Prоblem 4 (14 pоints) A binаry min-heаp suppоrts аn insert operation by adding a new element to the end of the heap array. The normal insert method will repeatedly swap the new element with its parent until the min-heap property is restored. A student proposes a modification to the standard insert operation in an effort to improve the runtime. The student observes that a newly inserted element often needs to move several levels upward before reaching its correct position. To reduce the number of swaps, the student proposes allowing the inserted element to move two levels upward at a time. The proposed idea is: If the inserted element has a grandparent and is smaller than its grandparent, swap the inserted element with its grandparent. Otherwise, perform the standard comparison with the parent and swap with the parent if the heap property is violated. The student claims that this modification is correct because moving an element closer to the root more quickly should reduce the number of swaps while still maintaining the heap property. The modified insert operation is shown below. The heap is stored in an array beginning at index 1. INSERT(A, x): append x to the end of A i = size(A) while i > 1: if grandparent of i exists: grandparent = floor(i/4) if A[i] < A[grandparent]: swap A[i] and A[grandparent] i = grandparent else: parent = floor(i/2) if A[i] < A[parent]: swap A[i] and A[parent] i = parent else: break else: parent = floor(i/2) if A[i] < A[parent]: swap A[i] and A[parent] i = parent else: break Task Determine whether this modified insert operation always maintains the min-heap property. You may answer in one of two ways: If you believe the algorithm is incorrect: Provide a counterexample. Your counterexample must include: A valid min-heap before the insertion. The value being inserted. The resulting heap produced by the modified algorithm. An explanation of why the resulting heap is not a valid min-heap. If you believe the algorithm is correct: Provide a justification explaining why moving the inserted element two levels upward cannot violate the min-heap property. A formal proof is not required but your explanation should address why the min-heap property is maintained.
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