## Revision c6e2ce5a thesis/ideas.tex

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thesis/ideas.tex
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        feel of

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\section{Experiment setup}

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 Cores is a hardware term that describes the number of independent central processing units in a single computing component (die or chip).

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       A Thread, or thread of execution, is a software term for the basic ordered sequence of instructions that can be passed through or processed by a single CPU core.

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       Intel® Hyper-Threading Technology (Intel® HT Technology) delivers two processing threads per physical core. Highly threaded applications can get more work done in parallel, completing tasks sooner.

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\section{Questions}

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    Then, 1 chapter about the result

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    Bibliography: 20 - 3.0

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\section{Scope}

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        The scope of the paper:

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        \begin{itemize}

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            \item no self-loop

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            \item no multigraph: graph with no parallel edges

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            \item unweighted and weighted graph

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            \item can apply to connected or disconnected graph. In the disconnected graph, certain vertices are not reachable from vertices in a different component. This yeilds the zero shortest paths counts, and as a result those 2 endpoints contributes 0 toward the betweenness.

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        \end{itemize}

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    connected graph: since the betweenness centrality is calculated based on shortest paths. In the disconnected graph, certain vertices are not reachable from vertices in a different component. This yields the zero shortest paths counts for betweenness centrality.

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\section{Notation}

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    A graph $G=(V,E)$ contains a set of vertices (nodes) $V$ and a set of edges (links) $E$. We assume we assume that al graphs are undirected and connected. \textbf{XXX check the modification for directed graph}

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    $m = |E|$ is the number of edges in graph $G$

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    $n = |V|$ is the number of vertices in graph $G$

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    Let $\omega$ be a weight function on the edges. We assume that $\omega(e) > 0, e \in E$ for weighted graphs. And we define $\omega(e) = 1, e \in E$ for unweighted graphs. If for $s, t \in V$ and ${s, t} \in E$, then we can also use $\omega(s, t)$ to denote the weight of the edge between $s$ and $t$.

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    The \emph{length} of a path is the sum of weights of all edges along the path. We use $d(s, t)$ to denote the \emph{distance} between vertices $s$ and $t$, i.e. the minimum length of any path connecting $s$ and $t$.

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\section{Result}

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        \begin{figure}[h]

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        \caption{}

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        \label{fig:histogram_num_of_nodes_CN200}

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        \centering

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        \includegraphics[scale=0.6]{images/histogram_num_of_nodes_CN200.png}

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        \end{figure}

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        \begin{figure}[h]

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        \caption{}

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        \label{fig:histogram_num_of_nodes_ER200}

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        \centering

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        \includegraphics[scale=0.6]{images/histogram_num_of_nodes_ER200.png}

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        \end{figure}

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        \begin{figure}[h]

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        \caption{}

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        \label{fig:histogram_num_of_nodes_PL200}

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        \centering

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        \includegraphics[scale=0.6]{images/histogram_num_of_nodes_PL200.png}

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        \end{figure}

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\section{Counting shortest paths}

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    In the end, I choose not to describe those algorithms in the final version of my thesis

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    \textbf{XXX What is the input}: the source vertex $s$

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        \textbf{XXX What is the output?}:

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        \subsection{Breadth-First Search for unweighted graph}

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            I understood the algorithm, but how can I explain it to another person?

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        \subsection{Dijkstra's algorithm for weighted graph}

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            The Dijkstra's algorithm can be modified so that when given a source vertex $s$, the output will be the set of predecessors and the number of shortest paths from $s$ to other $t \neq s \in V$

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            \begin{algorithm}

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            \caption{Dijkstra's algorithm} \label{algo:dijkstra}

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            \begin{algorithmic}[1]

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                \Procedure{Dijkstra for SSSP with source vertex s}{}

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                    \State $S \leftarrow$ empty stack

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                    \State $P[w] \leftarrow$ empty list, $w \in V$

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                    \State $\sigma[t] \leftarrow 0, t \in V$; $\sigma[s] \leftarrow 1$

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                    \State $d[t] \leftarrow -1, t \in V$; $d[s] \leftarrow 0$

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                    \State $Q \leftarrow$ empty queue

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                    \State enqueue $s \rightarrow Q$

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                    \While{$Q$ \emph{not empty}}

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                        dequeue $v \leftarrow Q$

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                        push $v \rightarrow S$

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                    \EndWhile

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                \EndProcedure

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            \end{algorithmic}

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            \end{algorithm}

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\clearpage

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\bibliography{references}{}

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\bibliographystyle{plain}


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