Theorem: The proposed Sudoku solving algorithm, which utilizes a 45-grid encoding and a 2-SAT solver to iteratively identify and eliminate invalid '1' placements, has a polynomial-time complexity. Proof: * Sudoku Encoding: * The Sudoku puzzle is encoded using 45 grids, where each grid corresponds to a specific number (1-9) and a specific position within a 3x3 subgrid. * Each grid contains '1's representing the presence of that number in the corresponding cells and '2's representing its absence. * Each grid must have exactly two '1's, ensuring the Sudoku constraints are satisfied. * Algorithm Description: a) Initialization: All possible '1' placements across all 45 grids are considered. b) Iteration: i. 2-SAT Solver: For each grid, a 2-SAT instance is constructed based on the current '1' placements and the Sudoku constraints (row, column, and subgrid). The 2-SAT solver is used to identify at least one invalid '1' placement within that grid. ii. Elimination: The identified invalid '1' placement is marked, permanently eliminating that possibility. iii. Constraint Propagation: The information from the invalid placement is propagated to other grids, potentially identifying more invalid placements. c) Termination: The algorithm terminates when a valid solution is found, which occurs when each grid contains exactly two '1's that satisfy all Sudoku constraints. * Complexity Analysis: * Constant Grid Size: Each grid has a constant size (9 cells). * Guaranteed Invalid Placement: In each iteration, at least one invalid '1' placement is guaranteed to be found in one of the grids (as long as the puzzle is solvable). * Limited Iterations: The total number of iterations is limited by the total number of possible '1' placements across all grids (45 grids * 9 cells/grid = 405). * Polynomial Time per Iteration: Each iteration involves: * Constructing a 2-SAT instance, which takes polynomial time. * Running the 2-SAT solver, which also takes polynomial time. * Marking the invalid placement and propagating constraints, which takes constant time. * Overall Complexity: * Since the number of iterations is limited by a constant (405), and each iteration takes polynomial time, the overall complexity of the algorithm is polynomial. Conclusion: The proposed Sudoku solving algorithm, which utilizes a 45-grid encoding and a 2-SAT solver to iteratively identify and eliminate invalid placements, has been proven to have a polynomial-time complexity. This result demonstrates that Sudoku, when encoded and solved using this method, can be solved efficiently. Further Considerations: * Optimizations: The algorithm can be further optimized by using heuristics to prioritize certain '1' placements or by employing more sophisticated constraint propagation techniques. * Generalization: While this proof focuses on Sudoku, the underlying principle of encoding a problem and using a constraint solver to iteratively eliminate invalid possibilities might be applicable to other NP-complete problems. This warrants further investigation and could have implications for the P vs. NP question.
Oh i see you check every possibilty. I made code that instead of checking every possibilty at every junction it checks for pairs and duos. Like two spaces in a row have 5/9 5/9 that means 5 and 9 cant go anywhere else.
Theorem: The proposed Sudoku solving algorithm, which utilizes a 45-grid encoding and a 2-SAT solver to iteratively identify and eliminate invalid '1' placements, has a polynomial-time complexity.
Proof:
* Sudoku Encoding:
* The Sudoku puzzle is encoded using 45 grids, where each grid corresponds to a specific number (1-9) and a specific position within a 3x3 subgrid.
* Each grid contains '1's representing the presence of that number in the corresponding cells and '2's representing its absence.
* Each grid must have exactly two '1's, ensuring the Sudoku constraints are satisfied.
* Algorithm Description:
a) Initialization: All possible '1' placements across all 45 grids are considered.
b) Iteration:
i. 2-SAT Solver: For each grid, a 2-SAT instance is constructed based on the current '1' placements and the Sudoku constraints (row, column, and subgrid). The 2-SAT solver is used to identify at least one invalid '1' placement within that grid.
ii. Elimination: The identified invalid '1' placement is marked, permanently eliminating that possibility.
iii. Constraint Propagation: The information from the invalid placement is propagated to other grids, potentially identifying more invalid placements.
c) Termination: The algorithm terminates when a valid solution is found, which occurs when each grid contains exactly two '1's that satisfy all Sudoku constraints.
* Complexity Analysis:
* Constant Grid Size: Each grid has a constant size (9 cells).
* Guaranteed Invalid Placement: In each iteration, at least one invalid '1' placement is guaranteed to be found in one of the grids (as long as the puzzle is solvable).
* Limited Iterations: The total number of iterations is limited by the total number of possible '1' placements across all grids (45 grids * 9 cells/grid = 405).
* Polynomial Time per Iteration: Each iteration involves:
* Constructing a 2-SAT instance, which takes polynomial time.
* Running the 2-SAT solver, which also takes polynomial time.
* Marking the invalid placement and propagating constraints, which takes constant time.
* Overall Complexity:
* Since the number of iterations is limited by a constant (405), and each iteration takes polynomial time, the overall complexity of the algorithm is polynomial.
Conclusion:
The proposed Sudoku solving algorithm, which utilizes a 45-grid encoding and a 2-SAT solver to iteratively identify and eliminate invalid placements, has been proven to have a polynomial-time complexity. This result demonstrates that Sudoku, when encoded and solved using this method, can be solved efficiently.
Further Considerations:
* Optimizations: The algorithm can be further optimized by using heuristics to prioritize certain '1' placements or by employing more sophisticated constraint propagation techniques.
* Generalization: While this proof focuses on Sudoku, the underlying principle of encoding a problem and using a constraint solver to iteratively eliminate invalid possibilities might be applicable to other NP-complete problems. This warrants further investigation and could have implications for the P vs. NP question.
Thanks it really cleared my backtracking concept
Glad it helped!
Can u make a video on graph theory like graphs algorithms and stuff ur explanation is very good
@@catcanhack Future upload :)
Nicely explained!
Hello from another UBC student!
hey whats up!?
hi, is this consider a depth first search also?
Hello , i'm from Tunisia , Good Job man
good explanation !
Hello from Indonesia, thanks for the explanation
I am liking your content....... remember my name Mudassir
thanks!
Oh i see you check every possibilty. I made code that instead of checking every possibilty at every junction it checks for pairs and duos. Like two spaces in a row have 5/9 5/9 that means 5 and 9 cant go anywhere else.