Question

How can a DAWG be created? I have found that there are two ways; one is converting a trie into a dawg and the other being creating a new DAWG straight away? Which one is the easiest? Can you please elaborate on the two and provide some links?

Answer 1

One way to think about the DAWG is as a minimum-state DFA for all of the words in your word list. As a result, the traditional algorithm for constructing a DAWG is the following:

1. Start off by constructing a trie for the collection of words.
2. Add a new node to the trie with edges from itself to itself on all inputs.
3. For each missing letter transition in the trie, add a transition from the start node to this new dead node.
4. (At this point, you now have a (probably non-minimum) DFA for the set of words.)
5. Minimize the DFA using the standard algorithm for DFA state minimization.

Once you have done this, you will be left with a DAWG for the set of words you are interested in.

The runtime of this algorithm is as follows. Constructing the initial DFA can be done by constructing a trie for all the original words (which takes time O(n), where n is the total number of characters in all the input strings), then filling in the missing transitions (which takes time O(n|Σ|), where |Σ| is the number of different characters in your alphabet). From there, the minimization algorithm runs in time O(n² |Σ|). This means that the overall runtime for the algorithm is O(n² |Σ|).

To the best of my knowledge, there is no straightforward algorithm for incrementally constructing DAWGs. Typically, you would build a DAWG for a set of words only if you already had all the words in advance. Intuitively, this is true because inserting a new word that has some suffixes already present in the DAWG might require a lot of restructuring of the DAWG to make certain old accepting states not accepting and vice-versa. Theoretically speaking, this results because inserting a new word might dramatically change the equivalence classes of the DFA’s distinguishability relation, which might require substantial changes to the DFA’s structure.

Hope this helps!