Skip to main content
SpringerLink
Log in

Planar and Grid Graph Reachability Problems

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

We study the complexity of restricted versions of s-t-connectivity, which is the standard complete problem for \(\mathsf{NL}\) . In particular, we focus on different classes of planar graphs, of which grid graphs are an important special case. Our main results are:

  • Reachability in graphs of genus one is logspace-equivalent to reachability in grid graphs (and in particular it is logspace-equivalent to both reachability and non-reachability in planar graphs).

  • Many of the natural restrictions on grid-graph reachability (GGR) are equivalent under \(\mathsf{AC}^{0}\) reductions (for instance, undirected GGR, outdegree-one GGR, and indegree-one-outdegree-one GGR are all equivalent). These problems are all equivalent to the problem of determining whether a completed game position in HEX is a winning position, as well as to the problem of reachability in mazes studied by Blum and Kozen (IEEE Symposium on Foundations of Computer Science (FOCS), pp. 132–142, [1978]). These problems provide natural examples of problems that are hard for \(\mathsf{NC}^{1}\) under \(\mathsf{AC}^{0}\) reductions but are not known to be hard for  \(\mathsf{L}\) ; they thus give insight into the structure of \(\mathsf{L}\) .

  • Reachability in layered planar graphs is logspace-equivalent to layered grid graph reachability (LGGR). We show that LGGR lies in \(\mathsf{UL}\) (a subclass of \(\mathsf{NL}\) ).

  • Series-Parallel digraphs (on which reachability was shown to be decidable in logspace by Jakoby et al.) are a special case of single-source-single-sink planar directed acyclic graphs (DAGs); reachability for such graphs logspace reduces to single-source-single-sink acyclic grid graphs. We show that reachability on such grid graphs \(\mathsf{AC}^{0}\) reduces to undirected GGR.

  • We build on this to show that reachability for single-source multiple-sink planar DAGs is solvable in \(\mathsf{L}\) .

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.

Similar content being viewed by others

References

  1. Allender, E., Mahajan, M.: The complexity of planarity testing. Inf. Comput. 189, 117–134 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Allender, E., Reinhardt, K., Zhou, S.: Isolation, matching, and counting: Uniform and nonuniform upper bounds. J. Comput. Syst. Sci. 59(2), 164–181 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Allender, E., Datta, S., Roy, S.: The directed planar reachability problem. In: Proc. 25th Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FST&TCS). Lecture Notes in Computer Science, vol. 1373, pp. 238–249. Springer, Berlin (2005)

    Chapter  Google Scholar 

  4. Allender, E., Barrington, D.A.M., Chakraborty, T., Datta, S., Roy, S.: Grid graph reachability problems. In: IEEE Conference on Computational Complexity, pp. 299–313 (2006)

  5. Barrington, D.A.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. J. Comput. Syst. Sci. 38, 150–164 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barrington, D.A.M., Lu, C.-J., Miltersen, P.B., Skyum, S.: Searching constant width mazes captures the AC0 hierarchy. In: 15th International Symposium on Theoretical Aspects of Computer Science (STACS). Lecture Notes in Computer Science, vol. 1373, pp. 73–83. Springer, Berlin (1998)

    Google Scholar 

  7. Blum, M., Kozen, D.: On the power of the compass (or, why mazes are easier to search than graphs). In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 132–142 (1978)

  8. Bourke, C., Tewari, R., Vinodchandran, N.V.: Directed planar reachability is in unambiguous logspace. In: IEEE Conference on Computational Complexity, pp. 217–221 (2007)

  9. Buhrman, H., Spaan, E., Torenvliet, L.: The relative power of logspace and polynomial time reductions. Comput. Complex. 3, 231–244 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Buss, S.: Polynomial-size Frege and resolution proofs of st-connectivity and Hex tautologies. Theor. Comput. Sci. 357, 35–52 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Buss, S.R., Hay, L.: On truth-table reducibility to SAT. Inf. Comput. 91(1), 86–102 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. Buss, S.R., Cook, S., Gupta, A., Ramachandran, V.: An optimal parallel algorithm for formula evaluation. SIAM J. Comput. 21(4), 755–780 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chakraborty, T., Datta, S.: One-input-face MPCVP is hard for L, but in LogDCFL. In: Conference on Foundations of Software Technology and Theoretical Computer Science (FST&TCS). Lecture Notes in Computer Science, vol. 4337, pp. 57–68. Springer, Berlin (2006)

    Chapter  Google Scholar 

  14. Cook, S.A., McKenzie, P.: Problems complete for deterministic logarithmic space. J. Algorithms 8, 385–394 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Datta, S., Kulkarni, R., Limaye, N., Mahajan, M.: Planarity, determinants, permanents, and (unique) matchings. In: Computer Science—Theory and Applications, Second International Symposium on Computer Science in Russia, (CSR 2007). Lecture Notes in Computer Science, vol. 4649, pp. 115–126. Springer, Berlin (2007)

    Google Scholar 

  16. Eppstein, D., Italiano, G.F., Tamassia, R., Tarjan, R.E., Westbrook, J.R., Yung, M.: Maintenance of a minimum spanning forest in a dynamic planar graph. J. Algorithms 13(1), 33–54 (1992). (Corrigendum in vol. 15, 1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Etessami, K.: Counting quantifiers, successor relations, and logarithmic space. J. Comput. Syst. Sci. 54(3), 400–411 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gross, J., Tucker, T.: Topological Graph Theory, 1st edn. Wiley, New York (1987)

