The Knight’s tour problem | Backtracking-1

• Difficulty Level : Hard
• Last Updated : 21 Oct, 2021

Backtracking is an algorithmic paradigm that tries different solutions until finds a solution that “works”. Problems that are typically solved using the 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 incrementally 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 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 the move number of Knight. 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++

 // C++ program for Knight Tour problem #include 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 = { 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;       /* 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, int yMove) {     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

C

 // C program for Knight Tour problem #include #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 = { 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;       /* 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; }

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;           /* 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;           /* 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

Python3

 # 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       # 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

C#

 // 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.

Javascript



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).

Auxiliary Space: O(N2)

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 the 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 the south or east direction, then checking the paths which lead their first is a better strategy.

Note that Backtracking is not the best solution for the Knight’s tour problem. See the 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