Beamer

% =====================================================
% Beamer Presentation Template
% Author: soheilh
% =====================================================

% Replace the \lipsum placeholders and the frame
% titles with your own content.

\documentclass[8pt,aspectratio=169]{beamer}
\usepackage{lipsum} % Only needed for the demo document
\usepackage[light]{beamertheme}

% =====================================================
% Presentation information
% =====================================================
\settitle{Presentation Title}
\setshorttitle{Title}
\setsecondtitle{Presentation Subtitle}
\setauthor{Author Name}
\setuniversity{Example University}
\setdepartment{Department of Example Studies}
\setlogo{figs/logo.png}
\setsupervisor{Prof. Name}
\setsupervisorfullname{Prof. Supervisor Name}
\setexaminer{Prof. Examiner Name}
\setdate{\today}

% =====================================================
% Section-navigation style
% =====================================================
% \enableroadmap          % plain "Outline" slide before each section
\enablesectiondivider   % numbered divider slide for each section

\begin{document}
    % ----------------- Title slide -----------------
    {
        \setbeamertemplate{headline}{}
        \begin{frame}
            \titlepage
        \end{frame}
    }
    % ----------------- Outline slide -----------------
    \begin{frame}{Outline}
        \tableofcontents
    \end{frame}

    % =================================================================
    \section{Introduction \& Problem Definition}
    % =================================================================
    \begin{frame}{Overview}
        \begin{itemize}\itemsep=6pt
            \item \textbf{First point:} \lipsum[1][1-2]
            \item \textbf{Second point:} a short display equation for variety:
            $$f(x) = \sum_{i=1}^{n} w_i \, x_i$$
        \end{itemize}

        \vspace{1.2em}
        \textbf{A secondary list:}
        \begin{itemize}\itemsep=4pt
            \item \lipsum[3][1-1]
            \item \lipsum[4][1-1]
            \item \lipsum[5][1-1]
        \end{itemize}
    \end{frame}

    \begin{frame}{Motivation}
        \lipsum[6][1-2]

        \vspace{1em}
        \begin{itemize}\itemsep=10pt
            \item<1-> \lipsum[7][1-1]
            \item<2-> \lipsum[8][1-1]
            \item<3-> \emphbb{A highlighted takeaway} -- \lipsum[9][1-1]
        \end{itemize}
    \end{frame}

    \begin{frame}{Contributions}
        \begin{itemize}\itemsep=10pt
            \item<1-> \textbf{Contribution 1:} \lipsum[8][1-1]
            \item<2-> \textbf{Contribution 2:} \lipsum[9][1-1]
            \item<3-> \textbf{Contribution 3:} \lipsum[10][1-1]
        \end{itemize}
        \vspace{1em}
        \onslide<4->{
            \begin{alertblock}{Take-home message}
                \lipsum[11][1-6]
            \end{alertblock}
        }
    \end{frame}

    \begin{frame}{Problem Definition}
        \vspace{-1.5em}
        \begin{block}{Placeholder Definition\hfill \small\normalfont(Example et al., 20XX)}
            \textbf{Setup:}
            \begin{itemize}
                \item Some ground set $X$ and a parameter $n$
                \item Auxiliary sets $A_1, A_2, \dots, A_m \subseteq X$
            \end{itemize}
            \smallskip
            \textbf{Goal:} \lipsum[10][1-1]
            $$\min_{S \subseteq X} \; g(S) \quad \text{subject to} \quad h_i(S) \ge b_i \;\; \forall i \in [m]$$
        \end{block}

        \begin{columns}[T]
            \begin{column}{0.53\textwidth}
                \textbf{Special cases:}
                \begin{itemize}\itemsep=4pt
                    \item Case A: $m=1,\ b_1 = 0$.
                    \item Case B: $m=1,\ b_1 = |X|$.
                \end{itemize}
            \end{column}
            \begin{column}{0.4\textwidth}
                \centering
                \fbox{\parbox{0.8\linewidth}{\centering\footnotesize\vspace{3em}figure placeholder\vspace{3em}}}
            \end{column}
        \end{columns}
    \end{frame}

    \begin{frame}{Notation}
        \begin{center}
            \renewcommand{\arraystretch}{1.25}
            \begin{tabular}{c l}
                \toprule
                \rowcolor{primary!20!background}
                \textbf{Symbol} & \textbf{Meaning} \\
                \midrule
                $X$ & ground set \\
                \rowcolor{success!10!background}
                $n$ & size parameter, $n = |X|$ \\
                $A_1,\dots,A_m$ & auxiliary sets / constraints \\
                \rowcolor{success!10!background}
                $S$ & a feasible solution \\
                $g(\cdot),\,h_i(\cdot)$ & objective / constraint functions \\
                \bottomrule
            \end{tabular}
        \end{center}
    \end{frame}

