A Wheeler automaton is a finite state automaton whose states admit a total Wheeler order, reflecting the co-lexicographic order of the strings labeling source-to-node paths). A Wheeler language is a regular language admitting an accepting Wheeler automaton. Wheeler languages admit efficient and elegant solutions to hard problems such as automata compression and regular expression matching, therefore deciding whether a regular language is Wheeler is relevant in applications requiring efficient solutions to those problems. In this paper, we show that it is possible to decide whether a DFA with n states and m transitions recognizes a Wheeler language in O(mn) time. This is a significant improvement over the running time $$O(n^{13} + m\log n)$$ of the previous polynomial-time algorithm (Alanko et al. Information and Computation 2021). A proof-of-concept implementation of this algorithm is available in a public repository. We complement this upper bound with a conditional matching lower bound stating that, unless the strong exponential time hypothesis (SETH) fails, the problem cannot be solved in strongly subquadratic time. The same problem is known to be PSPACE-complete when the input is an NFA (D’Agostino et al. Theoretical Computer Science 2023). Together with that result, our paper essentially closes the algorithmic problem of Wheeler language recognition.
