Divide and Conquer: Given a set, $S$ of $n$ numbers in an array, I am looking

for the pair of numbers in $S$ such that the difference between the pair is the

least. Design a divide and conquer algorithm. I need divide and conquer

algorithm, so you cannot sort. How much time does it take? Describe the algorithm that builds a max heap from an array of size $n$ in

linear time. Argue that it runs in Linear time. If a max heap can be built in linear time, can we sort an array of $n$ numbers

in lincar time? Prim's minimum spanning tree algorithm: Describe the most efficient im$-$

plementation of the algorithm and argue that it takes $O(m+nlogn)$

amortized cost, where $m$ is the number of edges and $n$ is the number

vertices in the graph.Linear time algorithm for selection problem, given $n$ numbers, find the $k^{\text{'}}$ th

largest in linear time: Typical implementation starts with partitioning the

$n$ numbers into $|$~$\frac{n}{5}$~$|$ groups and each group contains $5$ numbers or less

$($the last group$).$ My question is$,$ how about we partition the numbers into

$3$ in a group or $7$ in a group? Please explain.