What are the key differences between heapsort and mergesort, and which algorithm is more efficient in terms of time complexity and space complexity?

Heapsort and mergesort are both comparison-based sorting algorithms. The key differences between them are in their approach to sorting and their time and space complexity.

Heapsort uses a binary heap data structure to sort elements. It has a time complexity of O(n log n) in the worst-case scenario and a space complexity of O(1) since it sorts in place.

Mergesort, on the other hand, divides the array into two halves, sorts them recursively, and then merges them back together. It has a time complexity of O(n log n) in all cases and a space complexity of O(n) since it requires additional space for merging.

In terms of time complexity, both algorithms have the same efficiency. However, in terms of space complexity, heapsort is more efficient as it does not require additional space proportional to the input size.