another graph update

diff --git a/thesis2/.gitignore b/thesis2/.gitignore
index f019367..ebea2d2 100644 (file)
--- a/thesis2/.gitignore
+++ b/thesis2/.gitignore
@@ -5,7 +5,7 @@
 *.toc
 *.dvi
 *.bbl
-*.bbg
+*.blg
 *.pdf
 *.ps
 *.dot

diff --git a/thesis2/1.introduction.tex b/thesis2/1.introduction.tex
index 129e38d..c3c3588 100644 (file)
--- a/thesis2/1.introduction.tex
+++ b/thesis2/1.introduction.tex
@@ -290,6 +290,7 @@ techniques of recognizing data in text can still be used to interpret the
 isolated extracted parts from this project.
 
 \section{Directed Acyclic Graphs}
+\paragraph{Normal graphs}
 A graph($G$) is a mathematical structure to describe relations between nodes
 with edges. A standard graph is defined as the ordered pair: $G=(V,E)$.  In
 this ordered pair $V$ is the set of nodes and $E$ is set of undirected edges
@@ -309,28 +310,55 @@ Figure~\ref{fig:graphexample} is specified as:
        }
 \end{figure}
 
+\paragraph{Directed graphs}
+Directed graphs are special kind of graphs. A directed graph $G$ is also
+defined by the ordered pair: $G=(V,E)$. $V$ still defines the nodes and $E$
+still the edges but the inherent difference is that the edges are ordered
+tuples in stead of not ordered. Adding this property gives the edges a
+direction. Every edge has a specific start and end and are therefore called
+directed edges. A directed graph would look the same as the graph in
+Figure~\ref{fig:graphexample} but then the normal edges would be replaced by
+directional arrows that specifically go from one node to the other.
+
+\paragraph{Directed acyclic graphs}
+Directed Acyclic Graphs(DAGs) are a special kind of directed graphs. DAGs
+are also defined as $G=(V,E)$ but with a restriction on $E$. Namely that cycles
+are not allowed. Figure~\ref{fig:dagexample} shows two graphs. The left graph
+contains a cycle and the right graph does not. Only the right graph is a valid
+DAG. A cycle is defined by a sequence of edges where nodes are visited more
+then once. Adding the property of non-cyclicity to graphs lowers the
+computational complexity of checking if a node sequence is present in the
+graph to $\mathcal{O}(L)$ where $L$ is the length of the sequence.
 
-Directed graphs are special kinds of graphs. A directed graph is described as
-the ordered pair: $G=(V,A)$. Where $A$ is a set of ordered pairs that we will
-call \textit{directed edges}. Directed edges can be visualized as arrows in a
-graph whereas undirected edges are just lines connecting the nodes with no
-particular direction.
-
-Directed Acyclic Graphs(DAGs) are again a special kind of directed graph. DAGs
-don't allow cycles to be created. Every node is only reachable at maximum once
-when starting from an arbitrary point in the graph. DAGs have the nice property
-of checking if a sequence appears in the graph, checking if a sequence is
-present in a graph only has a computational complexity of $\mathcal{O}(L)$
-where $L$ is the length of the sequence.
-
-The type of graph used in the project is another special king of DAG called
-Directed Acyclic Word Graphs(DAWGs). This type of graph can be used to store
-large dictionaries of words. A DAWGs nodes basically represent indices of a
-word and the edges describe the character added when traversing. For example
-the graph in Figure~\ref{fig:2.1.1} can describe a dictionary that holds the
-words: \textit{abd, bad, bae}. Testing if a word is present in the DAWG is,
-just as for a DAG, falls als in the computational complexity class of
-$\mathcal{O}(L)$ meaning that it grows linearly with the length of the word.
+\begin{figure}[H]
+       \caption{Example DAG}
+       \label{fig:dagexample}
+       \centering
+       \digraph[]{dagexample}{
+               rankdir=LR
+               n01 -> n02
+               n02 -> n03
+               n03 -> n01
+               n11 -> n12
+               n12 -> n13
+               n12 -> n14
+       }
+\end{figure}
+
+\paragraph{Directed Acyclic Word Graphs}
+The type of graph used in the project is a special kind of DAG called
+Directed Acyclic Word Graphs(DAWGs). A DAWG can be defined by the ordered pair
+$G=(V,E)$ and is the same as a directed graph except for the edges. An edge in
+a DAWG is instead of a tuple a triple and consist of a starting point, an end
+point and a label. Because of the property of labeled edges data can be stored
+in a DAWG. When traversing a DAWG and saving all the edgelabels one can
+construct words. Because of properties of graphs a DAWG can store big sets of
+words in a small amount of storage because it can re-use edges to specify
+transitions. For example the graph in Figure~\ref{fig:2.1.1} can describe a
+dictionary that holds the words: \textit{abd, bad, bae}. Testing if a word is
+present in the DAWG is, just as for a DAG, falls in the computational
+complexity class of $\mathcal{O}(L)$ meaning that it grows linearly with the
+length of the word.
 
 \begin{figure}[H]
        \caption{Example DAWG}