Which ONE of the following options gives the worst-case time complexities for meld operation on instances of size $n$ of these data structures?

Let us analyze the worst-case time complexity of the meld operation for two instances, each of size $n$:

1. P: Unsorted doubly linked list with head and tail pointers

   • Since the list is unsorted and we have direct pointers to the head and tail of both lists, we can concatenate the two lists in $O(1)$ time by setting list1.tail->next = list2.head and list2.head->prev = list1.tail, followed by updating the overall tail pointer.

   • Hence, the time complexity is $\Theta(1)$.

2. Q: Min-heap implemented using an array

   • We can concatenate the two arrays of size $n$ into a single array of size $2n$ in $\Theta(n)$ time.

   • Then, we run the standard bottom-up heap construction algorithm (Floyd's buildHeap) on the combined array of size $2n$, which takes linear time $\Theta(n)$.

   • Hence, the overall time complexity is $\Theta(n)$.

3. R: Binary Search Tree (BST)

   • Perform an in-order traversal of both BSTs to obtain two sorted arrays of size $n$ in $\Theta(n)$ time.

   • Merge the two sorted arrays into a single sorted array of size $2n$ using standard two-pointer merging in $\Theta(n)$ time.

   • Construct a balanced BST from the merged sorted array in $\Theta(n)$ time using divide-and-conquer.

   • Hence, the overall time complexity is $\Theta(n)$.

Therefore, the correct worst-case time complexities are P: $\Theta(1)$, Q: $\Theta(n)$, R: $\Theta(n)$, matching Option (A).