Skip to main content
SpringerLink
Log in

Approximating Shortest Paths in Graphs

  • Conference paper
Book cover WALCOM: Algorithms and Computation (WALCOM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5431))

Included in the following conference series:

Abstract

Computing all-pairs distances in a graph is a fundamental problem of computer science but there has been a status quo with respect to the general problem of weighted directed graphs. In contrast, there has been a growing interest in the area of algorithms for approximate shortest paths leading to many interesting variations of the original problem.

In this article, we trace some of the fundamental developments like spanners and distance oracles, their underlying constructions, as well as their applications to the approximate all-pairs shortest paths.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Awerbuch, B., Baratz, A., Peleg, D.: Efficient broadcast and light weight spanners. Tech. Report CS92-22, Weizmann Institute of Science (1992)

    Google Scholar 

  2. Aingworth, D., Chekuri, C., Indyk, P., Motwani, R.: Fast estimation of diameter and shortest paths(without matrix multiplication). SIAM Journal on Computing 28, 1167–1181 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Althöfer, I., Das, G., Dobkin, D.P., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete and Computational Geometry 9, 81–100 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Awerbuch, B.: Complexity of network synchronization. Journal of Association of Computing Machinery 32(4), 804–823 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bollobás, B., Coppersmith, D., Elkin, M.: Sparse distance preserves and additive spanners. In: Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 414–423 (2003)

    Google Scholar 

  6. Baswana, S., Goyal, V., Sen, S.: All-pairs nearly 2-approximate shortest paths in O(n 2polylog n) time. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 666–679. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  7. Baswana, S., Gaur, A., Sen, S., Upadhyaya, J.: Distance Oracle for unweighted graphs: breaking the quadratic barrier with constant additive error. In: Proceedings of the ICALP (1) pp. 609–621 (2008)

    Google Scholar 

  8. Baswana, S., Kavitha, T.: Faster Algorithms for Approximate Distance Oracles and All-Pairs Small Stretch Paths. In: Proc. 47th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 591–602. IEEE, Los Alamitos (2006)

    Google Scholar 

  9. Baswana, S., Telikepalli, K., Mehlhorn, K., Pettie, S.: New construction of (α,β)-spanners and purely additive spanners. In: Proceedings of 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 672–681 (2005)

    Google Scholar 

  10. Baswana, S., Sen, S.: A simple linear time algorithm for computing a (2k − 1)-spanner of O(kn 1 + 1/k) size in weighted graphs. In: Proceedings of the 30th International Colloquium on Automata, Languages and Programming (ICALP), pp. 384–396 (2003)

    Google Scholar 

  11. Baswana, S., Sen, S.: Approximate distance oracles for unweighted graphs in \(\tilde{O}(n^2)\) time. In: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 271–280 (2004)

    Google Scholar 

  12. Baswana, S., Sen, S.: Approximate distance oracles for unweighted graphs in expected O(n 2) time. ACM Transactions on Algorithms 2, 557–577 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Baswana, S., Sen, S.: A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. Random Structures and Algorithms 30, 532–563 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chan, T.M.: More algorithms for all-pairs shortest paths in weighted graphs. In: Proceedings of 39th Annual ACM Symposium on Theory of Computing, pp. 590–598 (2007)

    Google Scholar 

  15. Cohen, E.: Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing 28, 210–236 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cohen, E., Zwick, U.: All-pairs small stretch paths. Journal of Algorithms 38, 335–353 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation 9, 251–280 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorothms. MIT Press, Cambridge (1990)

    Google Scholar 

  19. Dor, D., Halperin, S., Zwick, U.: All pairs almost shortest paths. Siam Journal on Computing 29, 1740–1759 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Elkin, M.: Computing almost shortest paths. ACM Transactions on Algorithms (TALG) 1, 282–323 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Elkin, M., Peleg, D.: Strong inapproximability of the basic k-spanner problem. In: Proc. of 27th International Colloquium on Automata, Languages and Programming, pp. 636–648 (2000)

    Google Scholar 

  22. Elkin, M., Peleg, D.: (1 + ε,β)-spanner construction for general graphs. SIAM Journal of Computing 33, 608–631 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Erdős, P.: Extremal problems in graph theory. In: Theory of Graphs and its Applications (Proc. Sympos. Smolenice,1963), pp. 29–36. Publ. House Czechoslovak Acad. Sci., Prague (1964)

    Google Scholar 

  24. Fredman, M.L., Komlós, J., Szemerédi, E.: Storing a sparse table with O(1) worst case time. Journal of Association of Computing Machinery 31, 538–544 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pettie, S.: A new approach to all-pairs shortest paths on real-weighted graphs. Theoretical Computer Science 312, 47–74 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Peleg, D., Schaffer, A.A.: Graph spanners. Journal of Graph Theory 13, 99–116 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Roditty, L., Thorup, M., Zwick, U.: Deterministic constructions of approximate distance oracles and spanners. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 261–272. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  28. Shoshan, A., Zwick, U.: All pairs shortest paths in undirected graphs with integer weights. In: Proceedings of the 36th Annual Symposium on Foundations of Computer Science (FOCS), pp. 605–615 (1999)

    Google Scholar 

  29. Thorup, M., Zwick, U.: Compact routing schemes. In: Proceedings of 13th ACM Symposium on Parallel Algorithms and Architecture, pp. 1–10 (2001)

    Google Scholar 

  30. Thorup, M., Zwick, U.: Approximate distance oracles. Journal of Association of Computing Machinery 52, 1–24 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Thorup, M., Zwick, U.: Spanners and emulators with sublinear distance errors. In: Proceedings of 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 802–809 (2006)

    Google Scholar 

  32. Zwick, U.: Exact and approximate distances in graphs - A survey. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 33–48. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  33. Zwick, U.: All-pairs shortest paths using bridging sets and rectangular matrix multiplication. Journal of Association of Computing Machinery 49, 289–317 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Comp. Sc. & Engg., Indian Institute of Technology Delhi, New Delhi, 110016, India

    Sandeep Sen

Authors
  1. Sandeep Sen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Indian Statistical Institute, 700 108, Kolkata, India

    Sandip Das

  2. Japan Advanced Institute of Science and Technology, 923-1292, Ishikawa, Japan

    Ryuhei Uehara

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Sen, S. (2009). Approximating Shortest Paths in Graphs. In: Das, S., Uehara, R. (eds) WALCOM: Algorithms and Computation. WALCOM 2009. Lecture Notes in Computer Science, vol 5431. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00202-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-00202-1_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00201-4

  • Online ISBN: 978-3-642-00202-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation