What is the time complexity for building a heap from an unsorted array?

Building a heap from an unsorted array can be efficiently executed using a method known as the "heapify" process, which involves arranging the elements of the array in a way that satisfies the heap property, either a max-heap or min-heap.

This process operates in linear time complexity, or O(n), because the approach takes advantage of the properties of binary heaps. When building a heap, you start from the bottom non-leaf nodes of the heap and work your way upward to the root node. The key insight is that the number of nodes that need to be moved decreases as you move up the tree; most of the nodes are leaves, which do not need any rearrangement.

Furthermore, the cost of heapifying each node decreases as you move from the bottom to the top of the heap. For instance, nodes at the deepest levels of the heap do not require any swaps, while nodes higher up require more work but are fewer in number. This results in an aggregate linear time complexity across all nodes, leading to an overall efficient O(n) performance for the heap building process.

This is why the time complexity for building a heap from an unsorted array is O(n).