To find the minimum element in a rotated sorted array, we can take advantage of the array's sorted structure using a smart approach that avoids checking every element.
Understanding the Problem
A rotated sorted array is one that has been shifted from its original sorted position. For example, the sorted array [1, 2, 3, 4, 5]
might become [4, 5, 1, 2, 3]
. Our goal is to find the smallest element in this modified structure efficiently.
What Are the Possible Cases?
1. The array is not rotated at all: If the first element is less than the last element, then the array is still sorted in order. The minimum is simply the first element.
2. The array is rotated: Somewhere in the array, there is a drop from a large number to a smaller one. That drop is where the rotation happened, and the smaller number is the minimum.
3. Small arrays: For arrays with one or two elements, we can directly check them. If there’s one element, it’s the answer. If there are two, the smaller of the two is the answer.
4. Empty array: If the array has no elements, there's no minimum to return. In such cases, we return -1
as a signal that the input is invalid or empty.
How Binary Search Helps
Since the array is mostly sorted (but rotated), we can use binary search to narrow down the search range:
- If the middle element is greater than the last element, it means the minimum must be on the right side.
- If the middle element is less than or equal to the last element, then the minimum could be at mid or to the left.
We continue this process until our search space is narrowed down to one element — and that’s the minimum!
Why This Approach is Efficient
Binary search cuts the search range in half at every step. So instead of looking at all n
elements, we find the answer in O(log n)
time — making this method ideal for large arrays.
This strategy gives us a clear and optimal way to find the minimum element in any rotated sorted array.