    MATH  Google Scholar 

  19. Halldórsson, M., Radhakrishnan, J., Subrahmanyam, K.V.: Directed vs. undirected monotone contact networks for threshold functions. In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 604–613 (1993)

  20. Henzinger, M.R., Klein, P., Rao, S., Subramanian, S.: Faster shortest-path algorithms for planar graphs. J. Comput. Syst. Sci. 55(1), 3–23 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Husfeldt, T.: Fully dynamic transitive closure in plane dags with one source and one sink. In: Proc. 3rd ESA. Lecture Notes in Computer Science, vol. 955, pp. 199–212. Springer, Berlin (1995)

    Google Scholar 

  22. Immerman, N.: Languages that capture complexity classes. SIAM J. Comput. 16(4), 760–778 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  23. Immerman, N.: Nondeterministic space is closed under complementation. SIAM J. Comput. 17, 935–938 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  24. Immerman, N.: Descriptive Complexity. Springer Graduate Texts in Computer Science. Springer, Berlin (1998)

    Google Scholar 

  25. Jakoby, A., Tantau, T.: Personal communication (2006)

  26. Jakoby, A., Tantau, T.: Logspace algorithms for computing shortest and longest paths in series-parallel graphs. In: Conference on Foundations of Software Technology and Theoretical Computer Science (FST&TCS), pp. 216–227 (2007)

  27. Jakoby, A., Liskiewicz, M., Reischuk, R.: Space efficient algorithms for directed series-parallel graphs. J. Algorithms 60, 85–114 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Ladner, R., Lynch, N.: Relativization of questions about log space reducibility. Math. Syst. Theory 10, 19–32 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  29. Lange, K.-J.: An unambiguous class possessing a complete set. In: 14th International Symposium on Theoretical Aspects of Computer Science (STACS). Lecture Notes in Computer Science, vol. 1200, pp. 339–350. Springer, Berlin (1997)

    Google Scholar 

  30. Limaye, N., Mahajan, M., Sarma, J.M.N.: Evaluating monotone circuits on cylinders, planes, and torii. In: Proc. 23rd Symposium on Theoretical Aspects of Computing (STACS). Lecture Notes in Computer Science, pp. 660–671. Springer, Berlin (2006)

    Google Scholar 

  31. Mahajan, M., Varadarajan, K.R.: A new NC-algorithm for finding a perfect matching in bipartite planar and small genus graphs. In: ACM Symposium on Theory of Computing (STOC), pp. 351–357 (2000)

  32. Mohar, B., Thomassen, C.: Graphs on Surfaces, 1st edn. John Hopkins University Press, Baltimore (2001)

    MATH  Google Scholar 

  33. Nisan, N., Ta-Shma, A.: Symmetric logspace is closed under complement. Chicago J. Theor. Comput. Sci. (1995)

  34. Papakostas, A.: Upward planarity testing of outerplanar dags. In: Graph Drawing (GD94). Lecture Notes in Computer Science, vol. 894, pp. 298–306. Springer, Berlin (1995)

    Google Scholar 

  35. Reingold, O.: Undirected ST-connectivity in log-space. In: Proceedings of the 37th Symposium on Foundations of Computer Science, pp. 376–385. IEEE Computer Society Press (2005)

  36. Reinhardt, K., Allender, E.: Making nondeterminism unambiguous. SIAM J. Comput. 29, 1118–1131 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  37. Szelepcsényi, R.: The method of forced enumeration for nondeterministic automata. Acta Inf. 26, 279–284 (1988)

    Article  MATH  Google Scholar 

  38. Thierauf, T., Wagner, F.: The isomorphism problem for planar 3-connected graphs is in unambiguous logspace. In: 25th Annual Symposium on Theoretical Aspects of Computer Science (STACS), pp. 633–644 (2008)

  39. Thomassen, C.: Embeddings of graphs with no short noncontractible cycles. J. Comb. Theory Ser. B 48(2), 155–177 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  40. Yang, H.: An NC algorithm for the general planar monotone circuit value problem. In: SPDP: 3rd IEEE Symposium on Parallel and Distributed Processing. ACM Special Interest Group on Computer Architecture (SIGARCH) and IEEE Computer Society (1991)

  41. Yannakakis, M.: Embedding planar graphs in four pages. J. Comput. Syst. Sci. 38(1), 36–67 (1989)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Rutgers, The State University of NJ, Piscataway, NJ, USA

    Eric Allender

  2. Computer Science Department, University of Massachusetts Amherst, Amherst, NJ, USA

    David A. Mix Barrington

  3. Computer and Information Science Department, University of Pennsylvania, Philadelphia, PA, USA

    Tanmoy Chakraborty

  4. Chennai Mathematical Institute, Chennai, India

    Samir Datta

  5. IBM India Research Lab, New Delhi, India

    Sambuddha Roy

Authors
  1. Eric Allender
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David A. Mix Barrington
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tanmoy Chakraborty
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Samir Datta
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Sambuddha Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric Allender.

Additional information

E. Allender supported in part by NSF Grant CCF-0514155.

D.A. Mix Barrington supported in part by NSF Grant CCR-9988260.

S. Roy supported in part by NSF Grant CCF-0514155.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Allender, E., Mix Barrington, D.A., Chakraborty, T. et al. Planar and Grid Graph Reachability Problems. Theory Comput Syst 45, 675–723 (2009). https://doi.org/10.1007/s00224-009-9172-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-009-9172-z

Keywords

Search

Navigation