Skip to main content
SpringerLink
Log in

Faster 2-Disjoint-Shortest-Paths Algorithm

  • Conference paper
  • First Online:
Computer Science – Theory and Applications (CSR 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12159))

Included in the following conference series:

Abstract

Consider the following kDSP problem: given a graph G and k pairs of terminal vertices \((s_1, t_1), (s_2, t_2), \ldots , (s_k, t_k)\), check if there exists a k-tuple of pairwise disjoint shortest \(s_i\)–\(t_i\) paths between these k pairs of terminal vertices. Algorithmically, the case of two vertex-disjoint paths turns out to be the most interesting one. For this setting, Eilam-Tzoreff established an algorithm running in \(\mathcal {O}(|V|^8)\) time, which uses dynamic programming (DP) and applies to both directed and undirected graphs and arbitrary positive edge weights (lengths). In this paper, we examine the DP relations arising in this problem and reduce the time complexity to \(\mathcal {O}(|V|^6)\) for the unit-length case and to \(\mathcal {O}(|V|^7)\) for the case of general weights.

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 EPUB and 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

References

  1. Björklund, A., Husfeldt, T.: Shortest two disjoint paths in polynomial time. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 211–222. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43948-7_18

    Chapter  Google Scholar 

  2. Björklund, A., Husfeldt, T.: Counting shortest two disjoint paths in cubic planar graphs with an NC algorithm. ArXiv abs/arXiv:1806.07586 (2018)

  3. Datta, S., Iyer, S., Kulkarni, R., Mukherjee, A.: Shortest \(k\)-disjoint paths via determinants. In: Ganguly, S., Pandya, P. (eds.) 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), vol. 122, pp. 19:1–19:21. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl (2018). https://doi.org/10.4230/LIPIcs.FSTTCS.2018.19. http://drops.dagstuhl.de/opus/volltexte/2018/9918

  4. Eilam-Tzoreff, T.: The disjoint shortest paths problem. Discrete Appl. Math. 85(2), 113–138 (1998). https://doi.org/10.1016/S0166-218X(97)00121-2. http://www.sciencedirect.com/science/article/pii/S0166218X97001212

    Article  MathSciNet  MATH  Google Scholar 

  5. Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphismproblem. Theor. Comput. Sci. 10(2), 111–121 (1980). https://doi.org/10.1016/0304-3975(80)90009-2. http://www.sciencedirect.com/science/article/pii/0304397580900092

    Article  MATH  Google Scholar 

  6. Gallai, T.: Maximum-minimum Sätze und verallgemeinerte Faktoren vonGraphen. Acta Math. Acad. Scientiarum Hung. 12(1), 131–173 (1964). https://doi.org/10.1007/BF02066678

    Article  MathSciNet  MATH  Google Scholar 

  7. Gustedt, J.: The general two-path problem in time \(\cal{O}(m \log n)\). Technical report 394, TU Berlin (1994)

    Google Scholar 

  8. Hirai, H., Namba, H.: Shortest \((a+b)\)-path packing via Hafnian. Algorithmica 80(8), 2478–2491 (2018). https://doi.org/10.1007/s00453-017-0334-0

    Article  MathSciNet  MATH  Google Scholar 

  9. Lynch, J.F.: The equivalence of theorem proving and the interconnection problem. SIGDA Newsl. 5(3), 31–36 (1975). https://doi.org/10.1145/1061425.1061430

    Article  Google Scholar 

  10. Robertson, N., Seymour, P.D.: Disjoint paths–a survey. SIAM J. Algebraic Discrete Methods 6(2), 300–305 (1985). https://doi.org/10.1137/0606030

    Article  MathSciNet  MATH  Google Scholar 

  11. Williams, V.V.: Multiplying matrices faster than Coppersmith-Winograd. In: Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, STOC 2012, pp. 887–898. ACM, New York (2012). https://doi.org/10.1145/2213977.2214056

Download references

Acknowledgements

I would like to express gratitude to Maxim Babenko for pointing me to the original article [4] and suggesting to investigate the possibility of optimization, for a lot of helpful discussions and for helping me with the preparation of the paper text.

Author information

Authors and Affiliations

  1. Department of Mathematical Logic and Algorithms, Moscow State University, Moscow, Russia

    Maxim Akhmedov

  2. Yandex LLC, Moscow, Russia

    Maxim Akhmedov

Authors
  1. Maxim Akhmedov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maxim Akhmedov .

Editor information

Editors and Affiliations

  1. University of Trier, Trier, Germany

    Henning Fernau

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Akhmedov, M. (2020). Faster 2-Disjoint-Shortest-Paths Algorithm. In: Fernau, H. (eds) Computer Science – Theory and Applications. CSR 2020. Lecture Notes in Computer Science(), vol 12159. Springer, Cham. https://doi.org/10.1007/978-3-030-50026-9_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-50026-9_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-50025-2

  • Online ISBN: 978-3-030-50026-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation