Skip to main content
Log in

On k-Path Covers and their applications

The VLDB Journal Aims and scope Submit manuscript

Abstract

For a directed graph G with vertex set V, we call a subset \(C\subseteq V\) a k-(All-)Path Cover if C contains a node from any simple path in G consisting of k nodes. This paper considers the problem of constructing small k-Path Covers in the context of road networks with millions of nodes and edges. In many application scenarios, the set C and its induced overlay graph constitute a very compact synopsis of G, which is the basis for the currently fastest data structure for personalized shortest path queries, visually pleasing overlays of subsampled paths, and efficient reporting, retrieval and aggregation of associated data in spatial network databases. Apart from a theoretic investigation of the problem, we provide efficient algorithms that produce very small k-Path Covers for large real-world road networks (with a posteriori guarantees via instance-based lower bounds). We also apply our algorithms to other (social, collaboration, web, etc.) networks and can improve in several instances upon previous approaches.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Notes

  1. http://www.openstreetmap.org.

  2. Meanwhile an erratum was published by the authors of [18] which can be found here: http://www.cse.cuhk.edu.hk/~taoyf/paper/sigmod11-skip-erratum.

  3. \(G^{-1}\) has the same vertex set as G but all edges reversed.

  4. http://www.dis.uniroma1.it/challenge9/download.shtml.

  5. http://snap.stanford.edu/data/.

References

  1. Abraham, I., Delling, D., Fiat, A., Goldberg, A.V., Werneck, R.F.: VC-dimension and shortest path algorithms. In: International Colloquium on Automata, Languages, and Programming (ICALP), pp. 690–699. Springer, Berlin (2011)

  2. Abraham, I., Delling, D., Goldberg, A.V., Werneck, R.F.: A hub-based labeling algorithm for shortest paths in road networks. In: Symposium on Experimental Algorithms (SEA), pp. 230–241. Springer, Berlin (2011)

  3. Bast, H., Funke, S., Sanders, P., Schultes, D.: Fast routing in road networks using transit nodes. Science 316(5824), 566 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brešar, B., Kardoš, F., Katrenič, J., Semanišin, G.: Minimum k-path vertex cover. Discrete Appl. Math. 159(12), 1189–1195 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14(1), 463–479 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Delling, D., Goldberg, A.V., Pajor, T., Werneck, R.F.F.: Customizable route planning. In: Symposium on Experimental Algorithms (SEA), pp. 376–387. Springer, Berlin (2011)

  7. Delling, D., Kobitzsch, M., Werneck, R.: Customizing driving directions with GPUs. In: Euro-Par Parallel Processing, vol. 8632, pp. 728–739. Springer, Berlin (2014)

  8. Dibbelt, J., Pajor, T., Wagner, D.: User-constrained multi-modal route planning. Networks 6, 10 (2012)

    Google Scholar 

  9. Dibbelt, J., Strasser, B., Wagner, D.: Customizable contraction hierarchies. arXiv preprint arXiv:1402.0402 (2014)

  10. Funke, S., Nusser, A., Storandt, S.: On k-path covers and their applications. PVLDB 7(10), 893–902 (2014). http://www.vldb.org/pvldb/vol7/p893-funke.pdf

  11. Funke, S., Storandt, S.: Polynomial-time construction of contraction hierarchies for multi-criteria objectives. In: Sanders, P., Norbert, Z. (eds) Algorithm Engineering and Experiments (ALENEX), pp. 41–54. SIAM (2013)

  12. Geisberger, R., Sanders, P., Schultes, D., Vetter, C.: Exact routing in large road networks using contraction hierarchies. Transp. Sci. 46(3), 388–404 (2012)

  13. Gutman, R.J.: Reach-based routing: a new approach to shortest path algorithms optimized for road networks. In: Lars, A., Giuseppe, F.I., Robert, S. (eds) Algorithm Engineering and Experiments (ALENEX), pp. 100–111 (2004)

  14. Haussler, D., Welzl, E.: Epsilon-nets and simplex range queries. In: Symposium on Computational Geometry (SCG), pp. 61–71. ACM (1986)

  15. Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of the Thiry-fourth Annual ACM Symposium on Theory of Computing. STOC ’02, pp. 767–775. ACM, New York, NY (2002)

  16. Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2- \(\varepsilon \). J. Comput. Syst. Sci. 74(3), 335–349 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ruan, N., Jin, R., Huang, Y.: Distance preserving graph simplification. In: Data Mining (ICDM), 2011 IEEE 11th International Conference on, pp. 1200–1205. IEEE (2011)

  18. Tao, Y., Sheng, C., Pei, J.: On k-skip shortest paths. In: ACM SIGMOD International Conference on Management of Data, pp. 421–432. ACM (2011)

  19. Vapnik, V.N., Chervonenkis, A.Y.: On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16(2), 264–280 (1971)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sabine Storandt.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Funke, S., Nusser, A. & Storandt, S. On k-Path Covers and their applications. The VLDB Journal 25, 103–123 (2016). https://doi.org/10.1007/s00778-015-0392-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00778-015-0392-3

Keywords

Navigation