From d170e330e95105e62ceda0b8476532ba408149f0 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 12 Sep 2025 17:04:36 +0200 Subject: [PATCH] Add A&D summary. Still WIP for fixing errors --- algorithms-and-datastructures/ad.pdf | Bin 0 -> 448058 bytes algorithms-and-datastructures/ad.tex | 89 ++++++ .../parts/datatypes/avl-tree.tex | 225 +++++++++++++++ .../parts/datatypes/binary-tree.tex | 264 ++++++++++++++++++ .../parts/datatypes/lists-dictionaries.tex | 186 ++++++++++++ algorithms-and-datastructures/parts/dp.tex | 159 +++++++++++ .../parts/graphs/all-pairs-sp.tex | 148 ++++++++++ .../parts/graphs/bfs-dfs.tex | 113 ++++++++ .../parts/graphs/connected-components.tex | 54 ++++ .../parts/graphs/intro.tex | 73 +++++ .../parts/graphs/matrix.tex | 67 +++++ .../parts/graphs/mst/boruvka.tex | 113 ++++++++ .../parts/graphs/mst/kruskal.tex | 134 +++++++++ .../parts/graphs/mst/prim.tex | 83 ++++++ .../parts/graphs/paths-walks-cycles.tex | 66 +++++ .../parts/graphs/shortest-path.tex | 169 +++++++++++ algorithms-and-datastructures/parts/intro.tex | 93 ++++++ .../parts/search/comparison-based-sort.tex | 175 ++++++++++++ .../parts/search/heap-sort.tex | 130 +++++++++ .../parts/search/quick-sort.tex | 59 ++++ .../parts/search/search.tex | 35 +++ 21 files changed, 2435 insertions(+) create mode 100644 algorithms-and-datastructures/ad.pdf create mode 100755 algorithms-and-datastructures/ad.tex create mode 100644 algorithms-and-datastructures/parts/datatypes/avl-tree.tex create mode 100644 algorithms-and-datastructures/parts/datatypes/binary-tree.tex create mode 100644 algorithms-and-datastructures/parts/datatypes/lists-dictionaries.tex create mode 100644 algorithms-and-datastructures/parts/dp.tex create mode 100644 algorithms-and-datastructures/parts/graphs/all-pairs-sp.tex create mode 100644 algorithms-and-datastructures/parts/graphs/bfs-dfs.tex create mode 100644 algorithms-and-datastructures/parts/graphs/connected-components.tex create mode 100644 algorithms-and-datastructures/parts/graphs/intro.tex create mode 100644 algorithms-and-datastructures/parts/graphs/matrix.tex create mode 100644 algorithms-and-datastructures/parts/graphs/mst/boruvka.tex create mode 100644 algorithms-and-datastructures/parts/graphs/mst/kruskal.tex create mode 100644 algorithms-and-datastructures/parts/graphs/mst/prim.tex create mode 100644 algorithms-and-datastructures/parts/graphs/paths-walks-cycles.tex create mode 100644 algorithms-and-datastructures/parts/graphs/shortest-path.tex create mode 100644 algorithms-and-datastructures/parts/intro.tex create mode 100644 algorithms-and-datastructures/parts/search/comparison-based-sort.tex create mode 100644 algorithms-and-datastructures/parts/search/heap-sort.tex create mode 100644 algorithms-and-datastructures/parts/search/quick-sort.tex create mode 100644 algorithms-and-datastructures/parts/search/search.tex diff --git a/algorithms-and-datastructures/ad.pdf b/algorithms-and-datastructures/ad.pdf new file mode 100644 index 0000000000000000000000000000000000000000..615c9ef6e3df7b61527866738f418e54a5744df5 GIT binary patch literal 448058 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the asymptotic complexity compared to the standard method. +\end{definition} + +\begin{properties}[]{Characteristics of Strassen’s Algorithm} + \begin{itemize} + \item \textbf{Standard Multiplication:} Requires \tco{n^3} time for two $n \times n$ matrices. + \item \textbf{Strassen’s Approach:} Reduces the complexity to \tco{n^{\log_2 7}} (approximately \tco{n^{2.81}}). + \item \textbf{Idea:} Uses divide-and-conquer to reduce the number of scalar multiplications from $8$ to $7$ in each recursive step. + \item Useful for applications involving large matrix multiplications. + \end{itemize} +\end{properties} + +\begin{usage}[]{Strassen's Algorithm} + Strassen's algorithm is used for matrix multiplication, reducing the computational complexity from \(O(n^3)\) to approximately \(O(n^{2.81})\). + + \begin{enumerate} + \item \textbf{Divide Matrices:} + \begin{itemize} + \item Split the input matrices \(A\) and \(B\) into four submatrices each: + \[ + A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}, \quad + B = \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix} + \] + \end{itemize} + + \item \textbf{Compute 7 Products:} + \begin{itemize} + \item Calculate seven intermediate products using combinations of the submatrices: + \begin{align*} + M_1 & = (A_{11} + A_{22})(B_{11} + B_{22}) \\ + M_2 & = (A_{21} + A_{22})B_{11} \\ + M_3 & = A_{11}(B_{12} - B_{22}) \\ + M_4 & = A_{22}(B_{21} - B_{11}) \\ + M_5 & = (A_{11} + A_{12})B_{22} \\ + M_6 & = (A_{21} - A_{11})(B_{11} + B_{12}) \\ + M_7 & = (A_{12} - A_{22})(B_{21} + B_{22}) + \end{align*} + \end{itemize} + + \item \textbf{Combine Results:} + \begin{itemize} + \item Use the intermediate products to compute the submatrices of the result \(C\): + \[ + C_{11} = M_1 + M_4 - M_5 + M_7, \quad + C_{12} = M_3 + M_5 + \] + \[ + C_{21} = M_2 + M_4, \quad + C_{22} = M_1 - M_2 + M_3 + M_6 + \] + \end{itemize} + + \item \textbf{Repeat Recursively:} + \begin{itemize} + \item If the submatrices are larger than a certain threshold, repeat the process recursively. + \end{itemize} + + \item \textbf{End:} + \begin{itemize} + \item The resulting matrix \(C\) is the product of \(A\) and \(B\). + \end{itemize} + \end{enumerate} +\end{usage} diff --git a/algorithms-and-datastructures/parts/graphs/mst/boruvka.tex b/algorithms-and-datastructures/parts/graphs/mst/boruvka.tex new file mode 100644 index 0000000..6225e60 --- /dev/null +++ b/algorithms-and-datastructures/parts/graphs/mst/boruvka.tex @@ -0,0 +1,113 @@ +\newpage +\subsubsection{Boruvka's algorithm} +\begin{definition}[]{Definition of Borůvka's Algorithm} + Borůvka's Algorithm is a greedy algorithm for finding the Minimum Spanning Tree (MST) of a connected, weighted graph. It repeatedly selects the smallest edge from each connected component and adds it to the MST, merging the components until only one component