Grid Graphs with Diagonal Edges and the Complexity of Xmas Mazes

  • Markus Holzer
  • Sebastian Jakobi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7288)


We investigate the computational complexity of some maze problems, namely the reachability problem for (undirected) grid graphs with diagonal edges, and the solvability of Xmas tree mazes. Simply speaking, in the latter game one has to move sticks of a certain length through a maze, ending in a particular game situation. It turns out that when the number of sticks is bounded by some constant, these problems are closely related to the grid graph problems with diagonals. If on the other hand an unbounded number of sticks is allowed, then the problem of solving such a maze becomes PSPACE-complete. Hardness is shown via a reduction from the nondeterministic constraint logic (NCL) of [E. D. Demaine, R. A. Hearn: A uniform framework or modeling computations as games. Proc. CCC, 2008] to Xmas tree mazes.


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  1. 1.
    Allender, E., Barrington, D.A.M., Chakraborty, T., Datta, S., Roy, S.: Planar and grid graph reachability problems. Theory Comput. Syst. 45(4), 675–723 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bourke, C., Tewari, R., Vinodchandran, N.V.: Directed planar reachability is in unambiguous log-space. ACM Trans. Comput. Theory 1(1), article 4 (2009)Google Scholar
  3. 3.
    Demaine, E.D., Hearn, R.A.: A uniform framework for modeling computations as games. In: Proc. Conf. Comput. Compl., pp. 149–162. Computer Society Press, College Park (2008)Google Scholar
  4. 4.
    Holzer, M., Jakobi, S.: On the complexity of rolling block and Alice mazes. IFIG Research Report 1202, Institut für Informatik, Justus-Liebig-Universität Gießen, Arndtstr. 2, D-35392 Gießen, Germany (2012)Google Scholar
  5. 5.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley (1994)Google Scholar
  6. 6.
    Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4), Article 17 (24 pages) (2008)Google Scholar
  7. 7.
    Savitch, W.J.: Relationships between nondeterministic and deterministic tape complexities. J. Comput. System Sci. 4(2), 177–192 (1970)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Markus Holzer
    • 1
  • Sebastian Jakobi
    • 1
  1. 1.Institut für InformatikUniversität GiessenGiessenGermany

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