Backtracking | Set 1 (The Knight’s tour problem)

Backtracking is an algorithmic paradigm that tries different solutions until finds a solution that “works”. Problems which are typically solved using backtracking technique have following property in common. These problems can only be solved by trying every possible configuration and each configuration is tried only once. A Naive solution for these problems is to try all configurations and output a configuration that follows given problem constraints. Backtracking works in incremental way and is an optimization over the Naive solution where all possible configurations are generated and tried.

For example, consider the following Knight’s Tour problem.
The knight is placed on the first block of an empty board and, moving according to the rules of chess, must visit each square exactly once.

Path followed by Knight to cover all the cells

Following is chessboard with 8 x 8 cells. Numbers in cells indicate move number of Knight.
knight-tour-problem
Let us first discuss the Naive algorithm for this problem and then the Backtracking algorithm.

Naive Algorithm for Knight’s tour
The Naive Algorithm is to generate all tours one by one and check if the generated tour satisfies the constraints.

while there are untried tours
{ 
   generate the next tour 
   if this tour covers all squares 
   { 
      print this path;
   }
}

Backtracking works in an incremental way to attack problems. Typically, we start from an empty solution vector and one by one add items (Meaning of item varies from problem to problem. In context of Knight’s tour problem, an item is a Knight’s move). When we add an item, we check if adding the current item violates the problem constraint, if it does then we remove the item and try other alternatives. If none of the alternatives work out then we go to previous stage and remove the item added in the previous stage. If we reach the initial stage back then we say that no solution exists. If adding an item doesn’t violate constraints then we recursively add items one by one. If the solution vector becomes complete then we print the solution.

Backtracking Algorithm for Knight’s tour
Following is the Backtracking algorithm for Knight’s tour problem.

If all squares are visited 
    print the solution
Else
   a) Add one of the next moves to solution vector and recursively 
   check if this move leads to a solution. (A Knight can make maximum 
   eight moves. We choose one of the 8 moves in this step).
   b) If the move chosen in the above step doesn't lead to a solution
   then remove this move from the solution vector and try other 
   alternative moves.
   c) If none of the alternatives work then return false (Returning false 
   will remove the previously added item in recursion and if false is 
   returned by the initial call of recursion then "no solution exists" )

Following are implementations for Knight’s tour problem. It prints one of the possible solutions in 2D matrix form. Basically, the output is a 2D 8*8 matrix with numbers from 0 to 63 and these numbers show steps made by Knight.

C

// C program for Knight Tour problem
#include<stdio.h>
#define N 8

int solveKTUtil(int x, int y, int movei, int sol[N][N],
                int xMove[],  int yMove[]);

/* A utility function to check if i,j are valid indexes
   for N*N chessboard */
bool isSafe(int x, int y, int sol[N][N])
{
    return ( x >= 0 && x < N && y >= 0 &&
             y < N && sol[x][y] == -1);
}

/* A utility function to print solution matrix sol[N][N] */
void printSolution(int sol[N][N])
{
    for (int x = 0; x < N; x++)
    {
        for (int y = 0; y < N; y++)
            printf(" %2d ", sol[x][y]);
        printf("\n");
    }
}

/* This function solves the Knight Tour problem using
   Backtracking.  This function mainly uses solveKTUtil()
   to solve the problem. It returns false if no complete
   tour is possible, otherwise return true and prints the
   tour.
   Please note that there may be more than one solutions,
   this function prints one of the feasible solutions.  */
bool solveKT()
{
    int sol[N][N];

    /* Initialization of solution matrix */
    for (int x = 0; x < N; x++)
        for (int y = 0; y < N; y++)
            sol[x][y] = -1;

    /* xMove[] and yMove[] define next move of Knight.
       xMove[] is for next value of x coordinate
       yMove[] is for next value of y coordinate */
    int xMove[8] = {  2, 1, -1, -2, -2, -1,  1,  2 };
    int yMove[8] = {  1, 2,  2,  1, -1, -2, -2, -1 };

    // Since the Knight is initially at the first block
    sol[0][0]  = 0;

    /* Start from 0,0 and explore all tours using
       solveKTUtil() */
    if (solveKTUtil(0, 0, 1, sol, xMove, yMove) == false)
    {
        printf("Solution does not exist");
        return false;
    }
    else
        printSolution(sol);

    return true;
}

/* A recursive utility function to solve Knight Tour
   problem */
int solveKTUtil(int x, int y, int movei, int sol[N][N],
                int xMove[N], int yMove[N])
{
   int k, next_x, next_y;
   if (movei == N*N)
       return true;

