Insertion sort on small arrays in merge sort

Question

Although merge sort runs in Θ(n lg n) worst-case time, and insertion sort runs in Θ(n 2 ) worst-case time, the constant factors in insertion sort make it faster for small n. Thus, it makes sense to use insertion sort within merge sort when subproblems become sufficiently small. Consider a modification to merge sort in which 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. What is the worst-case time to sort the n/k sublists (each of length k)?

b. Show that the sublists can be merged in Θ(n lg(n/k)) worst-case time.

c. What is the largest asymptotic (Θ-notation) value of k as a function of n for which the modified algorithm has the same asymptotic running time as standard merge sort? 

d.How should we choose k in practice?

Comments

https://cs.stackexchange.com/questions/68179/combining-merge-sort-and-insertion-sort

This might help best explanation in this

Answer 1

(a) In the insertion sort n element sorted time in the worst case is O(n2) time. In the problem k length  takes O(K2)  time .Hence

N/K sublist takes O(nk) times .

If we merge  sorted array using merge procedure it takes nlog(n/k) time .So total time=nk+nlog(n/k) time