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Container With Most Water: Study Note

1. Problem Summary

  • Given an array of heights, find two lines that form a container holding the most water
  • Problem type: Array, Two Pointers, Greedy
  • Difficulty level: Medium
  • Key constraints:
    • Input size: up to 10⁵ elements
    • Element values: up to 10⁴
    • Need O(n) time solution due to input size

2. Approaches & Strategies

  • Brute Force: Check all pairs of lines
    • Time: O(n²), Space: O(1)
    • Inefficient for large inputs
  • Two Pointer Technique (Optimal):
    • Start with widest container (left=0, right=n-1)
    • Move pointer at shorter line inward
    • Justification: Moving taller line inward can only decrease area
    • Time: O(n), Space: O(1)
  • Key Insight: Area = min(height[left], height[right]) × (right - left)

3. My Solution & Code

  • Used optimal two-pointer approach
  • Strategy: Start with maximum width, greedily move shorter pointer
  • More efficient than brute force by eliminating impossible candidates
from typing import List

class Solution:
    """
    Classic optimal solution:
    start with the outermost pair and move the pointer
    at the shorter line inward.  The area can only
    get smaller when the width shrinks, so we can safely
    discard the shorter line.
    """
    def maxArea(self, height: List[int]) -> int:
        left, right = 0, len(height) - 1
        max_water = 0

        while left < right:
            width = right - left
            h = min(height[left], height[right])
            max_water = max(max_water, width * h)

            # Move the pointer that is at the shorter line
            if height[left] < height[right]:
                left += 1
            else:
                right -= 1

        return max_water

4. Time & Space Complexity

  • Final Solution: O(n) time, O(1) space
  • Brute Force Alternative: O(n²) time, O(1) space
  • Trade-off: Linear time vs quadratic time, same space
  • Justification: Each element visited at most once

5. Mistakes & Insights

  • Early submissions (2022): Same correct logic but slower execution (700-1600ms)
  • Recent submission (2025): Same logic but faster (91ms, 78% percentile)
  • Improvement: Code structure and variable naming became cleaner
  • Key Insight: Moving the shorter line is safe because:
    • Width always decreases
    • Height cannot exceed shorter line
    • Keeping shorter line cannot improve area
  • Debugging history: Initially questioned brute force vs optimal approach
  • Learning: Recognized pattern of two-pointer optimization for array problems

6. What's Next

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