$(5$ points$)$ Design an algorithm for computing ${|}_{{?}_{?}}\sqrt[\phantom{2}]{n}{?}_{{?}_{?}}|$ for any positive integer $n.$ Besides assignment

and comparison, your algorithm may only use the four basic arithmetical operations.

$(5$ points$)$ Design an algorithm to find all the common elements in two sorted lists of numbers. For

example, for the lists $2,5,5,5$ and $2,2,3,5,5,7,$ the output should be $2,5,5.$ What is the maximum

number of comparisons your algorithm makes if the lengths of the two given lists are $m$ and $n,$

respectively?

$(5$ points$)$ Describe the standard algorithm for finding the binary representation of a positive

decimal integer in pseudocode.

$(5$ points$)$ Consider the following algorithm for finding the distance between the two closest

elements in an array of numbers. Make as many improvements as you can in this algorithmic

solution to the problem. If you need to$,$ you may change the algorithm altogether; if not, improve

the implementation given.

ALGORITHM MinDistance$($A$[0..$n $-1])$

$//$Input: Array $A[0..n-1]$ of numbers

$//$Output: Minimum distance between two of its elements

dmin larr$$

for ilarr$0$ to $n-1$ do

for jlarr$0$ to $n-1$ do

if $ij$ and dminlarr$|A[i]-A[j]||A[i]-A[j]|$

dminlarr$|A[i]-A[j]|$

return dmin