To design a Python algorithm to solve the problem of finding the maximum sum of a subarray using the Kadane's algorithm, follow these steps:

def kadane_algorithm(arr):
    max_sum = arr[0]  # Initialize the maximum sum with the first element of the array
    current_sum = arr[0]  # Initialize the current sum with the first element of the array

    # Iterate through the array starting from the second element
    for num in arr[1:]:
        # Update the current sum by either adding the current element or starting a new subarray
        current_sum = max(num, current_sum + num)
        # Update the maximum sum if the current sum is greater
        max_sum = max(max_sum, current_sum)

    return max_sum

# Example usage:
array = [-2, 1, -3, 4, -1, 2, 1, -5, 4]
max_subarray_sum = kadane_algorithm(array)
print("Maximum subarray sum using Kadane's algorithm:", max_subarray_sum)

In this algorithm:

• max_sum stores the maximum sum found so far.
• current_sum keeps track of the sum of the current subarray being processed.
• We iterate through the array, updating current_sum to either include the current element or start a new subarray from the current element.
• At each step, we update max_sum if current_sum is greater, thus ensuring we always have the maximum subarray sum encountered so far.
• Finally, we return max_sum as the result, which represents the maximum sum of a subarray within the given array.