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[AD] Update summary to new version of helpers

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2025-09-26 12:31:55 +02:00
parent 33f034fdd1
commit eecaab61fc
16 changed files with 110 additions and 109 deletions
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4
semester1/algorithms-and-datastructures/parts/graphs/mst/boruvka.tex
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@@ -8,8 +8,8 @@
\begin{itemize}
\item \textbf{Graph Type:} Works on undirected, weighted graphs.
\item \textbf{Approach:} Greedy, component-centric.
\item \textbf{Time Complexity:} \tct{(|V| + |E|) \log(|V|)}.
\item \textbf{Space Complexity:} Depends on the graph representation, typically \tct{E + V}.
\item \textbf{Time Complexity:} $\tct{(|V| + |E|) \log(|V|)}$.
\item \textbf{Space Complexity:} Depends on the graph representation, typically $\tct{E + V}$.
\item \textbf{Limitations:} Efficient for parallel implementations but less commonly used in practice compared to Kruskal's and Prim's.
\end{itemize}
\end{properties}

12
semester1/algorithms-and-datastructures/parts/graphs/mst/kruskal.tex
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@@ -8,9 +8,9 @@
\begin{itemize}
\item \textbf{Graph Type:} Works on undirected, weighted graphs.
\item \textbf{Approach:} Greedy, edge-centric.
\item \textbf{Time Complexity:} \tco{|E| \log (|E|)} (for sort), \tco{|V| \log(|V|)} (for union find data structure).\\
\timecomplexity \tco{|E| \log(|E|) + |V| \log(|V|)}
\item \textbf{Space Complexity:} Depends on the graph representation, typically \tct{E + V}.
\item \textbf{Time Complexity:} $\tco{|E| \log (|E|)}$ (for sort), $\tco{|V| \log(|V|)}$ (for union find data structure).\\
\timecomplexity $\tco{|E| \log(|E|) + |V| \log(|V|)}$
\item \textbf{Space Complexity:} Depends on the graph representation, typically $\tct{E + V}$.
\item \textbf{Limitations:} Requires sorting of edges, which can dominate runtime.
\end{itemize}
\end{properties}
@@ -127,8 +127,8 @@
\begin{properties}[]{Performance}
\begin{itemize}
\item \textsc{make}$(V)$: Initialize data structure \tco{n}
\item \textsc{same}$(u, v)$: Check if two components belong to the same set \tco{1} or \tco{n}, depending on if the representant is stored in an array or not
\item \textsc{union}$(u, v)$: Combine two sets, \tco{\log(n)}, in Kruskal we call this \tco{n} times, so total number (amortised) is \tco{n \log(n)}
\item \textsc{make}$(V)$: Initialize data structure $\tco{n}$
\item \textsc{same}$(u, v)$: Check if two components belong to the same set $\tco{1}$ or $\tco{n}$, depending on if the representant is stored in an array or not
\item \textsc{union}$(u, v)$: Combine two sets, $\tco{\log(n)}$, in Kruskal we call this $\tco{n}$ times, so total number (amortised) is $\tco{n \log(n)}$
\end{itemize}
\end{properties}

6
semester1/algorithms-and-datastructures/parts/graphs/mst/prim.tex
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@@ -12,10 +12,10 @@
\item \textbf{Approach:} Greedy, vertex-centric.
\item \textbf{Time Complexity:}
\begin{itemize}
\item With an adjacency matrix: \tct{V^2}.
\item With a priority queue and adjacency list: \tct{(|V| + |E|) \log(|V|)}.
\item With an adjacency matrix: $\tct{V^2}$.
\item With a priority queue and adjacency list: $\tct{(|V| + |E|) \log(|V|)}$.
\end{itemize}
\item \textbf{Space Complexity:} Depends on the graph representation, typically \tct{E + V}.
\item \textbf{Space Complexity:} Depends on the graph representation, typically $\tct{E + V}$.
\item \textbf{Limitations:} Less efficient than Kruskal's for sparse graphs using adjacency matrices.
\end{itemize}
\end{properties}