# Merge Sort Recurrence analysis for list of strings [duplicate]

I saw this question and tried to find out what the time complexity was:

Using the recurrence relation for Merge Sort:

$$T(n)\; =\; merge\_time\; +\; 2T(n/2)$$

Here, since we have a list of strings, the relevant merge operation is that of merging two sorted arrays of strings. Since comparing two strings is itself a $$O(n)$$ operation, this takes $$O(n^2)$$. Thus:

$$T(n)\; =\; O(n^2)\; +\; 2T(n/2)$$

Solving this, gives $$O(n^2)$$, not $$O(n^2logn)$$. This seems to suggest that this sorting can be done within quadratic time for all inputs. However, in this answer, it is explained why this is not the case. So, can somebody tell me what I am doing wrong here?