Analysis of a class of algorithms for problems on trace languages

  • A. Bertoni
  • M. Goldwurm
  • N. Sabadini
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 307)


The time complexity of a class of algorithms for problems on trace languages is studied both in the worst and in the average case. Membership problem for regular trace languages and the problem of counting the cardinality of a trace can be solved by algorithms belonging to such a class. These algorithms are described by a recursive procedure scheme whose input is a trace represented by a corresponding string. Under reasonable assumptions the worst case time complexity on traces of length n is ϑ(nα), where α is the cardinality of the largest clique in the concurrent alphabet <σ,C>.

The average case behaviour is studied under two different assumptions on the distribution of input set. In the first case equidistributed strings of length n are considered and it is proved that the average time complexity is ϑ<nk), where k is the number of connected components of the complementary alphabet <σ,Cc>. In the second case equally probable traces of length n are considered and it is proved that the average time complexity is ϑ(nh), where h is the multeplicity of the minimum modulus root of the polynomial σj=0,α(-)j Γj xj and each Γj is the number of j-cliques in <σ,C>.


Time Complexity Random Graph Maximum Clique Main Route Transition Diagram 
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  1. [1]
    Bertoni A., Brambilla M.,Mauri G., Sabadini N., “An application of the theory of free partially commutative monoids: asymptotic densities of trace languages”, Lect. Not. Comp. Sci. 118, Springer Berlin, 1981, pp. 205–215.Google Scholar
  2. [2]
    Bertoni A., Mauri G., Sabadini N., “Equivalence and membership problems for regular trace languages”, Proc. 9th ICALP, Lect. Not. Comp. Sci. 140 (Nielsen-Schmidt ed's), pp. 61–71, 1982.Google Scholar
  3. [3]
    Bertoni A., Mauri G., Sabadini N., “Representation of prefixes of a trace and membership problem for context free trace languages”, Int. Rep. Ist. Cib. Univ. Milano, 1985.Google Scholar
  4. [4]
    Cartier P., Foata D., “Problemes combinetories de commutation et rearrangements”, Lect. Not. in Math. 85, Springer, Berlin, 1969.Google Scholar
  5. [5]
    Erdos P., Renyi A., “On random graphs”, Publ. Math. Debrecen 6 (1959) pp. 290–297.Google Scholar
  6. [6]
    Grimmet G.R., McDiarmid C.J.H., “On colouring random graph”, Math. Proc. Camb. Philos. Soc. 77 (1975), pp.313–324.Google Scholar
  7. [7]
    Janicki R., “Synthesis of concurrent schemes”, Lect. Not. in Comp. Sci. 64, Springer Berlin (1978), pp. 298–307.Google Scholar
  8. [8]
    Lallement G., “Semigroups and combinatorial applications”, J. Wiley and sons, 1979.Google Scholar
  9. [9]
    Mazurkiewicz A., “Concurrent program schemes and their interpretations”, DAIMI, PB 78, Aarhus University, 1977.Google Scholar
  10. [10]
    Rozenberg G., Verraedt R., “Subset languages of Petri nets”, Proc. 3rd European workshop on applications and theory of Petri nets, Varenna Italy, 1982.Google Scholar
  11. [11]
    Szijàrtò M., “Trace languages and closure operations”, Automata Theoretic Letters, 1979/2, Dep. of Num. and Comp. Math., Eotvos Univ.,Budapest 1979.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • A. Bertoni
    • 1
  • M. Goldwurm
    • 1
  • N. Sabadini
    • 1
  1. 1.Dipartimento di Scienze dell'InformazïoneUniversità di MilanoItaly

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