remains. +\end{definition} + +\begin{properties}[]{Characteristics and Performance of Borůvka's Algorithm} + \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{Limitations:} Efficient for parallel implementations but less commonly used in practice compared to Kruskal's and Prim's. + \end{itemize} +\end{properties} + +\begin{algorithm} + \caption{Borůvka's Algorithm} + \begin{algorithmic}[1] + \Procedure{Boruvka}{$G = (V, E)$} + \State Initialize a forest $F$ with each vertex as a separate component. + \State Initialize an empty MST $T$. + \While{the number of components in $F > 1$} + \State Initialize an empty set $minEdges$. + \For{each component $C$ in $F$} + \State Find the smallest edge $(u, v)$ such that $u \in C$ and $v \notin C$. + \State Add $(u, v)$ to $minEdges$. + \EndFor + \For{each edge $(u, v)$ in $minEdges$} + \If{$u$ and $v$ are in different components in $F$} + \State Add $(u, v)$ to $T$. + \State Merge the components containing $u$ and $v$ in $F$. + \EndIf + \EndFor + \EndWhile + \State \Return $T$. + \EndProcedure + \end{algorithmic} +\end{algorithm} + +\begin{usage}[]{Borůvka's Algorithm} + Borůvka's algorithm finds the Minimum Spanning Tree (MST) by repeatedly finding the smallest outgoing edge from each connected component. + + \begin{enumerate} + \item \textbf{Initialize:} + \begin{itemize} + \item Assign each vertex to its own component. + \end{itemize} + + \item \textbf{Find Smallest Edges:} + \begin{itemize} + \item For each component, find the smallest outgoing edge. After combination, the other vertex will only be evaluated if it has an even lower weight edge outgoing from it. + \end{itemize} + + \item \textbf{Merge Components:} + \begin{itemize} + \item Add the selected edges to the MST. + \item Merge the connected components of the graph. + \end{itemize} + + \item \textbf{Repeat:} + \begin{itemize} + \item Repeat steps 2-3 until only one component remains (all vertices are connected). + \end{itemize} + + \item \textbf{End:} + \begin{itemize} + \item The MST is complete, and all selected edges form a connected acyclic graph. + \end{itemize} + \end{enumerate} +\end{usage} + + + +% \newpage +% \subsection{Example: Electrification of Möhrens} +% \begin{figure*}[ht] +% \centering +% \begin{tikzpicture} +% \tikzset{enclosed/.style={draw, circle, inner sep=0pt, minimum size =.5cm, circle}}; +% \node[enclosed, label={center: $\lightning$}] (power) at (0, 0) {}; +% \node[enclosed, label={center: A}] (A) at (-1, 1) {}; +% \node[enclosed, label={center: B}] (B) at (-1, 0) {}; +% \node[enclosed, label={center: C}] (C) at (-1, -1) {}; +% \node[enclosed, label={center: D}] (D) at (1, 1) {}; +% \node[enclosed, label={center: E}] (E) at (1, 0) {}; +% \node[enclosed, label={center: F}] (F) at (1, -1) {}; +% +% \draw (A) -- node[left] {\scriptsize 3} ++ (B); +% \draw (B) -- node[left] {\scriptsize 7} ++ (C); +% +% \draw (D) -- node[right] {\scriptsize 30} ++ (E); +% \draw (E) -- node[right] {\scriptsize 25} ++ (F); +% \draw plot[smooth] coordinates {(C) (1.5, -1.5) (1.5, 0.8) (D)}; +% \node[label={\scriptsize 20}] at (1.8, -1.8) {}; +% +% +% \draw (A) -- node[right] {\scriptsize 10} ++ (power); +% \draw (B) -- node[above,yshift=-0.1cm] {\scriptsize 5} ++ (power); +% \draw (C) -- node[right] {\scriptsize 15} ++ (power); +% \draw (D) -- node[left] {\scriptsize 40} ++ (power); +% \draw (E) -- node[above,yshift=-0.1cm] {\scriptsize 0} ++ (power); +% \draw (F) -- node[left] {\scriptsize 35} ++ (power); +% \end{tikzpicture} +% \end{figure*} +% +% \fhlc{Aquamarine}{Goal:} Find sub-graph for which the weights are minimal. +% +% $G = (V, E)$ is non-directed, connected. +% +% \fhlc{lightgray}{Weights of edges:} $\{w(e)\}\rvert_{e \in E}$ (non-negative)\\[0.2cm] +% \fhlc{lightgray}{Spanning edges $A\subseteq E$} Graph $(V, A)$ is connected\\[0.2cm] +% \fhlc{lightgray}{Weight:} $w(A) := \sum_{e \in A} w(e)$\\[0.2cm] diff --git a/algorithms-and-datastructures/parts/graphs/mst/kruskal.tex b/algorithms-and-datastructures/parts/graphs/mst/kruskal.tex new file mode 100644 index 0000000..4263290 --- /dev/null +++ b/algorithms-and-datastructures/parts/graphs/mst/kruskal.tex @@ -0,0 +1,134 @@ +\newpage +\subsubsection{Kruskal's algorithm} +\begin{definition}[]{Kruskal's Algorithm} + Kruskal's Algorithm is a greedy algorithm for finding the Minimum Spanning Tree (MST) of a connected, weighted graph. It sorts all the edges by weight and adds them to the MST in order of increasing weight, provided they do not form a cycle with the edges already included. +\end{definition} + +\begin{properties}[]{Characteristics and Performance} + \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{Limitations:} Requires sorting of edges, which can dominate runtime. + \end{itemize} +\end{properties} + +\begin{algorithm} + \caption{Kruskal's Algorithm} + \begin{algorithmic}[1] + \Procedure{Kruskal}{$G = (V, E)$} + \State Sort all edges $E$ in non-decreasing order of weight. + \State Initialize an empty MST $T$. + \State Initialize a disjoint-set data structure for $V$. + \For{each edge $(u, v)$ in $E$ (in sorted order)} + \If{$u$ and $v$ belong to different components in the disjoint set} + \State Add $(u, v)$ to $T$. + \State Union the sets containing $u$ and $v$. + \EndIf + \EndFor + \State \Return $T$. + \EndProcedure + \end{algorithmic} +\end{algorithm} + +\begin{usage}[]{Kruskal's Algorithm} + Kruskal's algorithm finds the Minimum Spanning Tree (MST) by sorting edges and adding them to the MST, provided they don't form a cycle. + + \begin{enumerate} + \item \textbf{Sort Edges:} + \begin{itemize} + \item Sort all edges in ascending order by their weights. + \end{itemize} + + \item \textbf{Initialize Disjoint Sets (union find):} + \begin{itemize} + \item Assign each vertex to its own disjoint set to track connected components. + \end{itemize} + + \item \textbf{Iterate Over Edges:} + \begin{itemize} + \item For each edge in the sorted list: + \begin{itemize} + \item Check if the edge connects vertices from different sets. + \item If it does, add the edge to the MST