$2-1$ Insertion sort on small arrays in merge sort

Although merge sort runs in $(nlgn)$ worst$-$case time and insertion sort

runs in $(n^2)$ worst$-$case time, the constant factors in insertion sort can

make it faster in practice for small problem sizes on many machines.

Thus it makes sense to coarsen the leaves of the recursion by using

insertion sort within merge sort when subproblems become sufficiently

small. Consider a modification to merge sort in which $\frac{n}{k}$ sublists of

length $k$ are sorted using insertion sort and then merged using the

standard merging mechanism, where $k$ is a value to be determined.

a$.$ Show that insertion sort can sort the $\frac{n}{k}$ sublists, each of length $k,$ in

$(nk)$ worst$-$case time.

b$.$ Show how to merge the sublists in $(nlg(\frac{n}{k}))$ worst$-$case time.

c$.$ Given that the modified algorithm runs in $(nk+nlg(\frac{n}{k}))$ worst$-$

case time, what is the largest value of $k$ as a function of $n$ for which

the modified algorithm has the same running time as standard merge

sort, in terms of $-$notation?

d$.$ How should you choose $k$ in practice?