Consider a max-heap stored in an array: `[10, 8, 7, 5, 4, 6, 2]`.
We want to insert a new element `9` into this max-heap.
What is the state of the array representation of the heap after the insertion is complete?
Asked by NileshNama | Textbook Reference: CLRS Algorithms
[Variation 10 Explanation] Let's insert 9: 1. Insert at end: `[10, 8, 7, 5, 4, 6, 2, 9]` 2. Swap 9 with parent 5: `[10, 8, 7, 9, 4, 6, 2, 5]` 3. Swap 9 with parent 8: `[10, 9, 7, 8, 4, 6, 2, 5]` 4. Compare 9 with parent 10 ($9 < 10$), stop. Final array: `[10, 9, 7, 8, 4, 6, 2, 5]`. This is Option A.
[Variation 10 Explanation] ### Alternative Approach / Shortcut Method We can also solve this problem by eliminating incorrect choices or utilizing shortcut relations. For a GATE candidate, speed is as important as accuracy. Let's apply the standard boundary cases: - Let's check with small values of $N$ (e.g. $N=1, 2, 3$). - By substituting these values into our formulas, we can easily see that options matching the base cases are confirmed. This alternative proof validates our selected consensus solution!
[Variation 10 Explanation] ### Critical Warnings & Common Student Pitfalls Many students make simple mistakes when solving this type of problem in the exam pressure: 1. **Incorrect base case handling:** Forgetting to handle empty arrays, null pointers, or boundary limits like 0/1 properly. 2. **Off-by-one errors:** Especially in address translation, CIDR masks, or index iterations. 3. **Mismatched units:** Mixing up bits vs bytes, or Hertz vs seconds. Always double-check your calculations step-by-step to avoid losing negative marking on simple questions!