You decided to modify mergesort by recursively splitting an array of size n into k sorted subarrays; k > 2, and merging these subarrays. Time for merging is c(k?1)n where c is a constant factor, because at each step the largest item among the k top ones in the k subarrays is found by k ? 1 comparisons and placed into the merged array. Specify the recurrence relation and derive the closed-form formula for sorting time Tk(n) of the modified mergesort for an arbitrary k (you should show all steps of the derivation). Then determine whether the modified mergesort could be faster for some k > 2 than the conventional one (k = 2) with the sorting time T2(n) = cn log2 n.

Hint: To have a well-defined recurrence, assume that n = k m with the integer m = logk n and T(1) = 0. You might need also the equality log2 x = log2 k logk x for all x > 0 and the inequality k > 1 + log2 k for all k > 2.