   /* Try all next moves from the current coordinate x, y */
   for (k = 0; k < 8; k++)
   {
       next_x = x + xMove[k];
       next_y = y + yMove[k];
       if (isSafe(next_x, next_y, sol))
       {
         sol[next_x][next_y] = movei;
         if (solveKTUtil(next_x, next_y, movei+1, sol,
                         xMove, yMove) == true)
             return true;
         else
             sol[next_x][next_y] = -1;// backtracking
       }
   }

   return false;
}

/* Driver program to test above functions */
int main()
{
    solveKT();
    return 0;
}

Java

// Java program for Knight Tour problem
class KnightTour {
    static int N = 8;

    /* A utility function to check if i,j are
       valid indexes for N*N chessboard */
    static boolean isSafe(int x, int y, int sol[][]) {
        return (x >= 0 && x < N && y >= 0 &&
                y < N && sol[x][y] == -1);
    }

    /* A utility function to print solution
       matrix sol[N][N] */
    static void printSolution(int sol[][]) {
        for (int x = 0; x < N; x++) {
            for (int y = 0; y < N; y++)
                System.out.print(sol[x][y] + " ");
            System.out.println();
        }
    }

    /* This function solves the Knight Tour problem
       using Backtracking.  This  function mainly
       uses solveKTUtil() to solve the problem. It
       returns false if no complete tour is possible,
       otherwise return true and prints the tour.
       Please note that there may be more than one
       solutions, this function prints one of the
       feasible solutions.  */
    static boolean solveKT() {
        int sol[][] = new int[8][8];

        /* Initialization of solution matrix */
        for (int x = 0; x < N; x++)
            for (int y = 0; y < N; y++)
                sol[x][y] = -1;

       /* xMove[] and yMove[] define next move of Knight.
          xMove[] is for next value of x coordinate
          yMove[] is for next value of y coordinate */
        int xMove[] = {2, 1, -1, -2, -2, -1, 1, 2};
        int yMove[] = {1, 2, 2, 1, -1, -2, -2, -1};

        // Since the Knight is initially at the first block
        sol[0][0] = 0;

        /* Start from 0,0 and explore all tours using
           solveKTUtil() */
        if (!solveKTUtil(0, 0, 1, sol, xMove, yMove)) {
            System.out.println("Solution does not exist");
            return false;
        } else
            printSolution(sol);

        return true;
    }

    /* A recursive utility function to solve Knight
       Tour problem */
    static boolean solveKTUtil(int x, int y, int movei,
                               int sol[][], int xMove[],
                               int yMove[]) {
        int k, next_x, next_y;
        if (movei == N * N)
            return true;

        /* Try all next moves from the current coordinate
            x, y */
        for (k = 0; k < 8; k++) {
            next_x = x + xMove[k];
            next_y = y + yMove[k];
            if (isSafe(next_x, next_y, sol)) {
                sol[next_x][next_y] = movei;
                if (solveKTUtil(next_x, next_y, movei + 1,
                                sol, xMove, yMove))
                    return true;
                else
                    sol[next_x][next_y] = -1;// backtracking
            }
        }

        return false;
    }

    /* Driver program to test above functions */
    public static void main(String args[]) {
        solveKT();
    }
}
// This code is contributed by Abhishek Shankhadhar


Output:
  0  59  38  33  30  17   8  63
 37  34  31  60   9  62  29  16
 58   1  36  39  32  27  18   7
 35  48  41  26  61  10  15  28
 42  57   2  49  40  23   6  19
 47  50  45  54  25  20  11  14
 56  43  52   3  22  13  24   5
 51  46  55  44  53   4  21  12

Note that Backtracking is not the best solution for the Knight’s tour problem. See below article for other better solutions. The purpose of this post is to explain Backtracking with an example.
Warnsdorff’s algorithm for Knight’s tour problem

References:
http://see.stanford.edu/materials/icspacs106b/H19-RecBacktrackExamples.pdf
http://www.cis.upenn.edu/~matuszek/cit594-2009/Lectures/35-backtracking.ppt
http://mathworld.wolfram.com/KnightsTour.html
http://en.wikipedia.org/wiki/Knight%27s_tour

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