Maximum Product Subarray

1. Clarify the problem: Before diving into solving the problem, let's clarify the requirements:

• We are given an integer array.
• We need to find the contiguous subarray within the array that has the largest product of its elements.
• The subarray must contain at least one element.

2. Analyze the problem: To solve this problem, we can use a dynamic programming approach to keep track of the maximum and minimum products seen so far. We iterate through the array and update the maximum and minimum products at each step.

3. Design an algorithm: Here is the algorithm to find the maximum product subarray:

1. Initialize two variables max_product and min_product to store the maximum and minimum products seen so far.
2. Initialize a variable result to store the maximum product.
3. Iterate through the array from left to right:
   • Calculate the new maximum product by taking the maximum of the current element, the maximum product multiplied by the current element, or the minimum product multiplied by the current element.
   • Calculate the new minimum product by taking the minimum of the current element, the maximum product multiplied by the current element, or the minimum product multiplied by the current element.
   • Update the result by taking the maximum of the current result and the new maximum product.
4. Return the result.

4. Explain your approach: The approach involves iterating through the array and keeping track of the maximum and minimum products seen so far. At each step, we calculate the new maximum and minimum products based on the current element and the previous maximum and minimum products. We update the result with the maximum product found so far. This approach allows us to handle both positive and negative numbers in the array.

5. Write clean and readable code:

python

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

    max_product = nums[0]
    min_product = nums[0]
    result = nums[0]

    for i in range(1, len(nums)):
        if nums[i] < 0:
            max_product, min_product = min_product, max_product

        max_product = max(nums[i], max_product * nums[i])
        min_product = min(nums[i], min_product * nums[i])

        result = max(result, max_product)

    return result

6. Test your code:

python

# Test case 1
nums1 = [2, 3, -2, 4]
# The contiguous subarray [2, 3] has the maximum product of 6.
print(maxProduct(nums1))  # Expected output: 6

# Test case 2
nums2 = [-2, 0, -1]
# The contiguous subarray [-2] has the maximum product of -2.
print(maxProduct(nums2))  # Expected output: -2

# Test case 3
nums3 = [-2, 3, -4]
# The contiguous subarray [3, -4] has the maximum product of 12.
print(maxProduct(nums3))  # Expected output: 12

# Add more test cases to validate the solution

7. Optimize if necessary: The provided solution is already efficient with a time complexity of O(N), where N is the length of the input array. The space complexity is O(1) since we are using a constant amount of extra space. There is no further optimization possible for this problem.

8. Handle error cases: The code assumes that the input is a valid list of integers. If the input is empty, the code will return 0 as the maximum product.

9. Discuss complexity analysis: The time complexity of the solution is O(N), where N is the length of the input array. We iterate through the array once. The space complexity is O(1) since we are using a constant amount of extra space to store variables. The solution has an optimal time and space complexity given the problem requirements.