Brute Force Solution

A naive solution would involve checking every possible subarray and finding the minimum length that satisfies the condition. This involves two nested loops. The outer loop iterates through all possible start indices, and the inner loop iterates through all possible end indices from the current start index.

Code

def min_subarray_len_brute_force(target, nums):
    min_len = float('inf')
    for i in range(len(nums)):
        current_sum = 0
        for j in range(i, len(nums)):
            current_sum += nums[j]
            if current_sum >= target:
                min_len = min(min_len, j - i + 1)
                break
    return min_len if min_len != float('inf') else 0

Time Complexity

O(n^2), where n is the length of the input array nums, due to the nested loops.

Space Complexity

O(1), as it uses only a constant amount of extra space.

Optimal Solution: Sliding Window

A more efficient approach involves using the sliding window technique. We maintain a window defined by two pointers, left and right. We expand the window by moving the right pointer until the sum of elements within the window is greater than or equal to the target. Once this condition is met, we try to shrink the window from the left side to find the minimal length.

Code

def min_subarray_len_sliding_window(target, nums):
    left = 0
    current_sum = 0
    min_len = float('inf')
    for right in range(len(nums)):
        current_sum += nums[right]
        while current_sum >= target:
            min_len = min(min_len, right - left + 1)
            current_sum -= nums[left]
            left += 1
    return min_len if min_len != float('inf') else 0

Time Complexity

O(n), where n is the length of the input array nums. Each element is visited at most twice (once by the right pointer and once by the left pointer).

Space Complexity

O(1), as it uses only a constant amount of extra space.

Edge Cases

• Empty Input Array: If the input array nums is empty, the function should return 0.
• No Solution: If no subarray sums up to the target, the function should return 0.
• All Positive Integers: The problem statement specifies that all integers are positive, which helps simplify the sliding window approach as we only need to expand the window to the right and shrink from the left.
• Target Greater Than Sum of All Elements: If the target is greater than the sum of all elements in the array, the function should return 0.