Talking about AC automata from trie

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Trie

Trie is a fun data structure. Each of its nodes is a letter, hence the name ' Letter tree '.

Out a picture to let everyone feel the next.

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It's a tree that's plugged in.

Ape,app,applicant,application,bake,ban,banana

The trie of such words. The red node indicates the accepted state.

Obviously, the search simply follows the chain, inserting only the edges to find the edges.

(delete simply removes the accepted state, or at this point it is deleted to its most recently accepted parent node when it has no child nodes.)

understood good writing efficiency.

AC automatic Machine

AC automata is a data structure based on trie. It is a real automaton.

The AC automaton, in short, is a trie with some strange things added.

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The solid line represents the path on the trie, and the dashed lines represent the fail pointer.

What is the fail pointer? When you are on this node, the path you want to take when the next character is mismatched. Very similar to the next array of KMP.

Its definition is basically the same. The longest suffix with the same prefix (the pointer to the last letter node of the prefix).

It's much the same. The calculation is the same. Jump to the S[p-1 (Next[k]) fail until you can match.

Simple, huh? It's like KMP.

So the question is again. How to output it?

Let's draw a picture and think about it.

Originally, follow the fail pointer to go down can ah ...

Then we have an algorithm that matches the AC automaton.

Match the first ' a ' and look down from root

Find a match, a match plus one. Look down for the next character to find null.

The fail pointer jumps back to root. Look for the character C.

Look for the character B. is not the accepted state, continue.

A is a receiving state. Walk back along the fail pointer to output the result.

-------------------

Matches C-fail-jump.

------------------------------

...........................

................... Eventually

Results

construct an AC automaton

So, how to construct an AC automaton?

It is obvious that you can go along with the fail pointer of the father at each join, go to the first found node with the same character clause, and point the fail to that node, and if there is no same character node in root, fail points to root.

This is an online approach. Offline Of course can be solved with BFS, so that when a node is processed, its father that layer of the node has been processed. The Fail node is only recently possible on the father's level. The complexity should be the same as the online approach.

Update: Made a mistake, ac automaton is not online. Cause the fail pointer may change.

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Talking about AC automata from trie