[Solved]: Deleting useless (dead) states from a finite automaton

Problem Detail: 

A useless state in a finite automaton is one from which no path leads to a final state, hence no (piece of a string) is recognized out of this state. Theoretically, the algorithm to determine the useful states is trivial: Let $G$ be the set of good (useful) states and let $\Omega$ be the set of all states. Initialize $G$ with all final states. Check all states $\Omega\setminus G$ for those that have a transition to a state in $G$ and add them to $G$. Repeat until nothing is added to $G$ any more.

A straightforward implementation mimicking the above, however, can be quite costly, looping over states and transitions over and over again. The number of loops checking $\Omega \setminus G$ is limited by the depth.

In a degenerate automaton containing transitions $a_0\to a_1 \to\dots\to a_n\to f$ where $f$ is a single final state and an additional transition $a_n\to u$ such that $u$ is useless, there would be around $n$ loops if you always loop over the $a_i$ in the order of $i$. But if you loop in decreasing order of $i$ shuffling found good states immediately into $G$, a single loop would suffice.

But this may lead to other degenerate situations (I am guessing).

I am looking for an algorithm to mark useful states, or remove useless states, that is not recursive (to prevent stack overflow, since I am looking at FAs with millions of states) and as efficient as possible.

Extra question: are there theoretical limits known for this algorithm?

Asked By : Harald

Answered By : Yuval Filmus

Reverse all edges in the graph, and add a new state pointing at all accepting states. Find the set of states reachable from the new state in linear time (using BFS/DFS). These are the useful states (according to your definition).

The running time of this algorithm is linear in the number of states plus number of transitions. This is about the best you can hope for (in terms of asymptotic worst-case running time).

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Question Source : http://cs.stackexchange.com/questions/49819

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