![Algorithm: Uses Backtracking
Index
20
function RECURSIVE - BACKTRACKING ( assignment, csp) returns a solution, or failure
if assignment is complete then return assignment
Var = SELECT-UNASSIGNED-VARIABLE (VARIABLES[csp], assignment, csp)
for each value in ORDER-DOMAIN-VALUES (var, assignment, csp) do
if value is consistent with assignment according to CONSTRAINTS [csp] then
add {var = value} to assignment
Result=RECURSIVE -BACKTRACKING(assignment, csp)
if result != failure then return result
remove {var = value} from assignment
return failure
Reference :[1]](https://image.slidesharecdn.com/sudokusolver-200610034347/85/Sudoku-solver-20-320.jpg)
Sudoku solver
- 1. 1 Department of Computer Science Gujarat University Ahmedabad - 380009 Sudoku solver Presented By: Fadia Pankti Project Guide: Dr. Nidhi Arora M.Sc.(AI & ML) – I Project – I December 2019
- 2. Table Of Contents 2 1.1 Objectives 1.2 Scope 1.3 What is Sudoku? 2.Problem Definition 3.Sudoku as constraint satisfaction problem 4. Solving methods 4.1 Brute force search methods 4.2 Backtracking 4.3 Forward checking 4.4 Constraint propagation 4.1 Brute force search methods 1. Introduction 5. Approach 6. Hardware & software requirements 7. Implementation 8. Testing & results 10. References 9. Conclusion & Future directions
- 3. Objectives a) To solve Sudoku problem considering it as Constraint Satisfaction Problem b) To get to the solution with minimum backtracking c) To reach the goal state in least possible time Index 3
- 4. Scope 4 Recent Scope: • Initially the board is given statically with different difficulty levels. • Remove cells to create a valid Sudoku games according to the difficulty level that is provided. Index
- 5. What is Sudoku ? • Sudoku is the Japanese abbreviation of a phrase meaning the digits must remain single, where Su means number, doku which translates as single or bachelor. Variants • Though the 9 × 9 grid with 3 × 3 inner boxes is by far the most common Sudoku game, many other variants exist. Index 5
- 6. Problem Definition • A Sudoku puzzle is defined as a logic-based, number- placement puzzle. The objective is to fill a 9×9 grid with digits in such a way that each column, each row, and each of the nine 3×3 grids that make up the larger 9×9 grid contains all of the digits from 1 to 9. Rule • No number may appear more than once in any row across, any column down, or within any small 9-box square or omitted.(although, the can be repeated along the diagonals). Index 6
- 7. Sudoku As Constraint Satisfaction Problem • A constraint satisfaction problem (CSP) is a problem that requires its solution within some limitations/conditions also known as constraints. Sudoku is one them. It consists of the following: • A finite set of variables which stores the solution. • A set of discrete values known as domain from which the solution is picked. • A finite set of constraints. Index 7
- 8. Continue… The idea: • represent states as a vector of feature values. We have, k-features (or variables) • Each feature takes a value. Domain of possible values for the variables: • height = {short, average, tall}, • weight = {light, average, heavy}. • In CSPs, the problem is to search for a set of values for the features (variables) so that the values satisfy some conditions (constraints). Index 8
- 9. Example Index 9 • Variables - Each empty cells on grid • Domain : - Any Digits from1 to 9. • Constraints : - Same digit can't appear twice (or more) in the same row. - Same digit can't appear twice (or more) in the same column. - Same digit can't appear twice (or more) in the same square.
- 10. Goal Index 10 • Solutions are complete and consistent assignments • E.g. every cell is filled with domain(1 to 9) and within the constraints.
