$1)$ Consider

a$)$ What does this algorithm compute? $(10$ points$)$

b$)$ What is the time complexity of this algorithm $($apply the analysis framework$)?(10$ points$)$

$1.$ Consider the following recursive algorithm

Set up a recurrence relation for the algorithm$$s basic operation count and solve it$.$ Finally analysis the time complexity of this algorithm by applying the analysis framework. $(20$ points$)$

$1.$ Prove $12$n$($n$-$

$3)$ Prove $12$n$($n$-1)\backslash$Theta $($n$2).($Hint: prove it based on the formal definition of $\backslash$Theta $)(20$ points

$4)$

Consider the following $20$ functions

n $2$n nlog$($n$)$

n $-$ n$3+7$n$5$ n$2+$ log$($n$)$ n$2$

n$3$ log$($n$)$ n$!$

ln$($n$)(1/3)$n $(3/2)$n

$6$ ln$2$n $3$

Log$($n$+1)1002$n$-12$

$($n$-1)!3$n

Group these functions so that f$($n$)$ and g$($n$)$ are in the same group if and only if f$($n$)$ O$($g$($n$))$ and g$($n$)$ O$($f$($n$)).$ List the groups in increasing order. $(20$ points$)$