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# The Knight’s tour problem | Backtracking-1

• Difficulty Level : Hard
• Last Updated : 24 Sep, 2021

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. 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``[``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`

## 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``][``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`

## 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 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