum, 8092, Zürich, Switzerland

    Jörg Derungs & Peter Widmayer

  2. Fakultät für Informatik, Technische Universität München, Boltzmann str. 3, 85748, Garching, Germany

    Riko Jacob

Authors
  1. Jörg Derungs
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Riko Jacob
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peter Widmayer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Riko Jacob.

Additional information

This work has been partially supported under the EU programme COST 295 (DYNAMO) and by the Swiss SBF under contract number C05.0047.

This work was done while R. Jacob was staying at ETH Zürich.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Derungs, J., Jacob, R. & Widmayer, P. Approximate Shortest Paths Guided by a Small Index. Algorithmica 57, 668–688 (2010). https://doi.org/10.1007/s00453-008-9228-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-008-9228-5

Keywords

Search

Navigation