$1)$ O$\backslash$&S Problem $9\mathrm{.}17.$ The referenced Eq$.(8\mathrm{.}67)$ is the same as the DFT definition on page $4,$ LN$013.$

$9\mathrm{.}17.$ Suppose you have two programs for computing the DFT of a sequence $\backslash ($ x$[$n$]\backslash )$ that has $\backslash ($ N$=2^\{$v$\}\backslash )$ nonzero samples. Program A computes the DFT by directly implementing the definition of the DFT sum from Eq$.(8\mathrm{.}67)$ and takes $\backslash ($ N$^\{2\}\backslash )$ seconds to run. Program B implements the decimation$-$in$-$time FFT algorithm and takes $\backslash (10$ N $\backslash$log $\_\{2\}$ N $\backslash )$ seconds to run. What is the shortest sequence $\backslash ($ N $\backslash )$ such that Program B runs faster than Program A$?$