Question

Suppose you are given an array of distinct sorted integers that has been circularly shifted k positions to the right. Design an efficient algorithm for finding k.

a) Binary search.

b) Linear search.

c) Hashing.

d) Divide and conquer.

Answer 1

To find the number of shifts k in a circularly shifted sorted array, a modified binary search algorithm is efficient. It locates the smallest element in the array, which corresponds to the shift point and the value of k. This method is more efficient than linear search or other approaches.

Step-by-step explanation:

The question is about designing an efficient algorithm to find the number of positions k by which an array of distinct sorted integers has been circularly shifted. The binary search method can be modified to solve this problem efficiently. Since the array is sorted but has been circularly shifted, the point where the shift occurs is the smallest element. This point also represents the number of shifts k.

Here's how a modified binary search would work:

1. Find the middle element of the array.
2. If this middle element is greater than the first element, the smallest element (and hence the shift) must be to the right. Move the search to the right half of the array.
3. If the middle element is smaller than the last element, the shift must be to the left. Move the search to the left half of the array.
4. Repeat this process, halving the search area each time, until the smallest element is found.
5. The index of the smallest element is the value of k.

Using this method reduces the time complexity significantly compared to a linear search and does not require additional data structures like hashing or divide and conquer strategies.