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\rule{\textwidth}{.5pt}\\[0.4cm] \huge Assignment 1: Question 2 \\
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\paragraph{Acknowledgments.} \YourAck



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\section*{Solving recurrences}

Solve the following recurrence relations to obtain a closed-form big-$\Theta$ expression for $T(n)$.
In each question, assume $T(c)$ is bounded by a constant for any small constant $c$.

\begin{enumerate}[(a)]
\item $T(n) = 9\,T(\frac{n}{3}) + n^2$

\begin{solution}
(ENTER YOUR SOLUTION HERE.)
\end{solution}

\item $T(n) = 4\,T(\frac{n}{4}) + n \log n$

\begin{solution}
(ENTER YOUR SOLUTION HERE.)
\end{solution}

\item $T(n) = T(\frac{n}{4}) + T(\frac{3n}{4}) + n$

\begin{solution}
(ENTER YOUR SOLUTION HERE.)
\end{solution}

\item $T(n) = \sqrt{n} \cdot T(\sqrt{n}) + n$ \\
\emph{Hint.} The correct expression is somewhere between $\Omega(n)$ and $O(n \log n)$.

\begin{solution}
(ENTER YOUR SOLUTION HERE.)
\end{solution}

\end{enumerate}

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