Backtracking | The Knight’s tour problem

So what is "The Knight's tour problem"?

A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight's move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed, otherwise it is open.

The knight's tour problem is the mathematical problem of finding a knight's tour. Creating a program to find a knight's tour is a common problem given to computer science students. Variations of the knight's tour problem involve chessboards of different sizes than the usual 8 × 8, as well as irregular (non-rectangular) boards.

The knight's tour problem is an instance of the more general Hamiltonian path problem in graph theory. The problem of finding a closed knight's tour is similarly an instance of the Hamiltonian cycle problem. Unlike the general Hamiltonian path problem, the knight's tour problem can be solved in linear time.

So,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.


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;
   }
}


Now,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.
   
Sample program:
 
 
class KnightTourTest
{
static int N = 8;
static boolean isSafe(int x, int y, int sol[][])
{
return (x >= 0 && x < N && y >= 0 &&
y < N && sol[x][y] == -1);
}

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();
}
}
static boolean solveKT()
{
int sol[][] = new int[8][8];

for (int x = 0; x < N; x++)
for (int y = 0; y < N; y++)
sol[x][y] = -1;

int xMove[] = {2, 1, -1, -2, -2, -1, 1, 2};
int yMove[] = {1, 2, 2, 1, -1, -2, -2, -1};

sol[0][0] = 0;

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;
}

public static void main(String args[]) {
solveKT();
}
}

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.