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\title{B-Tree Delete Complexity Proof}
\author{Shyamal Suhana Chandra}
\date{Copyright (C) 2025}

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\maketitle

\section{Theorem: B-Tree Delete Complexity}

\textbf{Statement:} Deleting from a B-Tree with $n$ keys takes $O(\log n)$ time.

\section{Proof}

\begin{enumerate}
    \item Find node to delete: $O(\log n)$
    \item If internal node, find predecessor/successor: $O(\log n)$
    \item Delete from node: $O(1)$
    \item Handle underflow:
    \begin{itemize}
        \item Borrow from sibling: $O(1)$
        \item Merge with sibling: $O(t) = O(1)$
        \item Propagate upward: at most $O(\log n)$ levels
    \end{itemize}
    \item Total time: $O(\log n) + O(\log n) = O(\log n)$
\end{enumerate}

\textbf{Conclusion:} B-Tree delete has $O(\log n)$ time complexity.

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