Answer:

The time complexity of the operations in a Binary Min Heap depends on the number of elements in the heap and the structure of the heap. The given options are:

a) Inserting an item under the assumption that the heap has capacity to accommodate one more item.
b) Merging with another heap under the assumption that the heap has capacity to accommodate items of the other heap.
c) Deleting an item from heap D.
d) Decreasing the value of a key.

Understanding Binary Min Heap:
A Binary Min Heap is a complete binary tree where the value of each node is smaller than or equal to its children. This structure allows efficient access to the smallest element in the heap, which is always the root node. The heap maintains the heap property during all operations, ensuring the smallest element is always at the root.

a) Inserting an item:
When inserting an item into a Binary Min Heap, we start by placing the new element at the bottom-rightmost position (maintaining the complete binary tree property). Then, we compare the new element with its parent and swap them if necessary to maintain the heap property. This process continues until the element reaches its correct position in the heap.

The time complexity for inserting an item into a Binary Min Heap is O(log n), where n is the number of elements in the heap. This is because the height of a complete binary tree is log n, and in the worst case, we may need to swap the new element with its parent from the bottom to the root, which takes log n swaps.

b) Merging with another heap:
When merging two Binary Min Heaps, we assume that the heap has enough capacity to accommodate the items from both heaps. The merging process involves taking all the elements from one heap and inserting them into the other heap. This can be done by repeatedly calling the insert operation for each element.

The time complexity for merging two Binary Min Heaps is O(n log n), where n is the total number of elements in both heaps. Since we need to insert each element individually, the time complexity becomes the sum of inserting n elements, which is O(n log n).

c) Deleting an item:
When deleting an item from a Binary Min Heap, we always remove the root element, which is the smallest element in the heap. After removing the root, we replace it with the last element in the heap and perform a "heapify" operation to maintain the heap property. The "heapify" operation compares the element with its children and swaps it with the smaller child if necessary. This process continues recursively until the element reaches its correct position in the heap.

The time complexity for deleting an item from a Binary Min Heap is O(log n), where n is the number of elements in the heap. This is because the "heapify" operation may need to perform swaps from the root to the bottom of the heap, which takes log n swaps in the worst case.

d) Decreasing the value of a key:
When decreasing the value of a key in a Binary Min Heap, we locate the element in the heap and update its value. Then, we compare the updated element with its parent and swap them if necessary to maintain the heap property. This process continues recursively until the element reaches its correct position in the heap.

The time complexity for decreasing the value of a key in a Binary Min Heap is O(log