Time Complexity

Step 1: Understand the time complexities.
- **Bubble sort (worst case)** (A): The worst-case time complexity of bubble sort is ** **.
- **Deleting head node in singly linked list** (B): Deleting the head node takes ** ** time.
- **Binary search** (C): Binary search has a time complexity of ** **.
- **Worst case of merge sort** (D): Merge sort has a worst-case time complexity of ** **.

Step 2: Arrange in increasing order.
From lowest to highest, the correct order is: - **(C)** Binary search: - **(B)** Deleting head node: - **(D)** Merge sort: - **(A)** Bubble sort (worst case):

Step 3: Conclusion.
Thus, the correct order is **(C), (B), (D), (A)**.

Step 1: Analysis of Algorithms.
- **Binary Search** (A): The time complexity for binary search in a sorted array is ** **, which corresponds to the recurrence relation ** ** (I).
- **Merge Sort** (B): Merge Sort divides the array into two halves recursively, with a linear combination of the subproblems, leading to the recurrence ** ** (II).
- **Quick Sort (worst case partitioning)** (C): In the worst case, quick sort behaves like selection sort with recurrence ** ** (III).
- **Linear Search** (D): Linear search performs a sequential scan of the list, giving the recurrence ** ** (IV).

Step 2: Conclusion.
The correct matching is **(A) - (I), (B) - (II), (C) - (III), (D) - (IV)**.