and merge the sets. + \end{itemize} + \end{itemize} + + \item \textbf{Stop When Done:} + \begin{itemize} + \item Stop when the MST contains \(n-1\) edges (for a graph with \(n\) vertices). + \end{itemize} + + \item \textbf{End:} + \begin{itemize} + \item The MST is complete, and all selected edges form a connected acyclic graph. + \end{itemize} + \end{enumerate} +\end{usage} + + +\newpage +\fhlc{Aquamarine}{Union-Find} +\begin{usage}[]{Union-Find Data Structure - Step-by-Step Execution} + The Union-Find data structure efficiently supports two primary operations for disjoint sets: + \begin{itemize} + \item \textbf{Union:} Merge two sets into one. + \item \textbf{Find:} Identify the representative (or root) of the set containing a given element. + \end{itemize} + + \textbf{Steps for Using the Union-Find Data Structure:} + + \begin{enumerate} + \item \textbf{Initialization:} + \begin{itemize} + \item Create an array \(parent\), where \(parent[i] = i\), indicating that each element is its own parent (a singleton set). + \item Optionally, maintain a \(rank\) array to track the rank (or size) of each set for optimization. + \end{itemize} + + \item \textbf{Find Operation:} + \begin{itemize} + \item To find the representative (or root) of a set containing element \(x\): + \begin{enumerate} + \item Follow the \(parent\) array recursively until \(parent[x] = x\). + \item Apply \textbf{path compression} by updating \(parent[x]\) directly to the root to flatten the tree structure: + \[ + parent[x] = \text{Find}(parent[x]) + \] + \end{enumerate} + \end{itemize} + + \item \textbf{Union Operation:} + \begin{itemize} + \item To merge the sets containing \(x\) and \(y\): + \begin{enumerate} + \item Find the roots of \(x\) and \(y\) using the Find operation. + \item Compare their ranks: + \begin{itemize} + \item Attach the smaller tree under the larger tree to keep the structure shallow. + \item If ranks are equal, arbitrarily choose one as the root and increment its rank. + \end{itemize} + \end{enumerate} + \end{itemize} + + \item \textbf{Optimization Techniques:} + \begin{itemize} + \item \textbf{Path Compression:} Flattens the tree during Find operations, reducing the time complexity. + \item \textbf{Union by Rank/Size:} Ensures smaller trees are always attached under larger trees, maintaining a logarithmic depth. + \end{itemize} + + \item \textbf{End:} + \begin{itemize} + \item After performing Find and Union operations, the Union-Find structure can determine connectivity between elements or group them into distinct sets efficiently. + \end{itemize} + \end{enumerate} +\end{usage} + +\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)} + \end{itemize} +\end{properties} diff --git a/algorithms-and-datastructures/parts/graphs/mst/prim.tex b/algorithms-and-datastructures/parts/graphs/mst/prim.tex new file mode 100644 index 0000000..398d9df --- /dev/null +++ b/algorithms-and-datastructures/parts/graphs/mst/prim.tex @@ -0,0 +1,83 @@ +\newpage +\subsection{MST} +\subsubsection{Prim's algorithm} +\begin{definition}[]{Definition of Prim's Algorithm} + Prim's Algorithm is a greedy algorithm for finding the Minimum Spanning Tree (MST) of a connected, weighted graph. + It starts with an arbitrary node and iteratively adds the smallest edge connecting a vertex in the MST to a vertex outside the MST until all vertices are included. +\end{definition} + +\begin{properties}[]{Characteristics and Performance of Prim's Algorithm} + \begin{itemize} + \item \textbf{Graph Type:} Works on undirected, weighted graphs. + \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|)}. + \end{itemize} + \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} + +\begin{algorithm} + \caption{Prim's Algorithm} + \begin{algorithmic}[1] + \Procedure{Prim}{$G = (V, E)$, $start$} + \State Initialize a priority queue $Q$. + \State Initialize $key[v] \gets \infty$ for all $v \in V$, except $key[start] \gets 0$. + \State Initialize an empty MST $T$. + \State Add all vertices to $Q$ with their key values. + \While{$Q$ is not empty} + \State $u \gets$ ExtractMin($Q$). + \State Add $u$ to $T$. + \For{each $(u, v)$ in $E$} + \If{$v$ is in $Q$ and weight($u, v$) $< key[v]$} + \State $key[v] \gets$ weight($u, v$). + \State Update $Q$ with $key[v]$. + \EndIf + \EndFor + \EndWhile + \State \Return $T$. + \EndProcedure + \end{algorithmic} +\end{algorithm} + +\begin{usage}[]{Prim's Algorithm} + Prim's algorithm is used to find the Minimum Spanning Tree (MST) of a graph. It starts with a single node and grows the MST by adding the smallest edge that connects a vertex in the MST to a vertex outside it. + + \begin{enumerate} + \item \textbf{Initialize:} + \begin{itemize} + \item Pick any starting vertex as part of the MST. + \item Mark the vertex as visited and add it to the MST. + \item Initialize a priority queue to store edges by their weight. + \end{itemize} + + \item \textbf{Explore Edges:} + \begin{itemize} + \item Add all edges from the visited vertex to the priority queue. + \end{itemize} + + \item \textbf{Pick the Smallest Edge:} + \begin{itemize} + \item Choose the smallest-weight edge from the priority queue that connects a visited vertex to an unvisited vertex. + \end{itemize} + + \item \textbf{Add the New Vertex:} + \begin{itemize} + \item Mark the vertex connected by the chosen edge as visited. + \item Add the edge and the vertex to the MST. + \end{itemize} + + \item \textbf{Repeat:} + \begin{itemize} + \item Repeat steps 2-4 until all vertices are part of the MST or no more edges are available. + \end{itemize} + + \item \textbf{End:} + \begin{itemize} + \item The MST is complete when all vertices are visited, and the selected edges form a connected acyclic graph. + \end{itemize} + \end{enumerate} +\end{usage} diff --git a/algorithms-and-datastructures/parts/graphs/paths-walks-cycles.tex b/algorithms-and-datastructures/parts/graphs/paths-walks-cycles.tex new file mode 100644 index 0000000..df54aa5 --- /dev/null +++ b/algorithms-and-datastructures/parts/graphs/paths-walks-cycles.tex @@ -0,0 +1,66 @@ +\newpage +\subsection{Paths, Walks, Cycles} +\begin{definition}[]{Paths, Walks, and Cycles in Graphs} + \begin{itemize} + \item \textbf{Walk:} A sequence of vertices and edges in a graph, where each edge connects consecutive vertices in the sequence. Vertices and edges may repeat. + \item \textbf{Path:} A walk with no repeated vertices. In a directed graph, the edges must respect the direction. + \item \textbf{Cycle:} A path that starts and ends at the same vertex. For a simple cycle, all vertices (except the start/end vertex) and edges are distinct. + \item \textbf{Eulerian Walk:} A walk that traverses every edge of a graph exactly once. + \item \textbf{Closed Eulerian Walk (Eulerian Cycle):} An Eulerian walk that starts and ends at the same vertex. + \item \textbf{Hamiltonian Path:} A path that visits each vertex of a graph exactly once. Edges may or may not repeat. + \item \textbf{Hamiltonian Cycle:} A Hamiltonian path that starts and ends at the same vertex. + \end{itemize} +\end{definition} + +\begin{properties}[]{Eulerian Graphs} + \begin{itemize} + \item \textbf{Undirected Graph:} + \begin{itemize} + \item A graph has an Eulerian walk if it has exactly two vertices of odd degree (necessary condition). + \item A graph has a closed Eulerian walk if all vertices have even degree (necessary condition). + \end{itemize} + \item \textbf{Directed Graph:} + \begin{itemize} + \item A graph has an Eulerian walk if at most one vertex has \textit{in-degree} one greater than its \textit{out-degree}, and at most one vertex has \textit{out-degree} one greater than its \textit{in-degree}. All other vertices must have equal in-degree and out-degree. + \item A graph has a closed Eulerian walk if every vertex has equal in-degree and out-degree. + \end{itemize} + \end{itemize} +\end{properties} + +\begin{properties}[]{Hamiltonian Graphs} + \begin{itemize} + \item Unlike Eulerian walks, there is no simple necessary or sufficient condition for the existence of Hamiltonian paths or cycles. + \item A graph with $n$ vertices is Hamiltonian if every vertex has a degree of at least $\lceil n / 2 \rceil$ (Dirac's Theorem, sufficient condition). + \end{itemize} +\end{properties} + +\begin{example}[]{Eulerian and Hamiltonian} + \begin{center} + \begin{tikzpicture}[node distance=1.5cm, main/.style={circle, draw, fill=blue!20, minimum size=10mm, inner sep=0pt}] % Eulerian graph example + \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=3] {4}; + \node[main] (5) [above right of=4] {5}; + \draw (1) -- (2) -- (5) -- (4) -- (3) -- (1) -- (4) -- (2); + \draw (2) -- (3); + + % Hamiltonian graph example + \node[main] (6) [right of=5, xshift=3cm] {A}; + \node[main] (7) [above right of=6] {B}; + \node[main] (8) [below right of=6] {C}; + \node[main] (9) [right of=7] {D}; + \node[main] (10) [right of=8] {E}; + \draw (6) -- (7) -- (9) -- (10) -- (8) -- (6); + \draw (6) -- (10); + \draw (7) -- (8); + \end{tikzpicture} + \end{center} +\end{example} + +\begin{remarks}[]{Key Differences Between Eulerian and Hamiltonian Concepts} + \begin{itemize} + \item Eulerian paths are concerned with traversing every \textbf{edge} exactly once, while Hamiltonian paths are about visiting every \textbf{vertex} exactly once. + \item Eulerian properties depend on the degree of vertices, whereas Hamiltonian properties depend on overall vertex connectivity. + \end{itemize} +\end{remarks} diff --git a/algorithms-and-datastructures/parts/graphs/shortest-path.tex b/algorithms-and-datastructures/parts/graphs/shortest-path.tex new file mode 100644 index 0000000..5e9b29c --- /dev/null +++ b/algorithms-and-datastructures/parts/graphs/shortest-path.tex @@ -0,0 +1,169 @@ +\newpage +\subsection{Shortest path} +\subsubsection{Dijkstra's Algorithm} +\begin{definition}[]{Dijkstra's Algorithm} + \textbf{Dijkstra's Algorithm} is a graph search algorithm that finds the shortest paths from a source vertex to all other vertices in a graph with non-negative edge weights. +\end{definition} + +\begin{algorithm} + \caption{Dijkstra's Algorithm} + \begin{algorithmic}[1] + \Procedure{Dijkstra}{$G = (V, E), s$} \Comment{$s$ is the source vertex} + \State Initialize distances: $d[v] \gets \infty \; \forall v \in V, d[s] \gets 0$ + \State Initialize priority queue $Q$ with all vertices, priority set to $\infty$ + \While{$Q$ is not empty} + \State $u \gets \textbf{Extract-Min}(Q)$ + \For{each neighbor $v$ of $u$} + \If{$d[u] + w(u, v) < d[v]$} \Comment{If weight through current vertex is lower, update} + \State $d[v] \gets d[u] + w(u, v)$ + \State Update $Q$ with new distance of $v$ + \EndIf + \EndFor + \EndWhile + \State \textbf{Return} distances $d$ + \EndProcedure + \end{algorithmic} +\end{algorithm} + +\begin{properties}[]{Characteristics of Dijkstra's Algorithm} + \begin{itemize} + \item \textbf{Time Complexity:} + \begin{itemize} + \item \tco{|V|^2} for a simple implementation. + \item \tco{(|V| + |E|) \log |V|} using a priority queue. + \end{itemize} + \item Only works with non-negative edge weights. + \item Greedy algorithm that processes vertices in increasing order of distance. + \item Common applications include navigation systems and network routing. + \end{itemize} +\end{properties} + +\begin{usage}[]{Dijkstra's Algorithm} + Dijkstra's algorithm finds the shortest path from a source vertex to all other vertices in a weighted graph (non-negative weights). + + \begin{enumerate} + \item \textbf{Initialize:} + \begin{itemize} + \item Set the distance to the source vertex as 0 and to all other vertices as infinity. + \item Mark all vertices as unvisited. + \end{itemize} + + \item \textbf{Start from Source:} + \begin{itemize} + \item Select the unvisited vertex with the smallest tentative distance. + \end{itemize} + + \item \textbf{Update Distances:} + \begin{itemize} + \item For each unvisited neighbor of the current vertex: + \begin{itemize} + \item Calculate the tentative distance through the current vertex. + \item If the calculated distance is smaller than