Maximum Subarray

Clarify the problem:

• The problem requires finding the maximum sum of a contiguous subarray within an array of integers.
• We need to implement a function that takes an array of integers as input and returns the maximum sum.

Analyze the problem:

• Input: An array of integers.
• Output: The maximum sum of a contiguous subarray.
• Constraints: The array can be empty.

Design an algorithm:

• We can use the Kadane's algorithm to solve this problem efficiently.
• The algorithm maintains two variables: max_sum and current_sum.
• We iterate over each element in the array and update current_sum by adding the current element or starting a new subarray if the current element is greater than the sum so far.
• We update max_sum whenever current_sum becomes greater than max_sum.
• Finally, we return the max_sum.

Explain your approach:

• We will implement a function called maxSubarraySum to solve the problem.
• The function will take an array of integers, nums, as input.
• We will initialize max_sum and current_sum to the first element of the array.
• We will iterate over the remaining elements in the array and update current_sum by adding the current element or starting a new subarray.
• Whenever current_sum becomes greater than max_sum, we update max_sum.
• Finally, we return the max_sum.

Write clean and readable code:

python

def maxSubarraySum(nums):
    if not nums:
        return 0

    max_sum = current_sum = nums[0]

    for i in range(1, len(nums)):
        current_sum = max(nums[i], current_sum + nums[i])
        max_sum = max(max_sum, current_sum)

    return max_sum

Test your code:

python

# Test case 1
nums = [-2, 1, -3, 4, -1, 2, 1, -5, 4]
# The maximum sum subarray is [4, -1, 2, 1], and the sum is 6
assert maxSubarraySum(nums) == 6

# Test case 2
nums = [1, 2, 3, 4, -5, 6, 7, 8]
# The maximum sum subarray is [1, 2, 3, 4, -5, 6, 7, 8], and the sum is 26
assert maxSubarraySum(nums) == 26

# Test case 3
nums = []
# The array is empty, so the maximum sum is 0
assert maxSubarraySum(nums) == 0

The code has been tested with multiple test cases, including cases with both positive and negative numbers and an empty array. The output for each test case matches the expected results.

Optimize if necessary:

• The current solution using Kadane's algorithm has an optimal time complexity of O(N), where N is the number of elements in the array.
• We iterate over each element once, calculating the maximum sum at each step.
• The space complexity is O(1) as we only need a constant amount of additional space for the max_sum and current_sum variables.

Handle error cases:

• The code handles the case when the input array is empty and returns 0 as the maximum sum.

Discuss complexity analysis:

• Let N be the number of elements in the input array.
• The time complexity of the solution is O(N) as we iterate over each element once.
• The space complexity is O(1) as we only use a constant amount of additional space.
• The code solves the problem optimally without any significant trade-offs.