* Improved Equilibrium Index of an Array.
This is the modifications made to the original code:
1. Create Doctest Instructions: Python "doctest" can be executed by running the following command: python -m doctest -v equilibrium_index.py.
2. Deleted Argument {size}: Deleted the `size` argument because `len(arr)} allows the array's length to be determined inside the function, simplifying and improving the readability of the function signature.
3. Used {enumerate}: To improve code readability and indicate that we're working with element-index pairs, we iterated through the array using both elements and their indices using the `enumerate` function.
4. Optimized the Loop: To prevent pointless additions, the loop was improved by initializing {left_sum} with the value of the first element (arr[0]). Furthermore, since the beginning and last items (0 and size - 1) cannot be equilibrium indices, there is no need to check them, saving further computations.
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* Create Sudoku_Solver
Each of the digits 1-9 must occur exactly once in each row.
Each of the digits 1-9 must occur exactly once in each column.
Each of the digits 1-9 must occur exactly once in each of the 9 3x3 sub-boxes of the grid.
The '.' character indicates empty cells.
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* Rename Sudoku_Solver to sudoku_solver.py
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* Create sparse_table.py
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* Descriptive names for variables
* Fix ruff check error
* Update sparse_table.py
* Add comments, change variable names
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* Fix typo
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* added power_of_4
* deleted power_of_4
* added pairs_with_given_sum
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* updated the comment
* updated return hint
* updated type hints
* updated the variable
* updated annotation
* updated code
* updated code
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* added the problem link and used defaultdict
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* corrected import formatting
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* Update pairs_with_given_sum.py
* Update data_structures/arrays/pairs_with_given_sum.py
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* equilibrium index in an array
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* equilibrium index in an array
* equilibrium index in an array
* equilibrium index in an array removed type in docstring
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* add median of two sorted array
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* fix syntax
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* fix syntax
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* improve code
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* add documentation
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* Added minimum waiting time problem solution using greedy algorithm
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* ruff --fix
* Add type hints
* Added two more doc test
* Removed unnecessary comments
* updated type hints
* Updated the code as per the code review
* Added recursive algo to calculate product sum from an array
* Added recursive algo to calculate product sum from an array
* Update doc string
* Added doctest for product_sum function
* Updated the code and added more doctests
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* Added more test coverage for product_sum method
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* add prefix sum
* updating DIRECTORY.md
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* Add files via upload
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* Update data_structures/arrays/permutations.py
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