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+\documentclass{article}
+\usepackage{float}
+\usepackage{hyperref}
+\hypersetup{
+    pdfborder={0 0 0}
+}
+\usepackage{graphicx}
+\graphicspath{{images/}}
+
+\setlength{\parindent}{0pt}
+\setlength{\parskip}{0.6em}
+\linespread{1.08}
+
+\title{Outline of EE}
+\author{Seungjun Lee}
+\date{February 22, 2026}
+
+\begin{document}
+\maketitle
+
+\begin{abstract}
+
+    \textbf{Title}:
+    When Asymptotic Complexity Fails: An Empirical and Cost-Based Study of Binary and Fibonacci Heaps in Python
+
+    \textbf{RQ}:
+    Why does the theoretical asymptotic advantage of Fibonacci heap in Dijkstra’s algorithm not translate into practical runtime improvements in Python implementations?
+
+    \textbf{Category}:
+    Computer Science
+
+\end{abstract}
+
+\newpage
+\tableofcontents
+\newpage
+
+\section{Introduction}
+
+\subsection*{Context}
+\begin{itemize}
+    \item Dijkstra’s algorithm widely used for shortest path problems
+    \item Priority queue choice affects theoretical time complexity
+    \item Pure Dijkstra: $O(E+ V^2)$
+    \item Binary heap: $O(E \log V)$
+    \item Fibonacci heap: $O(E + V \log V)$
+    \item Theoretical advantage not clearly observed in practical implementations
+\end{itemize}
+
+\subsection*{Outline of argument}
+\begin{itemize}
+    \item Graph structure influences decrease-key frequency
+    \item Decrease-key frequency determines theoretical advantage
+    \item Empirical runtime does not reflect the theory
+    \item Operation level cost explains the gap between the thoery and reality
+\end{itemize}
+
+\subsection*{Scope}
+\begin{itemize}
+    \item Python implementation only
+    \item Synthetic directed graphs (using outdegree-based generation)
+    \item Lognormal edge weight distribution
+    \item Nodes: 2000--16000, density: 0.00015--0.0384, stds: 1000--16000, trials: 200
+    \item Single-source single-target shortest path
+\end{itemize}
+
+\subsection*{Worthiness}
+\begin{itemize}
+    \item Evaluates limits of asymptotic complexity in practice
+    \item Provides empirical basis for heap selection
+    \item Bridges theory and implementation level cost analysis
+\end{itemize}
+
+\section{Theoretical Framework}
+
+\begin{itemize}
+    \item Formal runtime models:
+          \begin{itemize}
+              \item $T_B = c_1 E \log V$
+              \item $T_F = c_2 E + c_3 V \log V$
+          \end{itemize}
+    \item Role of decrease-key in Dijkstra
+    \item Relation between relax\_success and decrease-key calls
+    \item Theoretical crossover condition:
+          \[
+              T_F < T_B
+          \]
+\end{itemize}
+
+\section{Methodology}
+
+\subsection*{Development}
+\begin{itemize}
+    \item Real-world road dataset used to estimate distribution of real-world data
+    \item Lognormal parameters derived from real data
+    \item Synthetic graph generation with various nodes, density, std
+    \item Measurement metrics:
+          \begin{itemize}
+              \item reached
+              \item extract\_min\_calls
+              \item relax\_attempts
+              \item relax\_success
+              \item relax\_success\_ratio
+              \item runtime
+          \end{itemize}
+\end{itemize}
+
+\subsection*{Evidence}
+\begin{itemize}
+    \item Experimental configuration
+          \begin{verbatim}
+python -V
+Python 3.12.12
+
+pip -V
+pip 26.0.1
+
+python -c "import platform; print(platform.platform())"
+Linux-6.8.12-17-pve-x86_64-with-glibc2.41
+
+lscpu | head
+Architecture:               x86_64
+CPU op-mode(s):             32-bit, 64-bit
+Address sizes:              48 bits physical, 48 bits virtual
+Byte Order:                 Little Endian
+CPU(s):                     16
+On-line CPU(s) list:        1-3,8
+Off-line CPU(s) list:       0,4-7,9-15
+Vendor ID:                  AuthenticAMD
+Model name:                 AMD Ryzen 7 7700 8-Core Processor
+CPU family:                 25
+    \end{verbatim}
+\end{itemize}
+
+\subsection*{Analysis}
+\begin{itemize}
+    \item Justification of parameter intervals
+    \item Retain stability by 200 trials
+\end{itemize}
+
+\subsection*{Connection}
+\begin{itemize}
+    \item Method is the fundament of runtime investigation
+\end{itemize}
+
+\section{Real Data Analysis}
+
+\subsection*{Development}
+\begin{itemize}
+    \item Get real data from DIMACS
+    \item Extraction of edge weight distribution
+    \item Calculation of mean and std
+\end{itemize}
+
+\subsection*{Evidence}
+\begin{itemize}
+    \item Edge weight distribition
+          \begin{figure}[H]
+              \centering
+              \includegraphics[width=0.7\textwidth]{hist_original.png}
+          \end{figure}
+          \begin{figure}[H]
+              \centering
+              \includegraphics[width=0.7\textwidth]{hist_log.png}
+          \end{figure}
+    \item Summary statistics tableofcontents
