The Knight’s tour problem | Backtracking-1

Backtracking is an algorithmic paradigm that tries different solutions until finds a solution that “works”. Problems which are typically solved using backtracking technique have the 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 an 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. 

Problem Statement:
Given a N*N board with the Knight placed on the first block of an empty board. Moving according to the rules of chess knight must visit each square exactly once. Print the order of each the cell in which they are visited.

Example:

Input : 
N = 8
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

The path followed by Knight to cover all the cells
Following is a 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 the 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 works out then we go to the 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++

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// C++ program for Knight Tour problem
#include <bits/stdc++.h>
using namespace std;
 
#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 */
int 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++)
            cout << " " << setw(2) << sol[x][y] << " ";
        cout << endl;
    }
}
 
/* 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. */
int 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) == 0) {
        cout << "Solution does not exist";
        return 0;
    }
    else
        printSolution(sol);
 
    return 1;
}
 
/* 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 1;
 
    /* 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)
                == 1)
                return 1;
            else
                
               // backtracking
                sol[next_x][next_y] = -1;
        }
    }
    return 0;
}
 
// Driver Code
int main()
{
      // Function Call
    solveKT();
    return 0;
}
 
// This code is contributed by ShubhamCoder

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C

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// 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 */
int 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.  */
int 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) == 0) {
        printf("Solution does not exist");
        return 0;
    }
    else
        printSolution(sol);
 
    return 1;
}
 
/* 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 1;
 
    /* 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)
                == 1)
                return 1;
            else
                sol[next_x][next_y] = -1; // backtracking
        }
    }
 
    return 0;
}
 
/* Driver Code */
int main()
{
   
      // Function Call
    solveKT();
    return 0;
}

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Java

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// 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 Code */
    public static void main(String args[])
    {
        // Function Call
        solveKT();
    }
}
// This code is contributed by Abhishek Shankhadhar

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Python3

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# Python3 program to solve Knight Tour problem using Backtracking
 
# Chessboard Size
n = 8
 
 
def isSafe(x, y, board):
    '''
        A utility function to check if i,j are valid indexes
        for N*N chessboard
    '''
    if(x >= 0 and y >= 0 and x < n and y < n and board[x][y] == -1):
        return True
    return False
 
 
def printSolution(n, board):
    '''
        A utility function to print Chessboard matrix
    '''
    for i in range(n):
        for j in range(n):
            print(board[i][j], end=' ')
        print()
 
 
def solveKT(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.
    '''
 
    # Initialization of Board matrix
    board = [[-1 for i in range(n)]for i in range(n)]
 
    # move_x and move_y define next move of Knight.
    # move_x is for next value of x coordinate
    # move_y is for next value of y coordinate
    move_x = [2, 1, -1, -2, -2, -1, 1, 2]
    move_y = [1, 2, 2, 1, -1, -2, -2, -1]
 
    # Since the Knight is initially at the first block
    board[0][0] = 0
 
    # Step counter for knight's position
    pos = 1
 
    # Checking if solution exists or not
    if(not solveKTUtil(n, board, 0, 0, move_x, move_y, pos)):
        print("Solution does not exist")
    else:
        printSolution(n, board)
 
 
def solveKTUtil(n, board, curr_x, curr_y, move_x, move_y, pos):
    '''
        A recursive utility function to solve Knight Tour
        problem
    '''
 
    if(pos == n**2):
        return True
 
    # Try all next moves from the current coordinate x, y
    for i in range(8):
        new_x = curr_x + move_x[i]
        new_y = curr_y + move_y[i]
        if(isSafe(new_x, new_y, board)):
            board[new_x][new_y] = pos
            if(solveKTUtil(n, board, new_x, new_y, move_x, move_y, pos+1)):
                return True
 
            # Backtracking
            board[new_x][new_y] = -1
    return False
 
 
# Driver Code
if __name__ == "__main__":
     
    # Function Call
    solveKT(n)
 
# This code is contributed by AAKASH PAL

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C#

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// C# program for
// Knight Tour problem
using System;
 
class GFG {
    static int N = 8;
 
    /* A utility function to
    check if i,j are valid
    indexes for N*N chessboard */
    static bool 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++)
                Console.Write(sol[x, y] + " ");
            Console.WriteLine();
        }
    }
 
    /* 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 bool 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)) {
            Console.WriteLine("Solution does "
                              + "not exist");
            return false;
        }
        else
            printSolution(sol);
 
        return true;
    }
 
    /* A recursive utility function
    to solve Knight Tour problem */
    static bool 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
                    // backtracking
                    sol[next_x, next_y] = -1;
            }
        }
 
        return false;
    }
 
    // Driver Code
    public static void Main()
    {
        // Function Call
        solveKT();
    }
}
 
// This code is contributed by mits.

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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 

Time Complexity : 
There are N2 Cells and for each, we have a maximum of 8 possible moves to choose from, so the worst running time is O(8N^2).

Important Note:
No order of the xMove, yMove is wrong, but they will affect the running time of the algorithm drastically. For example, think of the case where 8th choice of the move is the correct one and before that our code ran 7 different wrong paths. It’s always a good idea a have a heuristic than to try backtracking randomly. Like, in this case, we know the next step would probably be in south or east direction, then checking the paths which leads their first is a better strategy.

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 
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 

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