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Clarifying Questions

1. Input Constraints: Can the input list of heights be empty?
2. Output Format: Should the function return just the maximum area, or both the maximum area and the indices that form this maximum area?
3. Input Range: Are there any constraints on the size of the input list?

Strategy

The problem is about finding two lines which together with the x-axis forms a container such that the container contains the most water.

Naive Solution

A basic brute-force approach would involve checking the area for every possible pair of lines. This approach would be:

• Time Complexity: (O(n^2))

Optimized Solution

A more efficient approach is the two-pointer technique:

1. Start with two pointers, one at the beginning (left) and one at the end (right) of the list.
2. Calculate the area with these two pointers.
3. Move the pointer which is at the shorter line towards the other pointer, as increasing the height while possibly decreasing the width can potentially give a larger area.
4. Repeat the steps until the two pointers meet.

This approach ensures that we check every possible combination in a linear manner:

• Time Complexity: (O(n))

Code

Here’s the implementation of the two-pointer technique in Python:

def maxArea(height):
    left, right = 0, len(height) - 1
    max_area = 0
    
    while left < right:
        # Calculate the area
        width = right - left
        current_height = min(height[left], height[right])
        current_area = width * current_height
        max_area = max(max_area, current_area)
        
        # Move the pointer at the shorter line
        if height[left] < height[right]:
            left += 1
        else:
            right -= 1
    
    return max_area

Test Cases

Let’s consider a few test cases to validate our approach:

1. maxArea([1,8,6,2,5,4,8,3,7]) should return 49.
2. maxArea([1,1]) should return 1.
3. maxArea([4,3,2,1,4]) should return 16.
4. maxArea([1,2,1]) should return 2.

Detailed Time Complexity Analysis

• Initialization and Pointers Setup: Constant time, (O(1)).
• Single Pass Loop: Linear time, (O(n)), where (n) is the number of elements in the input list.
• Calculation and Comparison: Each step inside the loop involves constant time operations, (O(1)).

Thus, the overall time complexity is (O(n)), and the space complexity is (O(1)) as we are using only a few extra variables.

Feel free to ask any further questions or let me know if there’s any other aspect you’d like to explore!

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