# Stochastic Analysis of Minimal Automata Growth for Generalized Strings

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## Abstract

Generalized strings describe various biological motifs that arise in molecular and computational biology. In this manuscript, we introduce an alternative but efficient algorithm to construct the minimal deterministic finite automaton (DFA) associated with any generalized string. We exploit this construction to characterize the typical growth of the minimal DFA (i.e., with the least number of states) associated with a random generalized string of increasing length. Even though the worst-case growth may be exponential, we characterize a point in the construction of the minimal DFA when it starts to grow linearly and conclude it has at most a polynomial number of states with asymptotically certain probability. We conjecture that this number is linear.

## Keywords

Aho-Corasick algorithm Deterministic finite automaton Generalized string Minimization Motif Polynomial growth## Mathematics Subject Classification (2010)

68Q25 68Q45 68Q87 68W40## Preview

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## Notes

### Acknowledgements

We are thankful to two anonymous referees for their careful reading of this paper and valuable suggestions. We are also very thankful to Dr. Dougherty for partially funding this research through her NSF EXTREEMS training grant.

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