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1 2 3 1bac9234 Quynh PX Nguyen %!TEX root = note.tex  %%%%%%%%%%%%%%%%%%  % WEEK 11  %%%%%%%%%%%%%%%%%%  \section{Week 11}  \subsection{ipkg = Itsy Package Management System}  b56a7ca2 Quynh PX Nguyen  ipkg, or the Itsy Package Management System, is a lightweight package management system designed for embedded devices that resembles Debian's dpkg.   opkg is the fork of ipkg   http://lists.openmoko.org/pipermail/devel/2008-July/000496.html  \subsection{Shortest Path Problems}   Single-source Shortest-Paths   \begin{itemize}   \item Dijkstra   \begin{itemize}   \item priority queue: $O(|V|^2)$   \item binary heap: $O(|E| log|V|)$   \item fibonacci heap: $O(|V| log|V| + |E|)$   \end{itemize}   \item Bellman-Ford: $O(|V|\times|E|)$, worst case: $O(|V|^3)$   \end{itemize}   Algorithms for the All-Pairs Problem   \begin{itemize}   \item Iteration $O(|V|^4)$   \item Iteration with Doubling Up $O(|V|^3 \times log|V|$   \item Floyd-Warshall Algorithm: $O(|V|^3)$   \end{itemize}  \subsection{Matrix}   \textbf{Multiplication} of $n \times n$-matrices   $A \otimes B = (c_{ij})$ where $c_{ij} = \oplus_{k=1}^{n} a_{ik} \otimes b_{kj}$   $k$-th \textbf{power} of matrix A:   $A^k = (d_{ij}$ where $d_{ij} = \oplus_{r = 0}^{k - 1} a_{ir} \otimes a_{rj}, A^0 = I$   \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}   The \textbf{closure} of the matrix A: $A^{*} = \oplus_{k >= 0} A^k$   \subsubsection{Other materials}   Check out slide for ICS 6D in \texttt{reading/w11 - MatrixMultiplication}. It details how the matrix multiplication works  \subsection{Algebraic Path Problem (APP)}   https://www.iam.unibe.ch/~run/talks/2008-06-05-Bern-Jonczy.pdf   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]   For example, Problem of computing the length of the shortest path (for all pairs): APP over graph \texttt{G} and tropical semiring \texttt{Trop}.   \textbf{XXX But I don't really understand...}