the current distance, update it. + \end{itemize} + \end{itemize} + + \item \textbf{Mark as Visited:} + \begin{itemize} + \item Mark the current vertex as visited. Once visited, it will not be revisited. + \end{itemize} + + \item \textbf{Repeat:} + \begin{itemize} + \item Repeat steps 2-4 until all vertices are visited or the shortest path to all vertices is determined. + \end{itemize} + + \item \textbf{End:} + \begin{itemize} + \item The algorithm completes when all vertices are visited or when the shortest paths to all reachable vertices are found. + \end{itemize} + \end{enumerate} +\end{usage} + + + + +\newpage +\subsubsection{Bellman-Ford Algorithm} +\begin{definition}[]{Bellman-Ford Algorithm} + The \textbf{Bellman-Ford Algorithm} computes shortest paths from a source vertex to all other vertices, allowing for graphs with negative edge weights. +\end{definition} + +\begin{algorithm} + \caption{Bellman-Ford Algorithm} + \begin{algorithmic}[1] + \Procedure{Bellman-Ford}{$G = (V, E), s$} \Comment{$s$ is the source vertex} + \State Initialize distances: $d[v] \gets \infty \; \forall v \in V, d[s] \gets 0$ + \For{$i \gets 1$ to $|V| - 1$} \Comment{Relax all edges $|V| - 1$ times} + \For{each edge $(u, v, w(u, v)) \in E$} + \If{$d[u] + w(u, v) < d[v]$} + \State $d[v] \gets d[u] + w(u, v)$ + \EndIf + \EndFor + \EndFor + \For{each edge $(u, v, w(u, v)) \in E$} \Comment{Check for negative-weight cycles} + \If{$d[u] + w(u, v) < d[v]$} + \State \textbf{Report Negative Cycle} + \EndIf + \EndFor + \State \Return distances $d$ + \EndProcedure + \end{algorithmic} +\end{algorithm} + +\begin{properties}[]{Characteristics of Bellman-Ford Algorithm} + \begin{itemize} + \item \textbf{Time Complexity:} \tco{|V| \cdot |E|}. + \item Can handle graphs with negative edge weights but not graphs with negative weight cycles. + \item Used for: + \begin{itemize} + \item Detecting negative weight cycles. + \item Computing shortest paths in graphs where Dijkstra’s algorithm is not applicable. + \end{itemize} + \end{itemize} +\end{properties} + +\begin{usage}[]{Bellman-Ford Algorithm} + The Bellman-Ford algorithm finds the shortest path from a source vertex to all other vertices in a weighted graph (handles negative weights). + + \begin{enumerate} + \item \textbf{Initialize:} + \begin{itemize} + \item Set the distance to the source vertex as 0 and to all other vertices as infinity. + \end{itemize} + + \item \textbf{Relax Edges:} + \begin{itemize} + \item Repeat for \(V-1\) iterations (where \(V\) is the number of vertices): + \begin{itemize} + \item For each edge, update the distance to its destination vertex if the distance through the edge is smaller than the current distance. + \end{itemize} + \end{itemize} + + \item \textbf{Check for Negative Cycles:} + \begin{itemize} + \item Check all edges to see if a shorter path can still be found. If so, the graph contains a negative-weight cycle. + \end{itemize} + + \item \textbf{End:} + \begin{itemize} + \item If no negative-weight cycle is found, the algorithm outputs the shortest paths. + \end{itemize} + \end{enumerate} +\end{usage} + + + +\begin{properties}[]{Comparison of Dijkstra and Bellman-Ford} + \begin{center} + \begin{tabular}{lcc} + \toprule + \textbf{Feature} & \textbf{Dijkstra's Algorithm} & \textbf{Bellman-Ford Algorithm} \\ + \midrule + Handles Negative Weights & No & Yes \\ + Detects Negative Cycles & No & Yes \\ + Time Complexity & \tco{(|V| + |E|) \log |V|} & \tco{|V| \cdot |E|} \\ + Algorithm Type & Greedy & Dynamic Programming \\ + \bottomrule + \end{tabular} + \end{center} +\end{properties} diff --git a/algorithms-and-datastructures/parts/intro.tex b/algorithms-and-datastructures/parts/intro.tex new file mode 100644 index 0000000..f0c68d1 --- /dev/null +++ b/algorithms-and-datastructures/parts/intro.tex @@ -0,0 +1,93 @@ +\newsection +\section{Introduction} + +\subsection{Sufficiency \& Necessity} +\begin{definition}[]{Sufficiency} + A condition $P$ is called \textit{sufficient} for $Q$ if knowing $P$ is true is enough evidence to conclude that $Q$ is true. + + This is equivalent to saying $Q \Rightarrow P$. +\end{definition} + +\begin{definition}[]{Necessity} + A condition $P$ is called \textit{necessary} for $Q$ if $Q$ cannot occur unless $P$ is true, but doesn't imply that $Q$ is true, only that it is false if $P$ is false. + + This is equivalent to saying $P \Rightarrow Q$ +\end{definition} + +\subsection{Asymptotic Growth} +$f$ grows asymptotically slower than $g$ if $\displaystyle\lim_{m \rightarrow \infty} \frac{f(m)}{g(m)} = 0$. +We can remark that $f$ is upper-bounded by $g$, thus $f \leq$\tco{g} and we can say $g$ is lower bounded by $f$, thus $g \geq$ \tcl{f}. +If two functions grow equally fast asymptotically, \tct{f} $= g$ + + +\subsection{Runtime evaluation} +Identify the basic operations (usually given by the task), then count how often they are called and express that as a function in $n$. +It is easier to note that in sum notation, then simplify that sum notation into a formula not containing any summation symbols. + + +% ──────────────────────────────────────────────────────────────────── +\subsection{Tips for Converting Summation Notation into Summation-Free Notation} + +\subsubsection{Identify the Pattern:} +\begin{itemize} + \item Examine the summand. + \item Look for patterns related to the index variable (usually $i$, $j$, etc.). Is it a linear function, a power of $i$, a combination? +\end{itemize} + + +\subsubsection{Arithmetic Series Formula} +If the summand is a simple arithmetic progression (e.g., $a + bi$ where $a$ and $b$ are constants), use the formula: +\[ + \sum_{i=m}^{n} (a + bi) = (n - m + 1)\left(a + b\frac{m + n}{2}\right) +\] + + +\subsubsection{Power Rule for Sums} +\begin{itemize} + \item For sums involving powers of $i$, you can use the following pattern: + \[ + \sum_{i=1}^{n} i^k = \frac{n^{k+1}}{k+1} + \] + \item Remember that this rule only applies when the index starts at 1. +\end{itemize} + + +\subsubsection{Telescoping Series} +Look