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Problem 60: Minimal Absent Words

Words that do not occur in files provide a useful technique to discard or discriminate files. They act like the set of factors of a file but admit a more compact trie representation as factor codes.

A word $w$, $|w|\gt 1$, is said to be absent or forbidden in a word $x$ if it does not occur in $x$. It is said to be minimal with this property if in addition both $w[0\dd |w|-2]$ and $w[1\dd |w|-1]$ do occur in $x$.

A natural method to compute the minimal words absent in $x$ is to start with its Factor automaton $\aFact(x)$, smallest deterministic automaton accepting all its factors. Each factor is spelt out along a path starting from the initial state and all states are terminal.

Design a linear-time algorithm to compute the trie of minimal absent words of a word $x$ from its Factor automaton $\aFact(x)$.
Use failure links of the automaton.

Design a linear-time algorithm to recover the Factor automaton of a word $x$ from the trie of its minimal absent words.
Use the Aho-Corasick technique.


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