Efficient algorithms on context-free graph languages

  • Clemens Lautemann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 317)


A number of different graph grammar types have been called ”context-free” in the literature. We consider two recent such formalisms, boundary node-label controlled (BNLC) and hyperedge replacement (HR) grammars, from a complexity-theoretical point of view. It is shown that all HR languages, the members of which satisfy a certain separability restriction, are contained in LOGCFL, the class of sets which are log-space reducible to context-free (string) languages. In particular, this implies the existence of efficient sequential as well as parallel recognition algorithms for these languages. Since HR grammars can simulate a large class of BNLC grammars, the same holds for an according class of BNLC languages. Thus, in a sense, a large class of BNLC and HR languages are ”close” to context-free string languages.

We then use these results for investigating the complexity of some graph-theoretical problems restricted to HR languages. It is shown that on HR languages which satisfy the abovementioned constraints, a number of problems (some of which are NP-complete in the general case) have polynomial-time sequential and very fast and feasible parallel solutions.


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  1. [CKS81]
    A.K. Chandra, D.C. Kozen, L.J. Stockmeyer, Alternation. JACM 28 (1981), pp. 114–133.Google Scholar
  2. [CER79]
    V.Claus, H.Ehrig, G.Rozenberg (eds.), Graph grammars and their application to Computer Science and Biology. LNCS 73, 1979.Google Scholar
  3. [Co81]
    S.A. Cook, Towards a complexity theory of synchronous parallel computation. L'Enseignement mathématique 27 (1981), pp. 99–124.Google Scholar
  4. [DK87]
    E.Dahlhaus, M.Karpinski, Parallel complexity for matching restricted to degree defined graph classes. Preprint, Universität Bonn, 1987.Google Scholar
  5. [DG78]
    P. Della Vigna, C. Ghézzi, Context-free graph grammars. Information and Control 37 (1978), pp. 207–233.Google Scholar
  6. [Ed65]
    J. Edmonds, Paths, trees, and flowers. Canad. J. Math. 17 (1965), pp. 449–467.Google Scholar
  7. [En87]
    J.Engelfriet, personal communication, Dec. 1987.Google Scholar
  8. [ENR83]
    H.Ehrig, M.Nagl, G.Rozenberg (eds.), Graph grammars and their application to Computer Science. LNCS 153, 1983.Google Scholar
  9. [Fe71]
    J. Feder, Plex languages. Inf. Sci. 3 (1971), pp. 225–241.Google Scholar
  10. [GJ79]
    M.Garey, D.Johnson, Computers and intractability. Freeman, 1979.Google Scholar
  11. [GK87]
    D.Grigoriev, M. Karpinski, The matching problem for bipartite graphs with polynomially bounded permanents is in NC. 28th Proc. IEEE FOCS, 1987.Google Scholar
  12. [HK87a]
    A. Habel, H.-J. Kreowski, Some structural aspects of hypergraph languages generated by hyperedge replacement. LNCS 247 (1987), pp. 207–219.Google Scholar
  13. [HK87b]
    A.Habel, H.-J.Kreowski, May we introduce to you: Hyperedge replacement. Proc. 3rd international workshop on graph grammars and their applications to Computer Science, to appear.Google Scholar
  14. [Hr88]
    J. Hromkovič, Two independent solutions of the 23-years old open problem in one year. EATCS Bulletin 34 (1988), pp. 310–313.Google Scholar
  15. [HU79]
    J.E.Hopcroft, J.D. Ullman, Introduction to automata theory, languages, and computation. Addison-Wesley, 1979.Google Scholar
  16. [Im87]
    N.Immerman, Nondeterministic space is closed under complement. TR 552, Yale University, July 1987, also in Proc. 3rd Ann. Conf. on Structure in Complexity Theory, 1988.Google Scholar
  17. [JR80]
    D. Janssens, G. Rozenberg, Restrictions, extensions, and variations of NLC grammars. Inf. Sci. 20 (1980), pp. 217–244.Google Scholar
  18. [Kr87]
    H.-J.Kreowski, Rule trees can help to escape hard graph problems. Preprint, Universität Bremen, 1987.Google Scholar
  19. [La87]
    C.Lautemann, Efficient algorithms on graphs represented by decomposition trees. Report No. 6/87, Fachbereich Mathematik/Informatik, Universität Bremen, 1987.Google Scholar
  20. [La88]
    C.Lautemann, Decomposition trees: structured graph representation and efficient algorithms. LNCS 299, pp. 28–39.Google Scholar
  21. [Le86]
    T. Lengauer, Efficient algorithms for finding minimum spanning forests of hierarchically defined graphs. LNCS 216 (1986), pp. 153–170.Google Scholar
  22. [LWV84]
    J.Y.-T. Leung, J. Witthof, O. Vornberger, On some variations of the bandwidth minimization problem. SIAM J. Comp. 13 (1984), pp. 650–667.Google Scholar
  23. [Pa72]
    T. Pavlidis, Linear and context-free graph grammars. JACM 19 (1972), pp. 11–22.Google Scholar
  24. [RW86a]
    G. Rozenberg, E. Welzl, Boundary NLC grammars — basic definitions, normal forms and complexity. Information and Control 69 (1986), pp. 136–167.Google Scholar
  25. [RW86b]
    G. Rozenberg, E. Welzl, Graph-theoretic closure properties of the family of boundary NLC languages. Acta Inf. 23 (1986), pp. 289–309.Google Scholar
  26. [RW87]
    G. Rozenberg, E. Welzl, Combinatorial properties of boundary NLC graph languages. Discr. Appl. Math. 16 (1987), pp. 59–73.Google Scholar
  27. [Ru80]
    W.L. Ruzzo, Tree-size bounded alternation. JCSS 20 (1980), pp. 218–235.Google Scholar
  28. [Sl82]
    A.O. Slisenko, Context-free graph grammars as a tool for describing polynomial-time subclasses of hard problems. Inf. Proc. Let. 14 (1982), pp. 52–56.Google Scholar
  29. [Su78]
    I.H. Sudborough, On the tape complexity of deterministic context-free languages. JACM 25 (1978), pp. 405–414.Google Scholar
  30. [Sz87]
    R. Szelepcsényi, The method of forcing for nondeterministic automata. EATCS Bulletin 33 (1987), pp. 96–99.Google Scholar
  31. [Ve87]
    H.Venkateswaran, Properties that characterize LOGCFL. Proc. 19th ACM-STOC (1987), pp. 141–150.Google Scholar
  32. [Vo88]
    W.Vogler, personal communication, March 1988.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Clemens Lautemann
    • 1
  1. 1.Fachbereich Mathematik/InformatikUniversität BremenBremen 33

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