for terms in consecutive elements of the summand that cancel out, leaving a simpler expression after expanding. This is particularly helpful for fractions and ratios. + + +\subsubsection{Geometric Series Formula} +For sums involving constant ratios (e.g., $a \cdot r^i$ where $r$ is the common ratio), use: +\[ + \sum_{i=0}^{n} a \cdot r^i = a \frac{1 - r^{n+1}}{1-r} +\] + + +\subsubsection{Gaussian Formula} +If $S$ is an arithmetic series with $n$ terms, then $S = \frac{n}{2} * (a + 1)$ + + +\subsubsection{Examples} +The only other way (other than learning these tips) in which you are going to get better at this is by parctising. +Work through examples, starting with simpler ones and moving towards more complex expressions. + + +\fhlc{Aquamarine}{Example:} + +Let's convert the summation: $\sum_{i=1}^{5} i$ + +\begin{enumerate} + \item \textbf{Pattern:} The summand is simply $i$, which represents a linear arithmetic progression. + + \item \textbf{Arithmetic Series Formula:} Applying the formula with $a = 1$, $b = 1$, $m = 1$, and $n = 5$: + \[ + \sum_{i=1}^{5} i = (5 - 1 + 1)\left(1 + 1 \cdot \frac{1 + 5}{2}\right) = 5 \cdot 3 = 15 + \] +\end{enumerate} + +Therefore, the summation evaluates to $15$. + +\subsection{Specific examples} +\begin{align*} + \frac{n}{\log(n)} \geq \Omega(\sqrt{n}) \Leftrightarrow \sqrt{n} \leq \text{\tco{\frac{n}{\log(n)}}} +\end{align*} diff --git a/algorithms-and-datastructures/parts/search/comparison-based-sort.tex b/algorithms-and-datastructures/parts/search/comparison-based-sort.tex new file mode 100644 index 0000000..383cb80 --- /dev/null +++ b/algorithms-and-datastructures/parts/search/comparison-based-sort.tex @@ -0,0 +1,175 @@ +\newpage +\subsection{Sort} +Sorted data proved to be much quicker to search through, but how do we sort efficiently? + +First, how to check if an array is sorted. This can be done in linear time: +\begin{algorithm} + \begin{spacing}{1.2} + \caption{\textsc{sorted(A)}} + \begin{algorithmic}[1] + \For{$i \gets 1, 2, \ldots, n - 1$} + \If{$A[i] > A[i + 1]$} \Return false \Comment{Item is unsorted} + \EndIf + \EndFor + \State \Return true + \end{algorithmic} + \end{spacing} +\end{algorithm} + +\tc{n} + + +\subsubsection{Bubble Sort} +\begin{algorithm} + \begin{spacing}{1.2} + \caption{\textsc{bubbleSort(A)}} + \begin{algorithmic}[1] + \For{$i \gets 1, 2, \ldots, n$} + \For{$j \gets 1, 2, \ldots, n$} + \If{$A[j] > A[j + 1]$} + \State exchange $A[j]$ and $A[j + 1]$ \Comment{Causes the element to ``bubble up''} + \EndIf + \EndFor + \EndFor + \end{algorithmic} + \end{spacing} +\end{algorithm} + +\tc{n^2} + + + +% ──────────────────────────────────────────────────────────────────── +\subsubsection{Selection Sort} +The concept for this algorithm is selecting an element (that being the largest one for each iteration, where the iteration variable determines the upper bound for the indices of the array) and swapping it with the last item (thus moving the largest elements up, whilst moving the smallest items down). This algorithm uses a similar concept to bubble sort, but saves some runtime compared to it, by reducing the number of comparisons having to be made. +\begin{algorithm} + \begin{spacing}{1.2} + \caption{\textsc{selectionSort(A)}} + \begin{algorithmic}[1] + \For{$i \gets n, n - 1, \ldots, 1$} + \State $k \gets$ Index of maximum element in $A[1, \ldots, i]$ \Comment{Runtime: $O(n)$} + \State exchange $A[k]$ and $A[i]$ + \EndFor + \end{algorithmic} + \end{spacing} +\end{algorithm} + +\tc{n^2} because we have runtime \tco{n} for the search of the maximal entry and run through the loop \tco{n} times, but we have saved some runtime elsewhere, which is not visible in the asymptotic time complexity compared to bubble sort. + + + +% ──────────────────────────────────────────────────────────────────── +\newpage +\subsubsection{Insertion Sort} +\begin{definition}[]{Insertion Sort} + Insertion Sort is a simple sorting algorithm that builds the final sorted array (or list) one element at a time. It iteratively takes one element from the input, finds its correct position in the sorted portion, and inserts it there. The algorithm starts with the first element, assuming it is sorted. It then picks the next element and inserts it into its correct position relative to the sorted portion. This process is repeated for all elements until the entire list is sorted. At each iteration step, the algorithm moves all elements that are larger than the currently picked element to the right by one, i.e. an element $A[i]$ to $A[i + 1]$ +\end{definition} + +\begin{properties}[]{Characteristics and Performance} +\begin{itemize} + \item \textbf{Efficiency:} Works well for small datasets or nearly sorted arrays. + \item \textbf{Time Complexity:} + \begin{itemize} + \item Best case (already sorted): \tcl{n\log(n)} + \item Worst case (reversed order): \tco{n^2} + \item Average case: \tct{n^2} + \end{itemize} + \item \textbf{Limitations:} Inefficient on large datasets due to its \tct{n^2} time complexity and requires additional effort for linked list implementations. +\end{itemize} +\end{properties} + +\begin{algorithm} + \begin{spacing}{1.2} + \caption{\textsc{insertionSort(A)}} + \begin{algorithmic}[1] + \Procedure{InsertionSort}{$A$} + \For{$i \gets 2$ to $n$} \Comment{Iterate over the array} + \State $key \gets A[i]$ \Comment{Element to be inserted} + \State $j \gets i - 1$ + \While{$j > 0$ and $A[j] > key$} + \State $A[j+1] \gets A[j]$ \Comment{Shift elements} + \State $j \gets j - 1$ + \EndWhile + \State $A[j+1] \gets key$ \Comment{Insert element} + \EndFor + \EndProcedure + \end{algorithmic} + \end{spacing} +\end{algorithm} + + + + +% ──────────────────────────────────────────────────────────────────── +\newpage +\subsubsection{Merge Sort} +\begin{definition}[]{Definition of Merge Sort} +Merge Sort is a divide-and-conquer algorithm that splits the input array into two halves, recursively sorts each half, and then merges the two sorted halves into a single sorted array. This process continues until