When analyzing the space complexity of an algorithm, we look at the amount of additional space required by the algorithm as the input size increases. In the case of heap sort, the space complexity can be evaluated in terms of the auxiliary space (i.e., extra space required excluding the input array) and the overall space complexity (including the input array).

Auxiliary Space Complexity of Heap Sort

Heap sort is an in-place sorting algorithm, which means it sorts the input array without using additional space other than the input array itself. Therefore, the auxiliary space complexity of heap sort is O(1), indicating that the extra space required is constant and independent of the input size.

Heap sort achieves this by using a binary heap data structure to store the elements of the array. The binary heap representation is built directly into the input array by swapping elements, eliminating the need for any additional data structures. The operations performed during the sorting process, such as building the heap, swapping elements, and re-heapifying after each swap, all occur within the original array.

Overall Space Complexity of Heap Sort

The overall space complexity of an algorithm includes both the auxiliary space and the space required by the input. In the case of heap sort, the overall space complexity is O(n), where ‘n’ is the size of the input array. This is because the input array itself contributes to the space complexity.

In heap sort, the input array is used to hold the elements of the heap, so the space occupied by the array is directly proportional to the input size. As the input size increases, the space complexity of heap sort grows linearly.

It is worth noting that the space complexity of heap sort can be improved to O(1) if the input array is not included in the overall space complexity analysis, focusing solely on the auxiliary space. However, in most practical cases, the overall space complexity, including the input array, is considered.

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