    \begin{frame}{Formal Model}
		\begin{columns}[T]
			\begin{column}{0.53\textwidth}
				\textbf{Variables:}
				\begin{itemize}\itemsep=2pt
					\item $x_i \in [0,1]$: a generic fractional decision variable
					\item $y_j \in [0,1]$: a generic auxiliary variable
				\end{itemize}
				\begin{mini*}
					{x,y}{\;\sum_{i} c_i x_i\;}{}{}
					\addConstraint{\sum_{i \in N(j)} x_i}{\geq y_j,}{\quad \forall j}
					\addConstraint{\sum_{i} x_i}{\leq n}{}
					\addConstraint{\sum_{j} y_j}{\geq q}{}
				\end{mini*}
			\end{column}
			\begin{column}{0.4\textwidth}
				\begin{block}{Block (A)}
					\lipsum[11][1-2]
				\end{block}
				\begin{alertblock}{Alert Block (B)}
					\lipsum[12][1-3]
				\end{alertblock}
				\begin{exampleblock}{Example Block (C)}
					\lipsum[13][1-3]
				\end{exampleblock}
			\end{column}
		\end{columns}
    \end{frame}

    % =================================================================
    \section{Related Work}
    % =================================================================
    \begin{frame}[t]
        \frametitle{Related Work}
        \small \lipsum[15][1-2]

        \vspace{0.5em}
        \begin{itemize}\itemsep=6pt
            \only<1>{\item \emphbb{\cite{hilbert1900}} \\ \small \lipsum[16][1-5]}
            \only<2->{\item \textbf{\cite{hilbert1900}} -- \lipsum[16][1-1]}

            \only<2>{\item \emphbb{\cite{turing1936}} \\ \small \lipsum[17][1-5]}
            \only<3->{\item \textbf{\cite{turing1936}} -- \lipsum[17][1-1]}

            \only<3>{\item \emphbb{\cite{turing1937}} \\ \small \lipsum[18][1-5]}
            \only<4->{\item \textbf{\cite{turing1937}} -- \lipsum[18][1-1]}

            \only<4>{\item \emphbb{\cite{church1936}} \\ \small \lipsum[19][1-5]}
            \only<5->{\item \textbf{\cite{church1936}} -- \lipsum[19][1-1]}

            \only<5>{\item \emphbb{\cite{cook1971}} \\ \small \lipsum[20][1-5]}
        \end{itemize}
    \end{frame}

    \begin{frame}{Comparison of Results}
        \begin{center}
            \renewcommand{\arraystretch}{1.3}
            \resizebox{0.7\linewidth}{!}{%
                \begin{tabular}{l c c c}
                    \toprule
                    \rowcolor{primary!20!background}
                    \textbf{Result} & \textbf{Metric A} & \textbf{Metric B} & \textbf{Scope} \\
                    \midrule
                    Author One et al. & 2.0 & $n-1$ & general case \\
                    Author One et al. & 1.7 & $n$ (exact) & special case only \\
                    Author Two et al. & 1.4 & $n$ (exact) & general case \\
                    \rowcolor{success!10!background}
                    \textbf{Author Three et al.} & \textbf{1.3} & $n$ (exact) & \textbf{general case} \\
                    Author Four et al. & 1.6 & $n+2$ (pseudo) & priority variant \\
                    \bottomrule
                \end{tabular}}
        \end{center}
    \end{frame}

    \begin{frame}{Result Progression}
        \begin{block}{Baseline Result}
            \lipsum[12][1-5]
        \end{block}

        \vspace{0.4em}
        \onslide<2->{
            \begin{exampleblock}{Improved Result}
                \lipsum[13][1-5]
            \end{exampleblock}
        }

        \vspace{0.4em}
        \onslide<3->{
            \begin{alertblock}{Further Refinement}
                \lipsum[14][1-5]
            \end{alertblock}
        }
    \end{frame}

    % =================================================================
    \section{Main Results \& Analysis}
    % =================================================================
    \begin{frame}{Case Analysis}
        \textbf{Main idea:} \lipsum[20][1-5]

        \vspace{0.5em}
        \begin{columns}[T]
            \begin{column}{0.48\textwidth}
                \begin{block}{Case 1: Situation A}
                    \begin{itemize}\itemsep=3pt
                        \item \lipsum[22][1-1]
                        \item \lipsum[23][1-1]
                        \item \lipsum[24][1-1]
                        \item \lipsum[25][1-1]
                    \end{itemize}
                    \vspace{1em}
                    \centering\emphbb{(Outcome A)}
                \end{block}
            \end{column}

            \begin{column}{0.48\textwidth}
                \begin{block}{Case 2: Situation B}
                    \begin{itemize}\itemsep=3pt
                        \item \textbf{Step 1:} \lipsum[24][1-1]
                        \item \textbf{Step 2:} \lipsum[25][1-1]
                        \item \textbf{Step 3:} \lipsum[26][1-1]
                        \item \textbf{Step 4:} \lipsum[27][1-1]
                    \end{itemize}
                    \vspace{1em}
                    \centering\emphbb{(Outcome B)}
                \end{block}
            \end{column}
        \end{columns}
    \end{frame}

    \begin{frame}{Error Bound}
        \begin{alertblock}{A Generic Error Bound}
            \lipsum[26][1-3]
            \[
            \text{Error} \;\le\; n\cdot\frac{\delta}{2}
            \]
            \lipsum[27][1-2]
        \end{alertblock}

