Graph Transformation in Constant Time

  • Mike Dodds
  • Detlef Plump
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4178)


We present conditions under which graph transformation rules can be applied in time independent of the size of the input graph: graphs must contain a unique root label, nodes in the left-hand sides of rules must be reachable from the root, and nodes must have a bounded outdegree. We establish a constant upper bound for the time needed to construct all graphs resulting from an application of a fixed rule to an input graph. We also give an improved upper bound under the stronger condition that all edges outgoing from a node must have distinct labels. Then this result is applied to identify a class of graph reduction systems that define graph languages with a linear membership test. In a case study we prove that the (non-context-free) language of balanced binary trees with backpointers belongs to this class.


Graph Transformation Outgoing Edge Graph Class Label Node Incoming Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnborg, S., Courcelle, B., Proskurowski, A., Seese, D.: An algebraic theory of graph reduction. Journal of the ACM 40(5), 1134–1164 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bakewell, A., Plump, D., Runciman, C.: Specifying pointer structures by graph reduction. Mathematical Structures in Computer Science (to appear); Preliminary version available as Technical Report YCS-2003-367, University of York (2003)Google Scholar
  3. 3.
    Bodlaender, H.L., van Antwerpen-de Fluiter, B.: Reduction algorithms for graphs of small treewidth. Inf. Comput. 167(2), 86–119 (2001)zbMATHCrossRefGoogle Scholar
  4. 4.
    Dodds, M., Plump, D.: Extending C for checking shape safety. In: Proceedings Graph Transformation for Verification and Concurrency. Electronic Notes in Theoretical Computer Science. Elsevier, Amsterdam (to appear, 2005)Google Scholar
  5. 5.
    Dörr, H. (ed.): Efficient Graph Rewriting and Its Implementation. LNCS, vol. 922. Springer, Heidelberg (1995)Google Scholar
  6. 6.
    Drewes, F., Habel, A., Kreowski, H.-J.: Hyperedge replacement graph grammars, ch. 2. In: Rozenberg, G. (ed.) Handbook of Graph Grammars and Computing by Graph Transformation. Foundations, vol. I, pp. 95–162. World Scientific, Singapore (1997)CrossRefGoogle Scholar
  7. 7.
    Engelfriet, J., Rozenberg, G.: Node replacement graph grammars, ch. 2. In: Rozenberg, G. (ed.) Handbook of Graph Grammars and Computing by Graph Transformation. Foundations, vol. I, pp. 1–94. World Scientific, Singapore (1997)CrossRefGoogle Scholar
  8. 8.
    Fradet, P., Métayer, D.L.: Shape types. In: Proceedings of the 1997 ACM Symposium on Principles of Programming Languages, pp. 27–39. ACM Press, New York (1997)Google Scholar
  9. 9.
    Fu, J.J.: Linear matching-time algorithm for the directed graph isomorphism problem. In: Staples, J., Katoh, N., Eades, P., Moffat, A. (eds.) ISAAC 1995. LNCS, vol. 1004, pp. 409–417. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  10. 10.
    Fu, J.J.: Pattern matching in directed graphs. In: Galil, Z., Ukkonen, E. (eds.) CPM 1995. LNCS, vol. 937, pp. 64–77. Springer, Heidelberg (1995)Google Scholar
  11. 11.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  12. 12.
    Habel, A., Plump, D.: Relabelling in graph transformation. In: Corradini, A., Ehrig, H., Kreowski, H.-J., Rozenberg, G. (eds.) ICGT 2002. LNCS, vol. 2505, pp. 135–147. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Lange, K.-J., Welzl, E.: String grammars with disconnecting or a basic root of the difficulty in graph grammar parsing. Discrete Applied Mathematics 16, 17–30 (1987)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Mike Dodds
    • 1
  • Detlef Plump
    • 1
  1. 1.Department of Computer ScienceThe University of York 

Personalised recommendations