Merge k sorted arrays of size n in less then O(nklogk) time complexity

Question

The question:

Merge k sorted arrays each with n elements into a single array of size nk in minimum time complexity. The algorithm should be a comparison-based algorithm. No assumption on the input should be made.

So I know about an algorithm that solves the problem in nklogk time complexity as mentioned here: https://www.geeksforgeeks.org/merge-k-sorted-arrays/ .

Though, my question is can we sort in less than nklogk , meaning, the runtime is o(nklogk) .

So I searched through the internet and found this answer: Merge k sorted arrays of size n in O(nk) time complexity

Which claims to divide an array of size K into singletons and merge them into a single array. But this is incorrect since one can claim that he found an algorithm that solves the problem in sqrt(n)klogk which is o(nklogk) but n=1 so we sort the array in KlogK time which doesn't contradict the lower bound on sorting an array.

So how can I contradict the lower bound on sorting an array? meaning, for an array of size N which doesn't have any assumptions on the input, sorting will take at least NlogN operations.

Answer 1

The lower bound of n log n only applies to comparison-based sorting algorithms (heap sort, merge sort, etc.). There are, of course, sorting algorithms that have better time complexities (such as counting sort ), however they are not comparison-based.