\subsection{Algorithm}
The algorithm of building DAWGs is an iterative process that goes roughly in
-three steps. We start with the null graph that can be described by
+two steps. We start with the null graph that can be described by
$G_0=(\{q_0\},\{q_0\},\{\}\{\})$ and does not contain any edges, one node and
$\mathcal{L}(G_0)=\emptyset$. The first word that is added to the graph will be
added in a naive way. We just create a new node for every transition of
using a stepwise approach.
\begin{enumerate}
\item
- Step one is finding the common prefix of the word in the graph and we chop
- it of the word. When the suffix that remains is empty and the last state of
- the common prefix is a final state we are done adding the word. When the
- suffix is non empty or the common prefix does not end in a final state we
- apply step two.
+ Say we add word $w$ to the grahp. Step one is finding the common prefix of
+ the word already in the graph. The common prefix is defined as the longest
+ subword $w'$ for which there is a $\delta^*(q_0, w')$. When the common
+ prefix is found we change the starting state to the last state of the
+ common prefix and remain with suffix $w''$ where $w=w'w''$.
\item
- We find the first confluence state in the common prefix and from there on
- we add the suffix. When there is a confluence state the route all the way
- back to the starting node will be cloned to make sure no extra words are
- added to the graph unwillingly.
+ We add the suffix to the graph starting from the last node of the common
+ prefix.
+ \item
+ When there are confluence nodes present within the common prefix they are
+ cloned starting from the last confluence node to avoid adding unwanted
+ words.
\item
- Finally if in the suffix propagating from the end to the beginning there
- are equal states we merge the state from the suffix to the equal state in
- the graph.
+ From the last node of the suffix up to the first node we replace the nodes
+ by checking if there are equivalent nodes present in the graph that we can
+ merge with.
\end{enumerate}
Pseudocode for the algorithm can be found in Listing~\ref{pseudodawg}.
\texttt{abcd}, \texttt{aecd} and \texttt{aecf}. In SG0 we begin with the null
graph.
-Adding the first entry \texttt{abcd} is trivial and just creates a single path
-and does not require any hard work. The result of adding the first word is
-visible in subgraph SG1.
-
-For the second entry \texttt{aecd} we will have to follow the steps described
-earlier. The common prefix is \texttt{a} and the common suffix is \texttt{bc}.
-Therefore the first node in which the common prefix starts needs to be copied
-and split off to make room for an alternative route. This is exactly what
-happens in SG2. Node $q_2$ is copied and gets a connection to node $q_3$. From
-the last node of the common prefix a route is built towards the first node,
-that is the copied node, of the common suffix resulting in an extra path
-$q_1\rightarrow q_5\rightarrow q_3$. Since there are no confluence nodes we can
-leave the prefix as it is. This results in subgraph SG2.
-
-The last entry to add is the word \texttt{aecf}. When this is done without
-confluence node checking we create extra paths in the graph that are unwanted.
-The common prefix is \texttt{aec} and the common suffix is \texttt{f}.
-\texttt{f} can not be merged into existing edges and thus a new edge will be
-created from the end of the common suffix to the end. This results in subgraph
-SG3. Note that we did not take care of confluence nodes in the common prefix
-and because of that the word \texttt{abcf} is also accepted even when it was
-not added to the graph. This is unwanted behaviour. Going back to the common
-prefix \texttt{aec}. When we check for confluence nodes we see that the last
-node of the common prefix is such a node and that we need to clone the path
-and separate the route to the final node. $q_3$ will be cloned into $q_6$ with
-the route of the common prefix. Tracking the route back we do not encounter any
-other confluence states and therefore we can merge in the suffix which results
-in the final DAWG visible in subgraph SG4. No new words are added.
+Adding the first entry \texttt{abcd} is trivial because just creates a single
+path and does not require any hard work. Because the common prefix we find in
+Step 1 is empty the suffix will be the entire word. There is also no
+possibility of merging nodes because there were no nodes to begin with. The
+result of adding the first word is visible in subgraph SG1.
+
+For the second entry \texttt{aecd} we will have to do some extra work.
+The common prefix found in Step 1 is \texttt{a} which we add to the graph. This
+leaves us in Step 2 with the suffix \texttt{ecd} which we add too. In Step 3 we
+see that there are no confluence nodes in our common prefix and therefore we
+do not have to clone nodes. In Step 4 we traverse from the last node back to
+the beginning of the suffix and find a common suffix \texttt{cd} and we can
+merge these nodes. In this way we can reuse the transition from $q_3$ to $q_4$.
+This leaves us with subgraph SG2.
+
+We now add the last entry which is the word \texttt{aecf}. When we do this
+without the confluence node checking we encounter an unwanted extra word. In
+Step 1 we find the common prefix \texttt{aec} and that leaves us with the
+suffix \texttt{f} which we will add. This creates subgraph SG3 and we notice
+there is an extra word present in the graph, namely the word \texttt{abcf}.
+This extra word appeared because of the confluence node $q_3$ which was present
+in the common prefix introducing an unwanted path. Therefore in Step 3 when we
+check for confluence node we would see that node $q_3$ is a confluence node and
+needs to be cloned off the original path, by cloning we take the added suffix
+with it to create an entirely new route. Tracking the route back again we do
+not encounter any other confluence nodes so we can start Step 4. For this word
+there is no common suffix and therefore no merging will be applied. This
+results in subgraph SG4 which is the final DAWG containing only the words
+added.
\begin{figure}[H]
\caption{Step 1}
repositories / bsc-thesis1415.git / blobdiff