On Prefix-Sorting Finite Automata
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Being able to efficiently test the membership of a word in a formal language is, arguably, one of the most fundamental problems in computer science. In this paper, we apply string-processing techniques (specifically, prefix-sorting) to speed up solutions for this problem on languages described by particular finite automata. Prefix sorting can be generalized to objects more complex than strings: the recent notion of Wheeler graph extends this concept to labeled graphs (for example, finite-state automata). A Wheeler graph admits a co-lexicographic ordering of its nodes and opens up the possibility of building fast and small data structures supporting path queries on the graph. However, while strings and trees can always be prefix-sorted in linear time, the situation on graphs is more complicated: not all graphs are Wheeler, and a recent result shows that the problem of identifying them is NP-complete even for acyclic NFAs. In this paper, we present the following results: (i) we show that the problem of recognizing and sorting Wheeler DFAs (even cyclic) can be solved offline in optimal linear time, (ii) when the DFA is acyclic, we give an online algorithm that, when fed with the DFA's states in any topological order, with logarithmic delay computes the co-lexicographic rank of a new incoming state or decides that such an ordering does not exist, and (iii) we show that also Wheeler 2-NFAs---i.e. those with at most two equally-labeled transitions leaving any state---admit a polynomial-time identification and prefix-sorting algorithm. Our algorithms (i)-(ii) generalize to labeled graphs existing prefix-sorting algorithms on strings and labeled trees that have been previously described in the literature.
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