Problem Description

The problem presents an array containing a sequence of numbers in ascending order that has been rotated between 1 and n times. A rotation means that the elements are shifted to the right by one position, and the last element is moved to the first position. The task is to find the minimum element in this rotated array. The challenge is to achieve this with an algorithm that runs in O(log n) time, which suggests that a binary search approach should be used because binary search has a logarithmic time complexity and is typically applied to sorted arrays to find a specific element quickly.

Intuition

The key to solving this problem lies in understanding how a rotation affects a sorted array. Despite the rotation, a portion of the array remains sorted. If the array is not rotated or has been rotated a full cycle (n times), then the smallest element would be at the beginning. If it is rotated, the array is composed of two increasing sequences, and the minimum element is the first element of the second sequence.

Therefore, we can use binary search to quickly identify the point where the transition from the higher value to the lower value occurs, which indicates the smallest element. The binary search is modified here to compare the middle element with the first element and decide where to move next:

1. If the middle element is greater than the first element, the smallest value must be to the right of the middle element. Hence, we search the right half of the array.
2. If the middle element is less than the first element, then the smallest value is somewhere to the left of the middle element, or it could be the middle element itself. Here, we search the left half.

By applying this logic recursively to the halves, the point at which the smallest element exists can be found efficiently, satisfying the required time complexity.

Learn more about Binary Search patterns.

Solution Approach

The implementation of the solution leverages a binary search algorithm to find the minimum element in the rotated array. Here are the detailed steps of the algorithm:

1. First, the solution checks if the first element of the array is less than or equal to the last element. If true, this indicates that the array is not rotated, or it is rotated a full cycle. Hence, the first element is already the minimum, and we can return it immediately.

   if nums[0] <= nums[-1]:
       return nums[0]

2. If the previous check fails, the solution sets two pointers, left and right, at the start and the end of the array, respectively. These pointers are used to dynamically narrow down the search region while performing the binary search.

   left, right = 0, len(nums) - 1

3. The algorithm enters a loop that continues as long as the left pointer is less than the right pointer. The purpose of this loop is to repeatedly narrow the search space until the minimum element is identified.

   while left < right:

4. Inside the loop, the solution calculates the mid point between left and right pointers. This mid point is used to compare the elements and decide which half of the array to search next.

   mid = (left + right) >> 1

5. The next step is to compare the element at the mid index with the first element in the array. If nums[mid] is greater than nums[0], we know that the smallest element must be to the right of mid, so we move the left pointer to mid + 1.

   if nums[0] <= nums[mid]:
       left = mid + 1

6. Otherwise, if nums[mid] is less than nums[0], the minimum element is to the left of mid, or it could be mid itself, so we move the right pointer to mid.

   else:
       right = mid

7. The loop continues until the left and right pointers converge to the index of the minimum element. At this point, we can return nums[left] as the minimum element of the array.

   return nums[left]

This approach guarantees that the running time will be O(log n) because it repeatedly eliminates half of the search space in each iteration, which is characteristic of a binary search.

Example Walkthrough

Let's consider a small example to illustrate the solution approach. Suppose we have the following rotated array:

nums = [4, 5, 6, 7, 0, 1, 2]

Initially, we check if the array is not rotated or has been rotated a full cycle:

if nums[0] <= nums[-1]:
    return nums[0]

However, nums[0] is 4 and nums[-1] is 2, so we do not return nums[0] because the array has been rotated.

Next, we set our left and right pointers:

left, right = 0, len(nums) - 1

Hence, left = 0 and right = 6.

Entering the loop, we calculate our mid index:

mid = (left + right) >> 1

This gets us mid = 3, with nums[mid] = 7.

We now compare nums[mid] with nums[0]. Since 7 (middle element) is greater than 4 (the first element), we update the left to mid + 1:

left = mid + 1

Now, left = 4 and right = 6.

On the next iteration, mid = (4 + 6) >> 1 = 5. The value nums[mid] is 1, which is less than nums[0].

Since nums[mid] is less than nums[0], we update the right pointer to mid:

right = mid

Now, left = 4 and right = 5.

Continuing with the loop, we calculate the next mid:

mid = (left + right) >> 1 = (4 + 5) >> 1 = 4

The value at mid index is 0, which is less than nums[0]. So, we update the right pointer to mid again:

right = mid

Since now both left and right are 4, the loop terminates, and we return the value nums[left] which is 0, the smallest element in the rotated array:

return nums[left]

Thus, following the provided approach, we efficiently find the minimum element in a rotated sorted array in O(log n) time.

Solution Implementation

Time and Space Complexity

The provided Python code implements a binary search algorithm to find the minimum element in a rotated sorted array. Here is the analysis of its time and space complexity:

Time Complexity:

The time complexity of the algorithm is O(log n), where n is the length of the input list nums. This is because the algorithm uses a binary search approach, where it repeatedly divides the search interval in half. At each step, the algorithm compares the middle element with the boundary elements to determine which half of the array to search next. The number of steps required to find the minimum will, therefore, be proportional to the logarithm of the array size.

Space Complexity:

The space complexity of the algorithm is O(1), as it uses only a constant amount of extra space. The variables left, right, and mid used for maintaining the bounds of the search space and no additional data structures are allocated that would depend on the size of the input list. Thus, the memory requirement remains constant irrespective of the input size.

Learn more about how to find time and space complexity quickly using problem constraints.