A local minimum in an array is an element that is less than or equal to its adjacent elements. If the element is at the beginning or end of the array, it only needs to be compared to its single neighbor.
A common approach to find a local minimum is using a modified binary search. This approach has a time complexity of O(log N). The logic is as follows:
- Compare the middle element with its neighbors.
- If the middle element is less than or equal to both its neighbors (or its single neighbor if it's an edge element), it's a local minimum.
- If the left neighbor is smaller, a local minimum must exist in the left half of the array.
- If the right neighbor is smaller, a local minimum must exist in the right half of the array.
- Recursively apply this process to the relevant half.
For a variation where a local minimum must be strictly less than its neighbors, the binary search approach might fail if there are duplicate values (e.g., [1, 1, 1, 0, 1, 1]). In such cases, a linear scan (O(N)) or a recursive approach that checks neighbors might be necessary to guarantee finding a strictly local minimum.
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