Compressed Tree Canonization

Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9135)

Abstract

Straight-line (linear) context-free tree (slt) grammars have been used to compactly represent ordered trees. Equivalence of slt grammars is decidable in polynomial time. Here we extend this result and show that isomorphism of unordered trees given as slt grammars is decidable in polynomial time. The result generalizes to isomorphism of unrooted trees and bisimulation equivalence. For non-linear slt grammars which can have double-exponential compression ratios, we prove that unordered isomorphism and bisimulation equivalence are \(\textsc {pspace}\)-hard and in \(\textsc {exptime}\).

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References

  1. 1.
    Aho, A., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)MATHGoogle Scholar
  2. 2.
    Balcázar, J., Gabarró, J., Sántha, M.: Deciding bisimilarity is P-complete. Formal Aspects of Computing 4, 638–648 (1992)MATHCrossRefGoogle Scholar
  3. 3.
    Brenguier, R., Göller, S., Sankur, O.: A comparison of succinctly represented finite-state systems. In: Koutny, M., Ulidowski, I. (eds.) CONCUR 2012. LNCS, vol. 7454, pp. 147–161. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  4. 4.
    Busatto, G., Lohrey, M., Maneth, S.: Efficient memory representation of XML document trees. Inf. Syst. 33(4–5), 456–474 (2008)CrossRefGoogle Scholar
  5. 5.
    Buss, S.R.: Alogtime algorithms for tree isomorphism, comparison, and canonization. Kurt Gödel Colloquium 97, 18–33 (1997)MathSciNetGoogle Scholar
  6. 6.
    Charikar, M., Lehman, E., Lehman, A., Liu, D., Panigrahy, R., Prabhakaran, M., Sahai, A., Shelat, A.: The smallest grammar problem. IEEE Trans. Inf. Theory 51(7), 2554–2576 (2005)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Das, B., Scharpfenecker, P., Torán, J.: Succinct encodings of graph isomorphism. In: Dediu, A.-H., Martín-Vide, C., Sierra-Rodríguez, J.-L., Truthe, B. (eds.) LATA 2014. LNCS, vol. 8370, pp. 285–296. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  8. 8.
    Galperin, H., Wigderson, A.: Succinct representations of graphs. Inf. Contr. 56, 183–198 (1983)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Jenner, B., Köbler, J., McKenzie, P., Torán, J.: Completeness results for graph isomorphism. J. Comput. Syst. Sci. 66(3), 549–566 (2003)MATHCrossRefGoogle Scholar
  10. 10.
    Lengauer, T., Wagner, K.W.: The correlation between the complexities of the nonhierarchical and hierarchical versions of graph problems. J. Comput. Syst. Sci. 44, 63–93 (1992)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Lindell, S.: A logspace algorithm for tree canonization (extended abstract). In: Proc. STOC 1992, pp. 400–404. ACM (1992)Google Scholar
  12. 12.
    Lohrey, M.: Algorithmics on SLP-compressed strings: a survey. Groups Complexity Cryptology 4(2), 241–299 (2012)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Lohrey, M., Maneth, S., Peternek, F.: Compressed tree canonization (2015). arXiv.org http://arxiv.org/abs/1502.04625
  14. 14.
    Lohrey, M., Maneth, S., Schmidt-Schauß, M.: Parameter reduction and automata evaluation for grammar-compressed trees. J. Comput. Syst. Sci. 78(5), 1651–1669 (2012)MATHCrossRefGoogle Scholar
  15. 15.
    Lohrey, M., Mathissen, C.: Isomorphism of regular trees and words. Inf. Comput. 224, 71–105 (2013)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Markus Lohrey
    • 1
  • Sebastian Maneth
    • 2
  • Fabian Peternek
    • 2
  1. 1.Universität SiegenSiegenGermany
  2. 2.University of EdinburghEdinburghUK

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