+          \begin{table}[H]
+              \centering
+              \begin{tabular}{ll}
+                  \hline
+                  Key         & Value    \\ \hline
+                  Total edges & 58333344 \\
+                  Mean        & 2950.322 \\
+                  Std         & 4071.694 \\
+                  Min         & 1        \\
+                  Max         & 368855   \\ \hline
+              \end{tabular}
+          \end{table}
+\end{itemize}
+
+\subsection*{Analysis}
+\begin{itemize}
+    \item Lognormal approximation calculation
+    \item Justification of synthetic distribution parameters
+\end{itemize}
+
+\subsection*{Connection}
+\begin{itemize}
+    \item Synthetic data is based on real-world data
+\end{itemize}
+
+\section{Relationship between variables}
+
+\subsection*{Development}
+\begin{itemize}
+    \item Density vs relax\_attempts
+    \item Sigma vs relax\_success\_ratio
+    \item Decrease-key vs speedup
+\end{itemize}
+
+\subsection*{Evidence}
+\begin{itemize}
+    \item Density vs relax\_attempts plot
+          \begin{figure}[H]
+              \centering
+              \includegraphics[width=0.7\textwidth]{density_vs_relax_attempts.png}
+          \end{figure}
+    \item Sigma vs relax\_success\_ratio platform
+          \begin{figure}[H]
+              \centering
+              \includegraphics[width=0.7\textwidth]{sigma_vs_relax_ratio.png}
+          \end{figure}
+    \item Decrease-key vs speedup correlation
+          \begin{figure}[H]
+              \centering
+              \includegraphics[width=0.7\textwidth]{speedup_corr.png}
+          \end{figure}
+\end{itemize}
+
+\subsection*{Analysis}
+\begin{itemize}
+    \item Density increases relax attempts nearly linearly
+    \item Higher variance increases relax\_success ratio
+    \item Speedup weakly correlated with decrease-key frequency
+\end{itemize}
+
+\subsection*{Connection}
+\begin{itemize}
+    \item Structural conditions necessary but insufficient for crossover
+\end{itemize}
+
+\section{Runtime}
+
+\subsection*{Development}
+\begin{itemize}
+    \item Multiple linear regression of runtime
+    \item Operation level cost estimation
+\end{itemize}
+
+\subsection*{Evidence}
+\begin{itemize}
+    \item Regression summary table
+          \begin{table}[H]
+              \small
+              \centering
+              \begin{tabular}{llllllll}
+                  \hline
+                  algorithm & intercept & $coef_{add}$  & $coef_{extract}$ & $coef_{relax}$ & $coef_{decrease}$ \\ \hline
+                  binary    & 0.000492  & -3.496595e-07 & 3.259253e-07     & 9.095429e-08   & 1.702164e-07      \\
+                  fibonacci & -0.001191 & 7.446706e-07  & 3.938917e-07     & 9.333419e-08   & 1.504052e-06      \\ \hline
+              \end{tabular}
+          \end{table}
+    \item $R^2$ comparison between models
+          \begin{table}[H]
+              \centering
+              \begin{tabular}{lll}
+                  \hline
+                  algorithm & $r_2$    & $n_{samples}$ \\ \hline
+                  binary    & 0.980930 & 54801         \\
+                  fibonacci & 0.980877 & 54801         \\ \hline
+              \end{tabular}
+          \end{table}
+\end{itemize}
+
+\subsection*{Analysis}
+\begin{itemize}
+    \item Fibonacci decrease-key cost significantly larger in Python
+    \item Constant factors dominate asymptotic differences
+\end{itemize}
+
+\subsection*{Connection}
+\begin{itemize}
+    \item Explains absence of empirical speedup
+\end{itemize}
+
+\section{Deriving the Crossover Condition}
+
+\subsection*{Development}
+\begin{itemize}
+    \item Substitute empirical coefficients into:
+          \[
+              T_F < T_B
+          \]
+\end{itemize}
+
+\subsection*{Evidence}
+\begin{itemize}
+    \item Derived inequality expression
+\end{itemize}
+
+\subsection*{Analysis}
+\begin{itemize}
+    \item Required conditions exceed practical graph scale
+    \item Python constant overhead prevents crossover
+\end{itemize}
+
+\subsection*{Connection}
+\begin{itemize}
+    \item Theoretical superiority does not always linked to practical improvement
+\end{itemize}
+
+\section{Conclusion}
+
+\subsection*{Conclusion}
+\begin{itemize}
+    \item Fibonacci heap does not outperform Binary heap in Python
+    \item Operation level costs dominate asymptotic complexity
+    \item Crossover condition impractical under tested environment
+\end{itemize}
+
+\subsection*{Limitation \& Evaulation}
+\begin{itemize}
+    \item Python-only implementation
+    \item Synthetic graph model assumptions
+    \item No low-level memory/cache analysis
+\end{itemize}
+
+\subsection*{Summarize main points}
+\begin{itemize}
+    \item Graph structure affects decrease-key frequency
+    \item Empirical runtime does not follow asymptotic expectation
+    \item Constant factors determine real-world performance
+\end{itemize}
+
+\end{document}