- 11. How to solve Sudoku ? I. Brute Force Search Method • BFS is one of the most general and basic techniques for searching for possible solutions to a problem. The idea is that we want to generate every possible move within the game and then test to see whether it solves our problem. • the BFS algorithm visits the empty cells in some order, filling in digits sequentially (from 1 to 9). Index 11
- 13. II. Backtracking • HOLD ON; what if a cell is discovered where none of the 9 digits is allowed?! … • Well, then there’s a quick and easy solution to this call Backtracking. • We should go back and change something in one of the previous cells and see if that works. • Thus the algorithm will leave this unsolvable cell blank for now and move back i.e backtrack to the previous cell. • But it is exhaustive search and will take lot of time to reach to solution. Index 13
- 15. III. Forward checking • The first improvement on backtracking search is forward checking. • Keep track of remaining legal values for unassigned variables. • It helps reducing the domain of the free variables that appears. • For example, in Sudoku, each time a value is assigned to a variable, the value is removed from the domain of the free variables that are either in the same line, in the same column or in the same square as the assigned variable. Index 15
- 16. Example Using Forward Checking Index 16
- 17. IV. Constraint Propagation Index 17 • place a four in the shaded box. It doesn't immediately conflict with any other locations and after placing it all of the squares still have possible values. • Next look at the square right below the shaded one, it must be a two. • Filling in a two there, however, leaves the lower left square with no possible value. • it only takes an extra two assignments to realize that the four was wrong, but what if the two isn't the next one.
- 18. Continue… • If the search is moving from left to right across the rows, it will assign the three empty squares to the right of the shaded one first. • Depending on the possible values that these squares have, there may be up to three layers of branching before reaching the conflict. • Each branch must now fail separately before the search realizes that the four was a bad choice. • The result is an increment in the time needed for the search to realize that the four was a bad choice. • The solution is to assign values that only have one possible choice immediately. Index 18
- 19. Continue… • In Constraint propagation we use the constraint to decrease the number of legal values for a variable, which in turn could reduce the legal values for another variable, and so on • In the example above after the four is assigned, constraint propagation realize that the space below the four must be two. It then notices that the lower left corner has no possible value and fails, returning to the search, which chooses another value for the shaded square. Index 19
- 20. Algorithm: Uses Backtracking Index 20 function RECURSIVE - BACKTRACKING ( assignment, csp) returns a solution, or failure if assignment is complete then return assignment Var = SELECT-UNASSIGNED-VARIABLE (VARIABLES[csp], assignment, csp) for each value in ORDER-DOMAIN-VALUES (var, assignment, csp) do if value is consistent with assignment according to CONSTRAINTS [csp] then add {var = value} to assignment Result=RECURSIVE -BACKTRACKING(assignment, csp) if result != failure then return result remove {var = value} from assignment return failure Reference :[1]
- 21. Hardware & Software configuration Index 21 Processor RAM Hard-disk Pentium or higher 512 MB or higher 2GB or more Technology Front End Python (version 3.7) Pycharm Hardware Configuration Software Configuration
- 22. Index 22 Result of brute force search method for easy puzzle Initial state Goal state
- 23. Index 23 Result Of recursive backtracking method for easy puzzle Initial state Goal state
- 24. Index 24 Result Of brute force search method for hard puzzle Initial state Goal state
- 25. Index 25 Result Of recursive backtracking method for hard puzzle Initial state Goal state
- 26. 26 Conclusion & Future Directions Conclusion • This study has shown that the Recursive backtracking algorithm is a feasible method to solve any Sudoku puzzles. The algorithm is also an appropriate method to find a solution faster and more efficient compared to the brute force algorithm. • The proposed algorithm is able to solve such puzzles with any level of difficulties in a short period of time. The performance of the Recursive backtracking algorithm is better than the brute force algorithm with respect to the computing time to solve any puzzle. • The brute force algorithm is guaranteed to have one solution. However, it does not adopt intelligent strategies to solve the puzzles. • This algorithm checks all possible solutions to the puzzle until a valid solution is found which is a time-consuming procedure resulting an inefficient solver.
- 27. 27 Future Directions Implementing GUI to show generated puzzle and solved puzzle board. Creating generator which can generate different puzzle, providing different levels like hard, easy, medium. Continue…
- 28. References 1. http://aima.cs.berkeley.edu/2nd-ed/newchap05.pdf 2. https://www.slideshare.net/TARUNKUMAR362/project-report-on-sudoku 3. https://www.ics.uci.edu/~yug10/uci/cs271/project/report/report.pdf 4. https://www.slideshare.net/yara2308/sudoku-38189890 5. https://steven.codes/blog/constraint-satisfaction-with-sudoku/ 6. https://www.codeproject.com/Articles/34403/Sudoku-as-a-CSP 7. https://norvig.com/sudoku.html 8. https://www.slideshare.net/sohamkoolkarni/compsci271-a- fall2016sudokusolverprojectreport Index 28 References
- 29. Thank you 29