Move A&D summary

semester1/algorithms-and-datastructures/parts/graphs/bfs-dfs.tex

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+\newpage
+\subsection{Topological Sorting / Ordering}
+\begin{definition}[]{Topological Ordering}
+    A \textbf{topological ordering} of a directed acyclic graph (DAG) $G = (V, E)$ is a linear ordering of its vertices such that for every directed edge $(u, v) \in E$, vertex $u$ comes before vertex $v$ in the ordering.
+\end{definition}
+
+\begin{properties}[]{Topological Ordering}
+    \begin{itemize}
+        \item A graph has a topological ordering if and only if it is a DAG.
+        \item The ordering is not unique if the graph contains multiple valid sequences of vertices.
+        \item Common algorithms to compute topological ordering:
+              \begin{itemize}
+                  \item \textbf{DFS-based Approach:} Perform a depth-first search and record vertices in reverse postorder.
+                  \item \textbf{Kahn’s Algorithm:} Iteratively remove vertices with no incoming edges while maintaining order.
+              \end{itemize}
+    \end{itemize}
+\end{properties}
+
+
+
+
+\newpage
+\subsection{Graph search}
+\subsubsection{DFS}
+\begin{definition}[]{Depth-First Search (DFS)}
+    \textbf{Depth-First Search} is an algorithm for traversing or searching a graph by exploring as far as possible along each branch before backtracking.
+\end{definition}
+
+\begin{algorithm}
+    \caption{Depth-First Search (Recursive)}
+    \begin{algorithmic}[1]
+        \Procedure{DFS}{$v$}
+            \State \textbf{Mark} $v$ as visited
+            \For{each neighbor $u$ of $v$}
+                \If{$u$ is not visited}
+                    \State \Call{DFS}{$u$}
+                \EndIf
+            \EndFor
+        \EndProcedure
+    \end{algorithmic}
+\end{algorithm}
+
+\begin{properties}[]{Depth-First Search}
+    \begin{itemize}
+        \item Can be implemented recursively or iteratively (using a stack).
+        \item Time complexity: \tco{|V| + |E|}, where $|V|$ is the number of vertices and $|E|$ is the number of edges.
+        \item Used for:
+              \begin{itemize}
+                  \item Detecting cycles in directed and undirected graphs.
+                  \item Finding connected components in undirected graphs.
+                  \item Computing topological ordering in a DAG.
+              \end{itemize}
+    \end{itemize}
+\end{properties}
+
+
+\subsubsection{BFS}
+\begin{definition}[]{Breadth-First Search (BFS)}
+    \textbf{Breadth-First Search} is an algorithm for traversing or searching a graph by exploring all neighbors of a vertex before moving to the next level of neighbors.
+\end{definition}
+
+\begin{algorithm}
+    \caption{Breadth-First Search (Iterative)}
+    \begin{algorithmic}[1]
+        \Procedure{BFS}{$start$}
+            \State \textbf{Initialize} queue $Q$ and mark $start$ as visited
+            \State \textbf{Enqueue} $start$ into $Q$
+            \While{$Q$ is not empty}
+                \State $v \gets \textbf{Dequeue}(Q)$
+                \For{each neighbor $u$ of $v$}
+                    \If{$u$ is not visited}
+                        \State \textbf{Mark} $u$ as visited
+                        \State \textbf{Enqueue} $u$ into $Q$
+                    \EndIf
+                \EndFor
+            \EndWhile
+        \EndProcedure
+    \end{algorithmic}
+\end{algorithm}
+
+\begin{properties}[]{Breadth-First Search}
+    \begin{itemize}
+        \item Implements a queue-based approach for level-order traversal.
+        \item Time complexity: \tco{|V| + |E|}.
+        \item Used for:
+              \begin{itemize}
+                  \item Finding shortest paths in unweighted graphs.
+                  \item Checking bipartiteness.
+              \end{itemize}
+    \end{itemize}
+\end{properties}
+
+\begin{example}[]{DFS and BFS Traversal}
+    \begin{center}
+        \begin{tikzpicture}[node distance=1.5cm, main/.style={circle, draw, fill=blue!20, minimum size=10mm, inner sep=0pt}]
+            % Graph vertices
+            \node[main] (1) {1};
+            \node[main] (2) [above right of=1] {2};
+            \node[main] (3) [below right of=1] {3};
+            \node[main] (4) [right of=2] {4};
+            \node[main] (5) [right of=3] {5};
+
+            % Edges
+            \draw (1) -- (2);
+            \draw (1) -- (3);
+            \draw (2) -- (4);
+            \draw (3) -- (5);
+            \draw (4) -- (5);
+        \end{tikzpicture}
+    \end{center}
+    \textbf{DFS Traversal:} Starting at $1$, a possible traversal is $1 \to 2 \to 4 \to 5 \to 3$.\\
+    \textbf{BFS Traversal:} Starting at $1$, a possible traversal is $1 \to 2 \to 3 \to 4 \to 5$.
+\end{example}