Write a solver for sudoku puzzles using a constraint satisfaction approach based on constraint propagation and backtracking, and one based on Relaxation Labeling. compare the approaches, their strengths and weaknesses.

A sudoku puzzle is composed of a square 9x9 board divided into 3 rows and 3 columns of smaller 3x3 boxes. The goal is to fill the board with digits from 1 to 9 such that

each number appears only once for each row column and 3x3 box; each row, column, and 3x3 box should containg all 9 digits. The solver should take as input a matrix withwhere empty squares are represented by a standars symbol (e.g., ".", "_", or "0"), while known square should be represented by the corresponding digit (1,...,9). For example: 37. 5.. ..6 ... 36. .12 ... .91 75. ... 154 .7. ..3 .7. 6.. .5. 638 ... .64 98. ... 59. .26 ... 2.. ..5 .64 Hints for Constraint Propagation and Backtracking:

Each cell should be a variable that can take values in the domain (1,...,9). The two types of constraints in the definition form as many types of constraints: Direct constraints impose that no two equal digits appear for each row, column, and box; Indirect constrains impose that each digit must appear in each row, column, and box. You can think of other types of (pairwise) constraints to further improve the constraint propagation phase. Note: most puzzles found in the magazines can be solved with only the constraint propagation step. Hints for Relaxation Labeling:

Each cell should be an object, the values between 1 and 9 labels. The compatibility rij(λ,μ) should be 1 if the assignments satisfy direct constraints, 0 otherwise.