Write and explain the recurrence relation of Merge Sort.

To write and explain the recurrence relation of Merge Sort, we need to understand the basic steps involved in the algorithm.

1. Divide: The first step of Merge Sort is to divide the given array into two halves. This is done recursively until we reach the base case, which is when the array has only one element.

2. Conquer: Once we have divided the array into smaller subarrays, we recursively sort each subarray. This is done by applying the Merge Sort algorithm to each subarray.

3. Combine: The final step is to merge the sorted subarrays to obtain the final sorted array. This is done by comparing the elements of the two subarrays and placing them in the correct order.

Now, let's define the recurrence relation for Merge Sort. Let T(n) represent the time complexity of Merge Sort for an array of size n.

The time complexity of Merge Sort can be expressed as follows:

T(n) = 2T(n/2) + O(n)

Here, 2T(n/2) represents the time taken to sort the two subarrays of size n/2, and O(n) represents the time taken to merge the two sorted subarrays.

The recurrence relation states that the time complexity of Merge Sort for an array of size n is equal to twice the time complexity of Merge Sort for an array of size n/2, plus the time taken to merge the two sorted subarrays.

By solving this recurrence relation, we can determine the overall time complexity of Merge Sort. It can be shown that the time complexity of Merge Sort is O(n log n), which means that the algorithm has a logarithmic time complexity with respect to the input size.

In summary, the recurrence relation for Merge Sort is T(n) = 2T(n/2) + O(n), and the overall time complexity of Merge Sort is O(n log n).