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[AD] Update summary to new version of helpers

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2025-09-26 12:31:55 +02:00
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semester1/algorithms-and-datastructures/parts/graphs/all-pairs-sp.tex
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@@ -27,7 +27,7 @@ We can also use $n$-times dijkstra or any other shortest path algorithm, or any
\begin{properties}[]{Characteristics of Floyd-Warshall Algorithm}
\begin{itemize}
\item \textbf{Time Complexity:} \tco{|V|^3}.
\item \textbf{Time Complexity:} $\tco{|V|^3}$.
\item Works for graphs with negative edge weights but no negative weight cycles.
\item Computes shortest paths for all pairs in one execution.
\end{itemize}
@@ -82,7 +82,7 @@ We can also use $n$-times dijkstra or any other shortest path algorithm, or any
\item Reweight edges to remove negative weights.
\item Use Dijkstra's algorithm for each vertex to find shortest paths.
\end{enumerate}
\item \textbf{Time Complexity:} \tco{|V| \cdot (|E| + |V| \log |V|)}.
\item \textbf{Time Complexity:} $\tco{|V| \cdot (|E| + |V| \log |V|)}$.
\item Efficient for sparse graphs compared to Floyd-Warshall.
\end{itemize}
\end{properties}
@@ -140,8 +140,8 @@ We can also use $n$-times dijkstra or any other shortest path algorithm, or any
\toprule
\textbf{Algorithm} & \textbf{Primary Use} & \textbf{Time Complexity} & \textbf{Remarks} \\
\midrule
Floyd-Warshall & AP-SP & \tco{|V|^3} & Handles negative weights \\
Johnson’s Algorithm & AP-SP (sparse graphs) & \tco{|V|(|E| + |V| \log |V|)} & Requires reweighting \\
Floyd-Warshall & AP-SP & $\tco{|V|^3}$ & Handles negative weights \\
Johnson’s Algorithm & AP-SP (sparse graphs) & $\tco{|V|(|E| + |V| \log |V|)}$ & Requires reweighting \\
\bottomrule
\end{tabular}
\caption{Comparison of the All-Pair Shortest path (AP-SP) algorithms discussed in the lecture}