# Analysis of a class of algorithms for problems on trace languages

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## Abstract

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(n^{α}), 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<n^{k}), where k is the number of connected components of the complementary alphabet <σ,C^{c}>. In the second case equally probable traces of length n are considered and it is proved that the average time complexity is ϑ(n^{h}), where h is the multeplicity of the minimum modulus root of the polynomial σ_{j=0,α}(-)^{j} Γ_{j} x^{j} and each Γ_{j} is the number of j-cliques in <σ,C>.

## Keywords

Time Complexity Random Graph Maximum Clique Main Route Transition Diagram## Preview

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## References

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