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Knights-Tour-Animation

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.[1] 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.[2]

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 is C implementation 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.
Following is C implementation 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.


#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])
{
    if ( x >= 0 && x < N && y >= 0 && y < N && sol[x][y] == -1)
        return 1;
    return 0;
}
 
/* 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();
    getchar();
    return 0;
}

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

Knight Graph

The m×n knight graph is a graph on mn vertices in which each vertex represents a square in an m×n chessboard, and each edge corresponds to a legal move by a knight (which may only make moves which simultaneously shift one square along one axis and two along the other).

The number of edges in the n×n knight graph is 4(n-2)(n-1) (8 times the triangular numbers), so for n=1, 2, ..., the first few values are 0, 0, 8, 24, 48, 80, 120, ... (OEIS A033996).

Knight graphs are bipartite and therefore are perfect.

The following table summarizes some named graph complements of knight graphs.

G G^_
(2,3)-knight graph (2,3)-queen graph
(3,3)-knight graph (3,3)-queen graph

The m×n knight graph is implemented in Mathematica as KnightTourGraph[m, n], and precomputed properties are available in using GraphData[{"Knight", {m, n}}].

Closed formulas for the numbers c_k of k-graph cycles of the n×n knight graph are given by c_k=0 for k odd and

c_4 = {0 for n<=3; 2(3n^2-18n+26) otherwise
(1)

(E. Weisstein, Nov. 16, 2014).

A knight's path is a sequence of moves by a knight such that each square of the board is visited exactly once. It is therefore a Hamiltonian path on the corresponding knight graph. Conrad et al. (1994) shows that a knight's path exists on an n×n board iff n>=5.

KnightsTours_700

 

 

 

 

 

 

 

 

 

 

more about at : http://mathworld.wolfram.com/KnightGraph.html

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