Question $5$: $(10$ Marks$)$

$($c$)[$S$]$ Strow the she following:

using the Marge$-$Sorting technique. You are be merge sorting algorithm.

$($d$)[5]$ Consider modifying the merge sort such that $\frac{n}{k}$ sub$-$lists each of length $k$ are sorted using the insertion sort.

$(1)$ Show that by using $\frac{n}{k}$ sub$-$lists then the insertion time complexity becomes $O(nk)$ worst$-$case.

$(2)$ Show also by using $\frac{n}{k}$ sub$-$lists the time complexity of the merge$-$sort becomes $O(nlo{g}_2(\frac{n}{k}))$ worst$-$case.

$(3)$ How should $k$ be chosen in practice.