Which one of the following sequences when stored in an array at locations $A[1], \ldots, A[10]$ forms a max-heap?

In a 1-indexed binary max-heap stored in an array $A[1 \dots 10]$, the max-heap property requires that for every child at index $i$ ($2 \le i \le 10$), its parent at index $\lfloor i/2 \rfloor$ must satisfy:

Equivalently, for each parent index $i$:

• Left child is at index $2i$, so $A[i] \ge A[2i]$

• Right child is at index $2i+1$, so $A[i] \ge A[2i+1]$

Let us check the given options:

• Option A: $A = [23, 17, 10, 6, 13, 14, 1, 5, 7, 12]$ For $i = 3$, $A[3] = 10$, its left child is $A[6] = 14$. Since $A[6] > A[3]$ ($14 > 10$), the max-heap property is violated.

• Option B: $A = [23, 17, 14, 7, 13, 10, 1, 5, 6, 12]$ Checking all parent-child relationships:

  • $i = 1$: $A[1] = 23 \ge A[2]=17$ and $A[1] = 23 \ge A[3]=14$

  • $i = 2$: $A[2] = 17 \ge A[4]=7$ and $A[2] = 17 \ge A[5]=13$

  • $i = 3$: $A[3] = 14 \ge A[6]=10$ and $A[3] = 14 \ge A[7]=1$

  • $i = 4$: $A[4] = 7 \ge A[8]=5$ and $A[4] = 7 \ge A[9]=6$

  • $i = 5$: $A[5] = 13 \ge A[10]=12$

  All conditions are satisfied, so (B) is a valid max-heap.

• Option C: $A = [23, 17, 14, 6, 13, 10, 1, 5, 7, 15]$ For $i = 5$, $A[5] = 13$, its child is $A[10] = 15$. Since $15 > 13$, it violates the max-heap property.

• Option D: $A = [23, 14, 17, 1, 10, 13, 16, 12, 7, 5]$ For $i = 4$, $A[4] = 1$, its left child is $A[8] = 12$. Since $12 > 1$, it violates the max-heap property.