You are given two sorted integer arrays A and B such that no integer is contained twice in the same array. A and B are nearly identical. However, B is missing exactly one number. What will be the order of time complexity for the best algorithm to find the missing number in B. Assume that n is the number of elements in A.
Question 9 options:

O(n)


O(logn)


O(nlogn)


O(n^2)

The problem states that we are given two sorted arrays, \(A\) and \(B\), where array \(B\) is missing exactly one number from array \(A\). To find the time complexity of the best algorithm to find the missing number, we need to consider efficient ways to identify the difference between the two arrays.

### Analysis of possible algorithms:
1. **Brute Force Approach (O(n))**:
   We could iterate through both arrays and compare each element of \(A\) and \(B\) until we find the mismatch, which will take **O(n)** time because it requires a linear scan through the arrays. Since the arrays are sorted, the first mismatch will reveal the missing number.

2. **Binary Search Approach (O(logn))**:
   Given that the arrays are sorted, a binary search could potentially be used to find the point where the two arrays diverge. This could allow finding the missing number in **O(logn)** time. However, this approach requires some more specific steps:
   - Compare the middle elements of \(A\) and \(B\).
   - If the middle elements are the same, the missing number is in the second half; if they differ, it is in the first half.
   - Recursively continue the binary search on the appropriate half.

Since binary search leverages the sorted nature of the arrays, this can be done in logarithmic time.

### Conclusion:
The best algorithm to find the missing number can be achieved using a binary search, which runs in **O(logn)** time. Therefore, the correct answer is:

**O(logn)**

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Would you like further details on how the binary search works or have any other questions?

Here are 5 related questions to deepen your understanding:
1. How would a linear scan algorithm for this problem be implemented?
2. Why is binary search more efficient than a linear scan in this case?
3. What other problems can be solved using a similar binary search approach?
4. How does the sorted nature of arrays help in reducing time complexity?
5. What is the difference between a logarithmic and linear time complexity?

**Tip:** When dealing with sorted arrays, always think about using binary search to optimize your solution.

You are given two sorted integer arrays A and B such that no integer is contained twice in the same array. A and B are nearly identical. However, B is missing exactly one number. What will be the order of time complexity for the best algorithm to find the missing number in B. Assume that n is the number of elements in A.

Discover how to find the missing number in two nearly identical sorted arrays with a detailed explanation of the binary search algorithm. Learn how this approach reduces time complexity to O(logn) in comparison to a linear scan. Ideal for students at the college level learning about algorithm efficiency and data structures.

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