the base case of a single element or an empty array is reached, as these are inherently sorted. +\end{definition} + +\begin{properties}[]{Characteristics and Performance of Merge Sort} +\begin{itemize} + \item \textbf{Efficiency:} Suitable for large datasets due to its predictable time complexity. + \item \textbf{Time Complexity:} + \begin{itemize} + \item Best case: \tcl{n \log n} + \item Worst case: \tco{n \log n} + \item Average case: \tct{n \log n} + \end{itemize} + \item \textbf{Space Complexity:} Requires additional memory for temporary arrays, typically \tct{n}. + \item \textbf{Limitations:} Not in-place, and memory overhead can be significant for large datasets. +\end{itemize} +\end{properties} + +\begin{algorithm} + \begin{spacing}{1.2} + \caption{Merge Sort} + \begin{algorithmic}[1] + \Procedure{MergeSort}{$A[1..n], l, r$} + \If{$l \geq r$} + \State \Return $A$ \Comment{Base case: already sorted} + \EndIf + \State $m \gets \floor{(l + r)/2}$ + \State $\Call{MergeSort}{A, l, m}$ \Comment{Recursive sort on left half} + \State $\Call{MergeSort}{A, m + 1, r}$ \Comment{Recursive sort on right half} + \State \Call{Merge}{$A, l, m, r$} + \EndProcedure + + \Procedure{Merge}{$A[1..n], l, m, r$} \Comment{Runtime: \tco{n}} + \State $result \gets$ new array of size $r - l + 1$ + \State $i \gets l$ + \State $j \gets m + 1$ + \State $k \gets 1$ + \While{$i \leq m$ and $j \leq r$} + \If{$A[i] \leq A[j]$} + \State $result[k] \gets A[i]$ + \State $i \gets i + 1$ + \Else + \State $result[k] \gets A[j]$ + \State $j \gets j + 1$ + \EndIf + \State $k \gets k + 1$ + \EndWhile + \State Append remaining elements of left / right site to $result$ + \State Copy $result$ to $A[l, \ldots, r]$ + \EndProcedure + \end{algorithmic} + \end{spacing} +\end{algorithm} + +\begin{table}[h!] + \centering + \begin{tabular}{lccccc} + \toprule + \textbf{Algorithm} & \textbf{Comparisons} & \textbf{Operations} & \textbf{Space Complexity} & \textbf{Locality} & \textbf{Time complexity}\\ + \midrule + \textit{Bubble-Sort} & \tco{n^2} & \tco{n^2} & \tco{1} & good & \tco{n^2}\\ + \textit{Selection-Sort} & \tco{n^2} & \tco{n} & \tco{1} & good & \tco{n^2}\\ + \textit{Insertion-Sort} & \tco{n \cdot \log(n)} & \tco{n^2} & \tco{1} & good & \tco{n^2}\\ + \textit{Merge-Sort} & \tco{n\cdot \log(n)} & \tco{n \cdot \log(n)} & \tco{n} & good & \tco{n \cdot \log(n)}\\ + \bottomrule + \end{tabular} + \caption{Comparison of four comparison-based sorting algorithms discussed in the lecture. Operations designates the number of write operations in RAM} +\end{table} + + diff --git a/algorithms-and-datastructures/parts/search/heap-sort.tex b/algorithms-and-datastructures/parts/search/heap-sort.tex new file mode 100644 index 0000000..a6ed404 --- /dev/null +++ b/algorithms-and-datastructures/parts/search/heap-sort.tex @@ -0,0 +1,130 @@ +\newpage +\subsubsection{Heap Sort} +\begin{definition}[]{Heap Sort} + Heap Sort is a comparison-based sorting algorithm that uses a binary heap data structure. It builds a max-heap (or min-heap) from the input array and repeatedly extracts the largest (or smallest) element to place it in the correct position in the sorted array. +\end{definition} + +\begin{properties}[]{Characteristics and Performance} + \begin{itemize} + \item \textbf{Efficiency:} Excellent for in-place sorting with predictable performance. + \item \textbf{Time Complexity:} + \begin{itemize} + \item Best case: \tcl{n \log n} + \item Worst case: \tco{n \log n} + \item Average case: \tct{n \log n} + \end{itemize} + \item \textbf{Space Complexity:} In-place sorting requires \tct{1} additional space. + \item \textbf{Limitations:} Inefficient compared to Quick Sort for most practical datasets. + \end{itemize} +\end{properties} + +\begin{algorithm} + \begin{spacing}{1.2} + \caption{Heap Sort} + \begin{algorithmic}[1] + \Procedure{HeapSort}{$A$} + \State $H \gets \Call{Heapify}{A}$ + \For{$i \gets \text{length}(A)$ to $2$} + \State $A[i] \gets \Call{ExtractMax}{A}$ + \EndFor + \EndProcedure + \end{algorithmic} + \end{spacing} +\end{algorithm} + +The lecture does not cover the implementation of a heap tree. See the specific section \ref{sec:heap-trees} on Heap-Trees + + +\newpage +\subsubsection{Bucket Sort} +\begin{definition}[]{Bucket Sort} + Bucket Sort is a distribution-based sorting algorithm that divides the input into a fixed number of buckets, sorts the elements within each bucket (using another sorting algorithm, typically Insertion Sort), and then concatenates the buckets to produce the sorted array. +\end{definition} + +\begin{properties}[]{Characteristics and Performance} + \begin{itemize} + \item \textbf{Efficiency:} Performs well for uniformly distributed datasets. + \item \textbf{Time Complexity:} + \begin{itemize} + \item Best case: \tcl{n + k} (for uniform distribution and $k$ buckets) + \item Worst case: \tco{n^2} (when all elements fall into a single bucket) + \item Average case: \tct{n + k} + \end{itemize} + \item \textbf{Space Complexity:} Requires \tct{n + k} additional space. + \item \textbf{Limitations:} Performance depends on the choice of bucket size and distribution of input elements. + \end{itemize} +\end{properties} + +\begin{algorithm} + \begin{spacing}{1.2} + \caption{Bucket Sort} + \begin{algorithmic}[1] + \Procedure{BucketSort}{$A, k$} + \State $B[1..n] \gets [0, 0, \ldots, 0]$ + \For{$j \gets 1, 2, \ldots, n$} + \State $B[A[j]] \gets B[A[j]] + 1$ \Comment{Count in $B[i]$ how many times $i$ occurs} + \EndFor + \State $k \gets 1$ + \For{$i \gets 1, 2, \ldots, n$} + \State $A[k, \ldots, k + B[i] - 1] \gets [i, i, \ldots, i]$ \Comment {Write $B[i]$ times the value $i$ into $A$} + \State $k \gets k + i$ \Comment{$A$ is filled until position $k - 1$} + \EndFor + \EndProcedure + \end{algorithmic} + \end{spacing} +\end{algorithm} + + +\newpage +\subsection{Heap trees} +\label{sec:heap-trees} +\subsubsection{Min/Max-Heap} +\begin{definition}[]{Min-/Max-Heap} + A Min-Heap is a complete binary tree where the value of each node is less than or equal to the values