algorithm - Why is the top down approach of heap construction less efficient than bottom up even though its order of growth is lower O(log n) over O(n)?

Question

How is the bottom up approach of heap construction of the order O(n) ? Anany Levitin says in his book that this is more efficient compared to top down approach which is of order O(log n). Why?

Answer 1

That to me seems like a typo.

There are two standard algorithms for building a heap. The first is to start with an empty heap and to repeatedly insert elements into it one at a time. Each individual insertion takes time O(log n), so we can upper-bound the cost of this style of heap-building at O(n log n). It turns out that, in the worst case, the runtime is Θ(n log n), which happens if you insert the elements in reverse-sorted order.

The other approach is the heapify algorithm, which builds the heap directly by starting with each element in its own binary heap and progressively coalescing them together. This algorithm runs in time O(n) regardless of the input.

The reason why the first algorithm requires time Θ(n log n) is that, if you look at the second half of the elements being inserted, you'll see that each of them is inserted into a heap whose height is Θ(log n), so the cost of doing each bubble-up can be high. Since there are n / 2 elements and each of them might take time Θ(log n) to insert, the worst-case runtime is Θ(n log n).

On the other hand, the heapify algorithm spends the majority of its time working on small heaps. Half the elements are inserted into heaps of height 0, a quarter into heaps of height 1, an eighth into heaps of height 2, etc. This means that the bulk of the work is spent inserting elements into small heaps, which is significantly faster.