6-1 Building a heap using insertion

We can build a heap by repeatedly calling $\text{MAX-HEAP-INSERT}$ to insert the elements into the heap. Consider the following variation of the $\text{BUILD-MAX-HEAP}$ procedure:

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BUILD-MAX-HEAP'(A)
    A.heap-size = 1
    for i = 2 to A.length
        MAX-HEAP-INSERT(A, A[i])

a. Do the procedures $\text{BUILD-MAX-HEAP}$ and $\text{BUILD-MAX-HEAP}'$ always create the same heap when run on the same input array? Prove that they do, or provide a counterexample.

b. Show that in the worst case, $\text{BUILD-MAX-HEAP}'$ requires $\Theta(n\lg n)$ time to build a $n$-element heap.

a. Consider the following counterexample.

  • Input array $A = \langle 1, 2, 3 \rangle$:
  • $\text{BUILD-MAX-HEAP}(A)$: $A = \langle 3, 2, 1 \rangle$.
  • $\text{BUILD-MAX-HEAP}'(A)$: $A = \langle 3, 1, 2 \rangle$.

b. Each insert step takes at most $O(\lg n)$, since we are doing it $n$ times, we get a bound on the runtime of $O(n\lg n)$.