algorithm - What is the worst-case time for insertion sort within merge sort? -

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recently stumbled upon problem introduction algorithms edition 3

problem 2-1:

although merge sort runs in o(n logn) worst-case time , insertion sort runs in o(n^2), latter runs faster small problem sizes. consider modification merge sort in n/k sublists of length k sorted using insertion sort , merged using standard merging mechanism.

(a) show insertion sort can sort n/k sublists, each of length k, in o(nk) worst-case time.

the answer given is:

ans: insertion sort takes (k^2) time per k-element list in worst case. therefore, sorting n/k lists of k elements each takes (k^2 n/k) = (nk) worst-case time

how (k^2 n/k) given data?? im not understanding @ , greatlly appreciate explanation.

the sublists of length k, therefore insertion sort takes k^2 each sublist. now, there n/k sublists in total, so, n/k * k^2 nk. key understanding here there n/k number of sublists, , insertion sort takes k^2 time sort each one.

another thing note, knowing merge sort has o(n logn) not important @ problem, because don't ask time sorting whole list, time sorting of sublists.


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