Exercise 6.5.8

The operation HEAP-DELETE(A,i) deletes the item in node $i$ from heap $A$. Give an implementation of HEAP-DELETE that runs in $\O(\lg{n})$ time for an $n$-element max-heap.

This is the pseudocode is as follows:

  A[i] = A[A.heap-size]
  A.heap-size -= 1

We just move the last element of the heap to the deleated position and then call MAX-HEAPIFY on it. This works, because the element is already smaller than its parent (because it was already under it on the heap), but might be larger than its children. MAX-HEAPIFY restored the heap property.