Skip to main content
SpringerLink
Log in

New Data Structures for Subgraph Connectivity

  • Conference paper
Automata, Languages and Programming (ICALP 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6198))

Included in the following conference series:

Abstract

We study the “subgraph connectivity” problem for undirected graphs with sublinear vertex update time. In this problem, we can make vertices active or inactive in a graph G, and answer the connectivity between two vertices in the subgraph of G induced by the active vertices. Two open problems in subgraph connectivity are solved in this paper. We give the first subgraph connectivity structure with worst-case sublinear time bounds for both updates and queries. Our worst-case subgraph connectivity structure supports \(\tilde{O}(m^{4/5})\) update time, \(\tilde{O}(m^{1/5})\) query time and occupies \(\tilde{O}(m)\) space, where m is the number of edges in the whole graph G.

In the second part of our paper, we describe another dynamic subgraph connectivity structure with amortized \(\tilde{O}(m^{2/3})\) update time, \(\tilde{O}(m^{1/3})\) query time and linear space, which improves the structure introduced by [Chan, Pătraşcu, Roditty, FOCS’08] that takes \(\tilde{O}(m^{4/3})\) space.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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. Chan, T.: Dynamic subgraph connectivity with geometric applications. SIAM J. Comput. 36(3), 681–694 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chan, T.M., Pǎtraşcu, M., Roditty, L.: Dynamic connectivity: Connecting to networks and geometry. In: Proceedings 49th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 95–104 (2008)

    Google Scholar 

  3. Coppersmith, D., Winograd, T.: Matrix multiplication via arithmetic progressions. In: Proc. 19th ACM Symp. on the Theory of Computing (STOC), pp. 1–6 (1987)

    Google Scholar 

  4. Demetrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths. J. ACM 51(6), 968–992 (2004)

    MATH  MathSciNet  Google Scholar 

  5. Demetrescu, C., Thorup, M., Chowdhury, R.A., Ramachandran, V.: Oracles for distances avoiding a failed node or link. SIAM J. Comput. 37(5), 1299–1318 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Demetrescu, C., Italiano, G.F.: Trade-offs for fully dynamic transitive closure on dags: breaking through the o(n2 barrier. J. ACM 52(2), 147–156 (2005)

    Article  MathSciNet  Google Scholar 

  7. Duan, R., Pettie, S.: Dual-failure distance and connectivity oracles. In: Proceedings 20th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 506–515 (2009)

    Google Scholar 

  8. Duan, R., Pettie, S.: Connectivity oracles for failure prone graphs. In: Proceedings 42nd Annual ACM Symposium on Theory of Computing, STOC (to appear, 2010)

    Google Scholar 

  9. Eppstein, D., Galil, Z., Italiano, G., Nissenzweig, A.: Sparsification – a technique for speeding up dynamic graph algorithms. J. ACM 44(5), 669–696 (1997)

    MATH  MathSciNet  Google Scholar 

  10. Frederickson, G.: Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput. 14(4), 781–798 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  11. Frigioni, D., Italiano, G.F.: Dynamically switching vertices in planar graphs. Algorithmica 28(1), 76–103 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Henzinger, M., King, V.: Randomized fully dynamic graph algorithms with polylogarithmic time per operation. J. ACM 46(4), 502–516 (1999)

    MATH  MathSciNet  Google Scholar 

  13. Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM 48(4), 723–760 (2001)

    MATH  MathSciNet  Google Scholar 

  14. King, V.: Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In: FOCS 1999: Proceedings of the 40th Annual Symposium on Foundations of Computer Science, Washington, DC, USA, IEEE Computer Society, Los Alamitos (1999)

    Google Scholar 

  15. King, V., Sagert, G.: A fully dynamic algorithm for maintaining the transitive closure. In: STOC 1999: Proceedings of the thirty-first annual ACM symposium on Theory of computing, pp. 492–498. ACM, New York (1999)

    Chapter  Google Scholar 

  16. Pǎtraşcu, M., Thorup, M.: Planning for fast connectivity updates. In: Proceedings 48th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 263–271 (2007)

    Google Scholar 

  17. Roditty, L., Zwick, U.: A fully dynamic reachability algorithm for directed graphs with an almost linear update time. In: Proceedings 36th ACM Symposium on Theory of Computing (STOC), pp. 184–191 (2004)

    Google Scholar 

  18. Roditty, L., Zwick, U.: On dynamic shortest paths problems. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 580–591. Springer, Heidelberg (2004)

    Google Scholar 

  19. Roditty, L., Zwick, U.: Improved dynamic reachability algorithms for directed graphs. SIAM J. Comput. 37(5), 1455–1471 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Sankowski, P.: Dynamic transitive closure via dynamic matrix inverse. In: Proceedings 45th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 509–517 (2004)

    Google Scholar 

  21. Thorup, M.: Decremental dynamic connectivity. J. Algor. 33(2), 229–243 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Thorup, M.: Near-optimal fully-dynamic graph connectivity. In: Proceedings 32nd ACM Symposium on Theory of Computing (STOC), pp. 343–350 (2000)

    Google Scholar 

  23. Thorup, M.: Worst-case update times for fully-dynamic all-pairs shortest paths. In: Proceedings 37th ACM Symposium on Theory of Computing (STOC), pp. 112–119 (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Michigan, Ann Arbor,  

    Ran Duan

Authors
  1. Ran Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Oxford University Computing Laboratory, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Samson Abramsky

  2. Université de Bordeaux (LaBRI) & INRIA, 351, cours de la Libération, 33405, Talence Cedex, France

    Cyril Gavoille

  3. INRIA, Centre de Recherche Bordeaux – Sud-Ouest, 351 cours de la Libération, 33405, Talence Cedex, France

    Claude Kirchner

  4. Heinz Nixdorf Institute, University of Paderborn, Fürstenallee 11, 33102, Paderborn, Germany

    Friedhelm Meyer auf der Heide

  5. University of Patras and RACTI, 26500, Patras, Greece

    Paul G. Spirakis

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Duan, R. (2010). New Data Structures for Subgraph Connectivity. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds) Automata, Languages and Programming. ICALP 2010. Lecture Notes in Computer Science, vol 6198. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14165-2_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-14165-2_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14164-5

  • Online ISBN: 978-3-642-14165-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation