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Sudoku Solver Using Backtracking
Sudoku
Backtrack
Backtracking

Lets today learn one concept and straight away implement it some real problem. The concept to learn is Backtracking.

This is not a new concept to us. We use this, follow this in our day to day life. This only proves that Computer Science and its concepts are very well related to real world only. Don't believe me?

Have an example: Suppose, You are at point A and want to reach at B. There are 3 roads say X, Y and Z to follow from A and you have no idea about which is the right path. What you will do? You will choose any one path out of three say X. But you eventually found that X does not lead to B. What will you do now? You will come back to A and follow path Y. Is it not right? Is it not what you do? And, for your information, you just followed the principle of Backtracking.

Backtracking is a technique to solve problems where multiple choices are there and we don't know the correct choice and hence we solve problem with trial and error i.e. trying each option until goal is achieved.

To have a look at general strategy used in Backtracking, have a look at this:

boolean Solve(choice = some_previous_choice):
{
    If this choice does not lead to some more other choices
    {
        If this choice is the desired goal state
            return true
        else
            return false
    }
    If this choice further has multiple options (choices) to be checked
    {
        For each further choice
        {
            check if this choice suceeds by calling Solve(current_choice)
            if Solve(this choice)
                return true
        }
    }
    return false
}

Don't see the above algorithm as word to word, but it just to understand what happens in backtracking. Now we will start to see how to solve sudoku and with this we will understand the Backtracking too. I will be talking to you with the comments in the code now. Lets understand it!

/*
    Program to solve the famous numbers placement puzzle 'Sudoku'.
    Uses the technique of backtracking.
    Gives you multiple solutions if there are possible.
*/

#include<iostream>
#include <cstdio>
using namespace std;
#include<math.h>

//This is the limit of size of sudoku board for now
//You can increase it, but at little cost on performance
#define MAX 25

//Function to get input from user as unsolved Sudoku Board
//Returns the size of board as per users choice
int takeInput(int sudoku[MAX][MAX])
{
    int size = 0;
    do 
    {
        cout<<"Choose the size of board from\nEnter 1 for 9 X 9\nEnter 2 for 16 X 16\nEnter 3 for 25 X 25\n";
        cout<<"Your Selection:\t";
        cin>>size;
        switch(size)
        {
            case 1:
                size = 9;
                break;
            case 2:
                size = 16;
                break;
            case 3:
                size = 25;
                break;
            default:
                cout<<"Invalid Selection"<<endl;
                break;
        }
    }while(!size); //until valid board size is selected

    cout<<"Enter the Initial "<<size<<" X "<<size<<" Sudoku Board:"<<endl;
    cout<<"Instruction: Enter 0 for blank"<<endl;
    for(int i=0;i<size;i++)
        for(int j=0;j<size; j++)
            cin>>sudoku[i][j];

    return size;
}

//Function to display solved Sudoku Board
void displaySolution(int sudoku[MAX][MAX], int size)
{
    for(int i=0;i<size;i++)
    {   
        for(int j=0;j<size; j++)
            printf("%4d", sudoku[i][j]);
        cout<<endl;
    }
    cout<<"\n\n*************************************************\n\n";
}

// Function to check if board is solved now i. e. it has no vacancies
// If the board is full then it is solved correctly
// This is true because, for each number we put in any cell,  we always check if all other vacant places can be 
// filled by some numbers such that board is always in valid state

bool isFull(int sudoku[MAX][MAX], int size)
{
    int i,j;    
    for(i=0;i<size;i++)
        for(j=0;j<size;j++)
            if(!sudoku[i][j])
                return false;
    return true;
}

//Function gives all different possible numbers those can be put on board such that board will be in valid 
// state, and this is done by checking numbers already appeared in row, column and block
//Function to find various possible values at position (r, c)
//Returns no. of possible values at that position
//and those values in array a[]
int findPossibleValues(int sudoku[MAX][MAX], int size, int a[], int r, int c)
{
    int n=0;
    int i,j;
    int s=(int)(sqrt(size));
    int b[MAX+1]={0};

    //Note no.s appeared in current row
    for(i=0; i<size; i++)
        b[sudoku[r][i]]=1;

    //Note no.s appeared in current column
    for(i=0; i<size; i++)
        b[sudoku[i][c]]=1;

    //Note no.s appeared in current block
    r=(r/s)*s, c=(c/s)*s;
    for(i=r; i<r+s; i++)
        for(j=c; j<c+s;j++)
            b[sudoku[i][j]]=1;

    //Fill array a[] with no.s unappeared in current row, column and block
    for(i=1;i<=size; i++)
        if(!b[i])
            a[n++]=i;

    return n;
}

//Function to solve the sudoku board
//Gives solved_flag as true if at least one solution is possible, false otherwise
//Gives multiple solutions with corresponding soultion no.
//Returns when user does not want more solutions

// This is our key function which solves the board
// Initially it checks if the board is full now
// If board is not full, then it finds the first position/cell which is vacant.
// Then it finds all possible values at that position
// Tries all these possible values in that cell
// For each trial/guess/choice, it tries to solve the updated board
// If no choice come out to be correct one for that vacant place, then definitely some previous choices are 
// wrong
// Hence it backtracks and tries to correct the previous choices
void SolveSudoku(int sudoku[MAX][MAX], int size, int &solution_num, bool &solved_flag, bool &enough)
{
    int i,j, a[MAX+1]={0}, n=0;

    if(enough) //true if user does not want more solutions
        return;

    if(isFull(sudoku, size))    //true if now sudoku board is solved completely
    {

        if(!solved_flag)
            cout<<"Sudoku Solved Successfully!"<<endl;
        solved_flag = 1;

        //show the solution
        cout<<"\n\nSolution no. "<<(solution_num++)<<endl;
        displaySolution(sudoku, size);

        //users' choice
        char more;
        cout<<"Do you want more solutions?"<<endl;
        cout<<"Press 1 for 'YES', anything OTHERWISE:\t";
        cin>>more;      
        cout<<"\n\n*************************************************\n\n";

        if(more != '1')
            enough = true;
        return;
    }

    //Find first vacant place/position
    int break_flag = 0;
    for(i=0;i<size;i++)
    {
        for(j=0;j<size;j++)
            if(!sudoku[i][j])
            {
                break_flag = 1;
                break;
            }
        if(break_flag)
            break;
    }

    //check possibilities at that vacant place
    n = findPossibleValues(sudoku, size, a, i, j);
    for(int l=0;l<n;l++)
    {
        //put value at vacant place
        sudoku[i][j]=a[l];
        //now solve the updated board
        SolveSudoku(sudoku, size, solution_num, solved_flag, enough);
    }

    **// Note this backtracking step, a very important step**
   // We come at this position, this step, this line when we have already checked all possible values at 
   // sudoku[i][j] and we couldn't find the solution
   // Put any value does not solves our board implies that we must have made wrong choice earlier
   // so we make this sudoku[i][j] again a vacant cell and try to correct our previous guesses/choices.
    sudoku[i][j]=0;
}

//Starting point
int main()
{
    //Input
    int sudoku[MAX][MAX] = {0}, size;
    size = takeInput(sudoku);

    int solution_num = 1;
    bool solved_flag = 0;
    bool enough = false;

    //Processing
    cout<<"Finding Solutions!\n\n"<<endl;
    SolveSudoku(sudoku, size, solution_num, solved_flag, enough);

    //Exit
    if(!solved_flag)
        cout<<"Invalid Board!"<<endl;

    cout<<"Exiting!\n\n"<<endl;

    return 0;
}

So, this is it for now. Guys go and implement it yourself, try to solve some similar problems involving backtracking, practice it.

Now once you have this code ready, give it a front end and make an App.

I hope you find this an interesting session. I would love to hear from you. Like it, share it, upvote it and also, don't forget to have look at my other notes here, you might like them too.

Thank you for now.

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