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Optimized Recursive Knapsack with Memoization #12550

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Summary
This PR improves the recursive implementation of the Knapsack Problem by adding memoization, reducing the time complexity from O(2ⁿ) to O(n × max_weight).

Changes Made:
✅ Added Memoization using a dictionary to store previously computed results.
✅ Optimized Recursion by avoiding redundant calculations.
✅ Improved Function Readability with clear parameter names and better docstrings.
✅ Maintained the Same Functionality but with better efficiency.
✅ Retained Doctests to ensure correctness.

Performance Improvement:
The previous approach was exponential O(2ⁿ) and inefficient for large inputs.
The new approach is O(n × max_weight), making it significantly faster.
Test Cases:
The function has been tested with sample cases:

assert knapsack([1, 2, 4, 5], [5, 4, 8, 6], 4, 5, 0) == 13
assert knapsack([3, 4, 5], [10, 9, 8], 3, 25, 0) == 27
Issue Reference:
Closes [#IssueNumber] (if applicable).

Additional Notes:
Looking forward to feedback! 🚀

adityakrmishra and others added 2 commits January 31, 2025 16:20
- Added memoization to avoid redundant calculations
- Improved function readability with better parameter naming
- Ensured correct handling of empty inputs
- Reduced time complexity from O(2ⁿ) to O(n * max_weight)
@algorithms-keeper algorithms-keeper bot added the tests are failing Do not merge until tests pass label Jan 31, 2025
@algorithms-keeper algorithms-keeper bot added the awaiting reviews This PR is ready to be reviewed label Jan 31, 2025
@algorithms-keeper algorithms-keeper bot removed the tests are failing Do not merge until tests pass label Jan 31, 2025
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