Final answer:
Building a heap from an unsorted array has a time complexity of O(n). This is proven by considering the work required to sift down nodes at each level of the heap, leading to a total work proportional to the number of elements, n.
Step-by-step explanation:
In computer science, particularly in algorithm analysis, when stating that the initial heaping takes O(n) operations, one is usually referring to the time complexity of building a heap from an unsorted array using the heapify process. This process involves organizing the elements of the array into a heap data structure, a type of binary tree where each node is greater than or equal to (in the case of a max heap) or less than or equal to (in the case of a min heap) the values of its children.
To prove that heap construction is O(n), we can use an argument based on the height of the tree and the number of nodes at each level. A heap can be seen as a complete binary tree, and it has ½n nodes at the lowest level, ¼n nodes at the second lowest, and so on, with the number of nodes halving as we move up each level. For each of these nodes, we might perform a sifting down operation. However, the operation is more efficient the closer a node is to the bottom of the tree because it has fewer levels to sift down.
By calculating the maximum work done at each level and summing it, the result is that the amount of work done is less than or equal to the sum of a geometric series that converges to a value proportional to n, thus the time complexity for building a heap is O(n).