of its children. + Conversely, a Max-Heap is a complete binary tree where the value of each node is greater than or equal to the values of its children. + In the characteristics below, $A$ is an array storing the value of a element +\end{definition} + +\begin{properties}[]{Characteristics} + \begin{itemize} + \item \textbf{Heap Property:} + \begin{itemize} + \item Min-Heap: $A[parent] \leq A[child]$ for all nodes. + \item Max-Heap: $A[parent] \geq A[child]$ for all nodes. + \end{itemize} + \item \textbf{Operations:} Both Min-Heaps and Max-Heaps support: + \begin{itemize} + \item \textbf{Insert:} Add an element to the heap and adjust to maintain the heap property. + \item \textbf{Extract Min/Max:} Remove the root element (minimum or maximum), replace it with the last element (bottom right most element), and adjust the heap. + \end{itemize} + \item \textbf{Time Complexity:} + \begin{itemize} + \item Insert: \tct{\log n}. + \item Extract Min/Max: \tct{\log n}. + \item Build Heap: \tct{n}. + \end{itemize} + \end{itemize} +\end{properties} + +\begin{example}[]{Min-Heap} + The following illustrates a Min-Heap with seven elements: + \begin{center} + \begin{forest} + for tree={ + circle, draw, fill=blue!20, minimum size=10mm, inner sep=0pt, % Node style + s sep=15mm, % Sibling separation + l sep=15mm % Level separation + } + [2 + [4 + [8] + [10] + ] + [6 + [14] + [18] + ] + ] + \end{forest} + \end{center} +\end{example} diff --git a/algorithms-and-datastructures/parts/search/quick-sort.tex b/algorithms-and-datastructures/parts/search/quick-sort.tex new file mode 100644 index 0000000..137c143 --- /dev/null +++ b/algorithms-and-datastructures/parts/search/quick-sort.tex @@ -0,0 +1,59 @@ +\newpage +\subsubsection{Quick Sort} +\begin{definition}[]{Quick Sort} + Quick Sort is a divide-and-conquer algorithm that selects a pivot element from the array, partitions the other elements into two subarrays according to whether they are less than or greater than the pivot, and then recursively sorts the subarrays. The process continues until the base case of an empty or single-element array is reached. +\end{definition} + +\begin{properties}[]{Characteristics and Performance} + \begin{itemize} + \item \textbf{Efficiency:} Performs well on average and for in-place sorting but can degrade on specific inputs. + \item \textbf{Time Complexity:} + \begin{itemize} + \item Best case: \tcl{n \log n} + \item Worst case: \tco{n^2} (when the pivot is poorly chosen) + \item Average case: \tct{n \log n} + \end{itemize} + \item \textbf{Space Complexity:} In-place sorting typically requires \tct{\log n} additional space for recursion. + \item \textbf{Limitations:} Performance depends heavily on pivot selection. + \end{itemize} +\end{properties} + +\begin{algorithm} + \begin{spacing}{1.2} + \caption{Quick Sort} + \begin{algorithmic}[1] + \Procedure{QuickSort}{$A, l, r$} + \If{$l < r$} + \State $k \gets \Call{Partition}{A, l, r}$ + \State \Call{QuickSort}{$A, l, k - 1$} \Comment{Sort left group} + \State \Call{QuickSort}{$A, k + 1, r$} \Comment{Sort right group} + \EndIf + \EndProcedure + + \Procedure{Partition}{$A, l, r$} + \State $i \gets l$ + \State $j \gets r - 1$ + \State $p \gets A[r]$ + \While{$i > j$} \Comment{Loop ends when $i$ and $j$ meet} + \While{$i < r$ and $A[i] \leq p$} + \State $i \gets i + 1$ \Comment{Search next element for left group} + \EndWhile + + \While{$i > l$ and $A[j] > p$} + \State $j \gets j - 1$ \Comment{Search next element for right group} + \EndWhile + + \If{$i \leq j$} + \State Exchange $A[i]$ and $A[j]$ + \EndIf + \EndWhile + \State Swap $A[i]$ and $A[r]$ \Comment{Move pivot element to correct position} + \State \Return $i$ + \EndProcedure + \end{algorithmic} + \end{spacing} +\end{algorithm} + +The tests $i < r$ and $j > l$ in the while loops in \textsc{Partition} catch the cases where there are no elements that can be added to the left or right group. + +The correct position for the pivot element is $k = j + 1 = i$, since all elements on the left hand side are smaller and all on the right hand side larger than $p$. diff --git a/algorithms-and-datastructures/parts/search/search.tex b/algorithms-and-datastructures/parts/search/search.tex new file mode 100644 index 0000000..a0dda2e --- /dev/null +++ b/algorithms-and-datastructures/parts/search/search.tex @@ -0,0 +1,35 @@ +\newsection +\section{Search \& Sort} +\subsection{Search} +\subsubsection{Linear search} +Linear search, as the name implies, searches through the entire array and has linear runtime, i.e. $\Theta(n)$. + +It works by simply iterating over an iterable object (usually array) and returns the first element (it can also be modified to return \textit{all} elements that match the search pattern) where the search pattern is matched. + +\tc{n} + +\subsubsection{Binary search} +If we want to search in a sorted array, however, we can use what is known as binary array, improving our runtime to logarithmic, i.e. $\Theta(\log(n))$. +It works using divide and conquer, hence it picks a pivot in the middle of the array (at $\floor{\frac{n}{2}}$) and there checks if the value is bigger than our search query $b$, i.e. if $A[m] < b$. This is repeated, until we have homed in on $b$. Pseudo-Code: + +\begin{algorithm} + \begin{spacing}{1.2} + \caption{\textsc{binarySearch(b)}} + \begin{algorithmic}[1] + \State $l \gets 1$, $r \gets n$ \Comment{\textit{Left and right bound}} + \While{$l \leq r$} + \State $m \gets \floor{\frac{l + r}{2}}$ + \If{$A[m] = b$} \Return m \Comment{\textit{Element found}} + \ElsIf{$A[m] > b$} $r \gets m - 1$ \Comment{\textit{Search to the left}} + \Else \hspace{0.2em} $l \gets m + 1$ \Comment{\textit{Search to the right}} + \EndIf + \EndWhile + \State \Return "Not found" + \end{algorithmic} + \end{spacing} +\end{algorithm} +\tc{\log(n)} + +Proving runtime lower bounds (worst case runtime) for this kind of algorithm is done using a decision tree. It in fact is $\Omega(\log(n))$ + +% INFO: If =0, then there is an issue with math environment in the algorithm