Algorithms and Data Structures

Part 2:

Write a class called SudokuBoard. This class provides the data structure and the functionality of a sudoku board.

An instance of this class creates a sudoku board as a linear array, called boardCells, whose size is computed using the two parameters it receives representing the boxWidth and the boxHeight of a Sudoku board.

Write methods for each of the following tasks. Each method takes a cell number in boardCells as a parameter, and returns

a) the row to which the cell belongs in the 2D form of the Sudoku board;

b) the column to which the cell belongs in the 2D form of the Sudoku board;

c) the value in the cell; and

d) the box in which the cell lies in the 2D form of the Sudoku board.

Write get methods for each of the variables boxWidth, boxHeight, boardSize (i.e. the number of rows or columns in the Sudoku board), and numberOfCells in the Sudoku board.

Write a toString() to print the sudoku board in 2D format.

Write a set method to set the value of a given cell in boardCells to a given integer.

Recall that the (i, j)th cell in an n X n (read: n by n) array corresponds to the cell at index i * n + j in a linear array.

Reducing Sudoku to SAT The constraints of a Sudoku puzzle can be modeled using a boolean formula in conjumctive normal form (cn). There are several ways to translate or encode a Sudoku problem into an equivalent SAT problem. The remaining of this section presents one of these encodings. The typical Sudoku puzzle that one finds in newspapers or pastime books consists ofa square grid of size 9, containing preset values in some of its 81 cells. The grid is subdivided into 9 3x3 boxes. The objective of the puzzle is to fill each blank cell so that each column, each row, and each box contains all the digits from 1 to 9 Create 729 (9x9x9) different bgoleanvariables where j j and k take on values from 1 to 9. Intuitively, the boolean variable uk is set to true if and only if the cell at row j and column j takes the value k. Recall the constraints on a Sudoku puzzle solution: 1. Each row must have all the digits 1 through 9 2. Each column must have all the digits 1 through 9, 3. Each 3x3 box must have all the digits 1 through 9, 4. Each cell must have exactly one value in the range 1 through !9 Constraints 1, 2, and 3 are similar except that each refers to a different subset of cells. We now present a SAT encoding for Constraint 1. Constraint, when applied to a rowi can be viewed as a conjunct (and) of the following nine constraints: 1.1. The number 1 is the value of exactly one cell in rowi 1.2The number 2 is the value of exactly one cell in row t For this project you will develop a SudokuToSAT reducer that receives an input puzzle and writes out a SAT instance in thecof format. Your reducer will: 1.9. The number 9 is the value of exactly one cell in row i 1. read an input Sudoku instance from a file, We will only address Constraint 1.1 since the other eight constraints can be translated in analogous manner. Constraint 1.1, in tun, can be broken down as a conjunct of two 2. translate the input Sudoku instance into an equivalent SAT instance Here are the details. 1.1.A At least one cell of row has the value 1 1.1.B. At most one cell of rowhas the value 1 Step 1 Your program will read from the command line the name of a file, and then read from this file the input Sudoku puzzle (instance). An input file for a Sudoku instance contains: Constraint 1.1A, for row i, yields the propositional clause: . Zero or more comment lines at the beginning of the file, each starting with the character 'c', followed by Constraint 1.1 B. for row yields a subomula comprising 36 clauses of the form * a line with "3 3" that indicates the dimensions ofeach box in the puzzle, followed b a sequence of 31 integers in the range from 0 to 9. These integers are the values in Sudoku grid in row-major order, with a 0 indicating a cell that does not have a preset value. ((notuui) or (not vi^i) combined together with the and operator, for all distinet combinations of j and k, j and k in the range 1 through 9. Can you think of a mathematical formula to find the number of such distinct combinations? Step 2 Your program will then translate the input Sudoku instance into an equivalent SAT instance. A total of 37 (ie. 1 +36 clauses from the translation of Constraint 1.1 for row ivalue1 A total of 37 * %333 clauses from Constraint l for rowi A total of333 9-2997 clauses from Constraint Translating a Sudoku instance into an equivalent SAT instance Reducing Sudoku to SAT The constraints of a Sudoku puzzle can be modeled using a boolean formula in conjumctive normal form (cn). There are several ways to translate or encode a Sudoku problem into an equivalent SAT problem. The remaining of this section presents one of these encodings. The typical Sudoku puzzle that one finds in newspapers or pastime books consists ofa square grid of size 9, containing preset values in some of its 81 cells. The grid is subdivided into 9 3x3 boxes. The objective of the puzzle is to fill each blank cell so that each column, each row, and each box contains all the digits from 1 to 9 Create 729 (9x9x9) different bgoleanvariables where j j and k take on values from 1 to 9. Intuitively, the boolean variable uk is set to true if and only if the cell at row j and column j takes the value k. Recall the constraints on a Sudoku puzzle solution: 1. Each row must have all the digits 1 through 9 2. Each column must have all the digits 1 through 9, 3. Each 3x3 box must have all the digits 1 through 9, 4. Each cell must have exactly one value in the range 1 through !9 Constraints 1, 2, and 3 are similar except that each refers to a different subset of cells. We now present a SAT encoding for Constraint 1. Constraint, when applied to a rowi can be viewed as a conjunct (and) of the following nine constraints: 1.1. The number 1 is the value of exactly one cell in rowi 1.2The number 2 is the value of exactly one cell in row t For this project you will develop a SudokuToSAT reducer that receives an input puzzle and writes out a SAT instance in thecof format. Your reducer will: 1.9. The number 9 is the value of exactly one cell in row i 1. read an input Sudoku instance from a file, We will only address Constraint 1.1 since the other eight constraints can be translated in analogous manner. Constraint 1.1, in tun, can be broken down as a conjunct of two 2. translate the input Sudoku instance into an equivalent SAT instance Here are the details. 1.1.A At least one cell of row has the value 1 1.1.B. At most one cell of rowhas the value 1 Step 1 Your program will read from the command line the name of a file, and then read from this file the input Sudoku puzzle (instance). An input file for a Sudoku instance contains: Constraint 1.1A, for row i, yields the propositional clause: . Zero or more comment lines at the beginning of the file, each starting with the character 'c', followed by Constraint 1.1 B. for row yields a subomula comprising 36 clauses of the form * a line with "3 3" that indicates the dimensions ofeach box in the puzzle, followed b a sequence of 31 integers in the range from 0 to 9. These integers are the values in Sudoku grid in row-major order, with a 0 indicating a cell that does not have a preset value. ((notuui) or (not vi^i) combined together with the and operator, for all distinet combinations of j and k, j and k in the range 1 through 9. Can you think of a mathematical formula to find the number of such distinct combinations? Step 2 Your program will then translate the input Sudoku instance into an equivalent SAT instance. A total of 37 (ie. 1 +36 clauses from the translation of Constraint 1.1 for row ivalue1 A total of 37 * %333 clauses from Constraint l for rowi A total of333 9-2997 clauses from Constraint Translating a Sudoku instance into an equivalent SAT instance