        \vspace{0.6em}
        \begin{exampleblock}{A Generic Running-Time Note}
            \begin{itemize}\itemsep=3pt
                \item \lipsum[28][1-1]
                \item \lipsum[29][1-1]
                \[
                \mathcal{O}\!\left(n\log n\right)
                \]
            \end{itemize}
        \end{exampleblock}
    \end{frame}

    \begin{frame}{Time Complexity}
        \begin{itemize}
            \item \lipsum[26][1-5]
            \item \lipsum[27][1-3]
        \end{itemize}

        \vspace{0.6em}
        \begin{exampleblock}{A Generic Running-Time Note}
            \lipsum[28][1-3]
            \[
            \mathcal{O}(n^{1+\epsilon}\sqrt{n}\log n + (k/\epsilon)^{1-1/d})
            \]
        \end{exampleblock}
    \end{frame}

    \begin{frame}{Dynamic-Programming Formulation}
        \textbf{Idea:} \lipsum[29][1-1]
        \vspace{0.5em}
        \begin{block}{State Space}
            \lipsum[30][1-1]
            \[
            T(n) = 2T(n/2) + O(n), \qquad T(n) = O(n \log n)
            \]
        \end{block}
        \vspace{0.5em}
        \textbf{Transitions from state $i$:}
        \begin{enumerate}
            \item \textbf{Skip $p_i$:}
            \[
            \binom{n}{k} = \frac{n!}{k!(n-k)!}, \qquad \sum_{k=0}^{n} \binom{n}{k} = 2^n
            \]
            \item \textbf{Open a cluster at $p_i$:} let $I_i$ add $\Delta_j$ new points of color $j$; jump to $\text{next}(i)$:
            \[
            \chi(G) \le \Delta(G) + 1, \qquad \Pr[X \geq (1+\delta)\mu] \leq \exp\!\left(-\frac{\delta^2 \mu}{2+\delta}\right)
            \]
        \end{enumerate}
    \end{frame}

    \begin{frame}{Algorithm Block}
        \begin{algobox}{Algorithm 1: Gonzalez's 2-Approximation}
            \begin{algorithm}[H]
                \KwInput{Point set $X \subset$ metric space $(X,d)$, number of centers $k$}
                \KwOutput{Center set $S$ with $|S| = k$}
                $S \leftarrow \{x_1\}$ \tcp*{pick arbitrary starting point}
                \For{$i \leftarrow 2$ \KwTo $k$}{
                    $x_i \leftarrow \arg\max_{x \in X} \, d(x, S)$ \tcp*{farthest point from current $S$}
                    $S \leftarrow S \cup \{x_i\}$\;
                }
                \Return{$S$}
            \end{algorithm}
        \end{algobox}
    \end{frame}
    
    \begin{frame}[fragile]{Code Block}
        \begin{codeblock}[title=Fibonacci with memoization]{C++}
#include <iostream>
#include <unordered_map>

std::unordered_map<int, long long> memo;

long long fib(int n) {
    if (n <= 1) return n;
    if (memo.count(n)) return memo[n];
    return memo[n] = fib(n - 1) + fib(n - 2);
}

int main() {
    for (int i = 0; i < 10; ++i)
        std::cout << fib(i) << " ";
}
        \end{codeblock}
    \end{frame}

    % =================================================================
    \section{Appendix}
    % =================================================================
    \begin{frame}{Extended Discussion}
        \small
        \lipsum[2]\\
        \lipsum[3]
    \end{frame}

	\begin{frame}{Supplementary Notes}
		\begin{columns}[T]
			\begin{column}{0.45\textwidth}
                \textbf{Aspect A:}\\
                \small
				\lipsum[3][1-15]
			\end{column}
			\begin{column}{0.45\textwidth}
				\textbf{Aspect B:}\\
                \small
                \lipsum[3][1-15]
			\end{column}
		\end{columns}
    \end{frame}

    % =================================================================
    \section{Conclusion}
    % =================================================================
    \begin{frame}{Conclusion}
        \begin{itemize}\itemsep=8pt
            \item \textbf{Summary:} \lipsum[8][1-1]
            \item \textbf{Contribution:} \lipsum[9][1-1]
            \item \textbf{Future Work:} \lipsum[10][1-1]
        \end{itemize}
    \end{frame}

    \setbeamertemplate{frametitle continuation}{}
    \setbeamertemplate{bibliography item}{}
    \begin{frame}[allowframebreaks]{References}
    	\footnotesize
    	\bibliography{bibs/refs}
    \end{frame}

    % =================================================================
    \section*{Q \& A}
    % =================================================================
    \begin{frame}
        \begin{center}
            \Huge{\textbf{Thank You}}

            \vspace{1em}
            \large{Any Questions?}
        \end{center}
    \end{frame}

\end{document}