Time and Space Complexity Analysis of Merge Sort

The Time Complexity of Merge Sort is O(n log n) in both the average and worst cases. The space complexity of Merge sort is O(n).

Aspect	Complexity
Time Complexity	O(n log n)
Space Complexity	O(n)

Time Complexity Analysis of Merge Sort:

Consider the following terminologies:

T(k) = time taken to sort k elements
M(k) = time taken to merge k elements

So, it can be written

T(N) = 2 * T(N/2) + M(N)
= 2 * T(N/2) + constant * N

These N/2 elements are further divided into two halves. So,

T(N) = 2 * [2 * T(N/4) + constant * N/2] + constant * N
= 4 * T(N/4) + 2 * N * constant
. . .
= 2^k * T(N/2^k) + k * N * constant

It can be divided maximum until there is one element left. So, then N/2^k = 1. k = log₂N

T(N) = N * T(1) + N * log₂N * constant
= N + N * log₂N

Therefore the time complexity is O(N * log₂N).

So in the best case, the worst case and the average case the time complexity is the same.

Space Complexity Analysis of Merge Sort:

Merge sort has a space complexity of O(n). This is because it uses an auxiliary array of size n to merge the sorted halves of the input array. The auxiliary array is used to store the merged result, and the input array is overwritten with the sorted result.