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latex/note_w11.tex
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\section{Week 11}
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\subsection{ipkg = Itsy Package Management System}
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    ipkg, or the Itsy Package Management System, is a lightweight package management system designed for embedded devices that resembles Debian's dpkg.
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    opkg is the fork of ipkg
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    http://lists.openmoko.org/pipermail/devel/2008-July/000496.html
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\subsection{Shortest Path Problems}
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    Single-source Shortest-Paths
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    \begin{itemize}
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        \item Dijkstra
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            \begin{itemize}
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                \item priority queue: $O(|V|^2)$
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                \item binary heap: $O(|E| log|V|)$
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                \item fibonacci heap: $O(|V| log|V| + |E|)$
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            \end{itemize}
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        \item Bellman-Ford: $O(|V|\times|E|)$, worst case: $O(|V|^3)$
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    \end{itemize}
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    Algorithms for the All-Pairs Problem
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    \begin{itemize}
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        \item Iteration $O(|V|^4)$
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        \item Iteration with Doubling Up $O(|V|^3 \times log|V|$
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        \item Floyd-Warshall Algorithm: $O(|V|^3)$
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    \end{itemize}
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\subsection{Matrix}
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    \textbf{Multiplication} of $n \times n$-matrices
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    $A \otimes B = (c_{ij})$ where $c_{ij} = \oplus_{k=1}^{n} a_{ik} \otimes b_{kj}$
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    $k$-th \textbf{power} of matrix A:
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    $A^k = (d_{ij}$ where $d_{ij} = \oplus_{r = 0}^{k - 1} a_{ir} \otimes a_{rj}, A^0 = I$
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    \textbf{XXX} I have the feeling that the definition from the \href{https://www.iam.unibe.ch/~run/talks/2008-06-05-Bern-Jonczy.pdf}{slide} is wrong. Wolfram gave a different definition \href{http://mathworld.wolfram.com/MatrixPower.html}{here}
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    The \textbf{closure} of the matrix A: $A^{*} = \oplus_{k >= 0} A^k$
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    \subsubsection{Other materials}
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        Check out slide for ICS 6D in \texttt{reading/w11 - MatrixMultiplication}. It details how the matrix multiplication works
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\subsection{Algebraic Path Problem (APP)}
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    https://www.iam.unibe.ch/~run/talks/2008-06-05-Bern-Jonczy.pdf
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    The Algebraic Path Problem consists in performing a special unary operation, called the closure , over a square matrix with entries in a semiring [Fink, 1992]
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    For example, Problem of computing the length of the shortest path (for all pairs): APP over graph \texttt{G} and tropical semiring \texttt{Trop}.
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    \textbf{XXX But I don't really understand...}

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