P $1.[3$ pts$]$ Use the method illustrated in class $($recursion trees in the textbook$)$ to derive the solution to the following recurrence. Show your work. $(Please$ draw the recursion tree$)$

T$($n$)=2$T$($n$/4)+\sqrt[\phantom{2}]{n}.$

P $2.[4$ pts$]$ Describe a divide and conquer algorithm to compute the square of an n$-$digit integer in O$(n^{{\mathit{l}\mathit{o}\mathit{g}}_{2}3}$time, by reducing to the squaring of three $$n$/2-$digit integers. Adding two numbers with k digits, and shifting a number with k digits take O$($k$)$ time.

Your submission should include the following points:

$$ Problem statement. You need not provide an example.

$$ The main idea of the algorithm. Argue the correctness of your algorithm here,

by showing how the algebraic derivations are used by your algorithm.

$$ Algorithm pseudo$-$code.

$$ Running time analysis. You can directly use the solution of the recurrence

relation worked in class $($aka Master Theorem $/$ recursion trees section in the

text$).(Please$ show all the steps for the $4$ points$)$

Hint: use the identity xy $=\frac{x^2+y^2-(x-y{)}^2}{2}.$

P $3.[3$ pts$]$ Describe a divide and conquer algorithm to compute the square of an n$-$digit integer in O$(n^{{\mathit{l}\mathit{o}\mathit{g}}_{3}6}$time, by reducing to the squaring of six $$n$/3-$digit integers.

Include the same points in your submission as for Problem P $2..$ Is this algorithm

asymptotically faster that your algorithm from P $2.?$

Hint: use the expression for $(x+y+z{)}^2.$

$(Please$ show all the $4$ steps for this question from $P2)$

P $4.[3$ pts$]($BONUS$)$ Describe a divide and conquer algorithm to compute the square of an n$-$digit integer in O$(n^{{\mathit{l}\mathit{o}\mathit{g}}_{3}5}$time, by reducing to the squaring of five $$n$/3-$digit integers. To simplify the analysis, we assume that the numbers in the recursive calls do not require more bits than the terms that make up these numbers. This means thatthe number of digits of the integers passed to the recursive calls do not exceed $$n$/3$

Include the same points in your submission as for Problem P $2..$ Is this algorithm

asymptotically faster that your algorithm from P $2.?$

Hint: investigate the expression $($x $+$ y $+$ z${)}^2+($x $$ y $+$ z${)}^2.$

$($Please show all the $4$ steps for this question from P$2)$