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Warnsdorff’s algorithm for Knight’s tour problem

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  • Difficulty Level : Expert
  • Last Updated : 24 Nov, 2022
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Problem : A knight is placed on the first block of an empty board and, moving according to the rules of chess, must visit each square exactly once. 

Following is an example path followed by Knight to cover all the cells. The below grid represents a chessboard with 8 x 8 cells. Numbers in cells indicate move number of Knight. 
 

knight-tour-problem

We have discussed Backtracking Algorithm for solution of Knight’s tour. In this post Warnsdorff’s heuristic is discussed. 
Warnsdorff’s Rule: 

  1. We can start from any initial position of the knight on the board.
  2. We always move to an adjacent, unvisited square with minimal degree (minimum number of unvisited adjacent).

This algorithm may also more generally be applied to any graph. 

Some definitions:  

  • A position Q is accessible from a position P if P can move to Q by a single Knight’s move, and Q has not yet been visited.
  • The accessibility of a position P is the number of positions accessible from P.

Algorithm:  

  1. Set P to be a random initial position on the board
  2. Mark the board at P with the move number “1”
  3. Do following for each move number from 2 to the number of squares on the board: 
    • let S be the set of positions accessible from P.
    • Set P to be the position in S with minimum accessibility
    • Mark the board at P with the current move number
  4. Return the marked board — each square will be marked with the move number on which it is visited.

Below is implementation of above algorithm.  

C++




// C++ program to for Knight's tour problem using
// Warnsdorff's algorithm
#include <bits/stdc++.h>
#define N 8
 
// Move pattern on basis of the change of
// x coordinates and y coordinates respectively
static int cx[N] = {1,1,2,2,-1,-1,-2,-2};
static int cy[N] = {2,-2,1,-1,2,-2,1,-1};
 
// function restricts the knight to remain within
// the 8x8 chessboard
bool limits(int x, int y)
{
    return ((x >= 0 && y >= 0) && (x < N && y < N));
}
 
/* Checks whether a square is valid and empty or not */
bool isempty(int a[], int x, int y)
{
    return (limits(x, y)) && (a[y*N+x] < 0);
}
 
/* Returns the number of empty squares adjacent
   to (x, y) */
int getDegree(int a[], int x, int y)
{
    int count = 0;
    for (int i = 0; i < N; ++i)
        if (isempty(a, (x + cx[i]), (y + cy[i])))
            count++;
 
    return count;
}
 
// Picks next point using Warnsdorff's heuristic.
// Returns false if it is not possible to pick
// next point.
bool nextMove(int a[], int *x, int *y)
{
    int min_deg_idx = -1, c, min_deg = (N+1), nx, ny;
 
    // Try all N adjacent of (*x, *y) starting
    // from a random adjacent. Find the adjacent
    // with minimum degree.
    int start = rand()%N;
    for (int count = 0; count < N; ++count)
    {
        int i = (start + count)%N;
        nx = *x + cx[i];
        ny = *y + cy[i];
        if ((isempty(a, nx, ny)) &&
           (c = getDegree(a, nx, ny)) < min_deg)
        {
            min_deg_idx = i;
            min_deg = c;
        }
    }
 
    // IF we could not find a next cell
    if (min_deg_idx == -1)
        return false;
 
    // Store coordinates of next point
    nx = *x + cx[min_deg_idx];
    ny = *y + cy[min_deg_idx];
 
    // Mark next move
    a[ny*N + nx] = a[(*y)*N + (*x)]+1;
 
    // Update next point
    *x = nx;
    *y = ny;
 
    return true;
}
 
/* displays the chessboard with all the
  legal knight's moves */
void print(int a[])
{
    for (int i = 0; i < N; ++i)
    {
        for (int j = 0; j < N; ++j)
            printf("%d\t",a[j*N+i]);
        printf("\n");
    }
}
 
/* checks its neighbouring squares */
/* If the knight ends on a square that is one
   knight's move from the beginning square,
   then tour is closed */
bool neighbour(int x, int y, int xx, int yy)
{
    for (int i = 0; i < N; ++i)
        if (((x+cx[i]) == xx)&&((y + cy[i]) == yy))
            return true;
 
    return false;
}
 
/* Generates the legal moves using warnsdorff's
  heuristics. Returns false if not possible */
bool findClosedTour()
{
    // Filling up the chessboard matrix with -1's
    int a[N*N];
    for (int i = 0; i< N*N; ++i)
        a[i] = -1;
 
    // Random initial position
    int sx = rand()%N;
    int sy = rand()%N;
 
    // Current points are same as initial points
    int x = sx, y = sy;
    a[y*N+x] = 1; // Mark first move.
 
    // Keep picking next points using
    // Warnsdorff's heuristic
    for (int i = 0; i < N*N-1; ++i)
        if (nextMove(a, &x, &y) == 0)
            return false;
 
    // Check if tour is closed (Can end
    // at starting point)
    if (!neighbour(x, y, sx, sy))
        return false;
 
    print(a);
    return true;
}
 
// Driver code
int main()
{
    // To make sure that different random
    // initial positions are picked.
    srand(time(NULL));
 
    // While we don't get a solution
    while (!findClosedTour())
    {
    ;
    }
 
    return 0;
}


Java




// Java program to for Knight's tour problem using
// Warnsdorff's algorithm
import java.util.concurrent.ThreadLocalRandom;
 
class GFG
{
    public static final int N = 8;
 
    // Move pattern on basis of the change of
    // x coordinates and y coordinates respectively
    public static final int cx[] = {1, 1, 2, 2, -1, -1, -2, -2};
    public static final int cy[] = {2, -2, 1, -1, 2, -2, 1, -1};
 
    // function restricts the knight to remain within
    // the 8x8 chessboard
    boolean limits(int x, int y)
    {
        return ((x >= 0 && y >= 0) &&
                 (x < N && y < N));
    }
 
    /* Checks whether a square is valid and
    empty or not */
    boolean isempty(int a[], int x, int y)
    {
        return (limits(x, y)) && (a[y * N + x] < 0);
    }
 
    /* Returns the number of empty squares
    adjacent to (x, y) */
    int getDegree(int a[], int x, int y)
    {
        int count = 0;
        for (int i = 0; i < N; ++i)
            if (isempty(a, (x + cx[i]),
                           (y + cy[i])))
                count++;
 
        return count;
    }
 
    // Picks next point using Warnsdorff's heuristic.
    // Returns false if it is not possible to pick
    // next point.
    Cell nextMove(int a[], Cell cell)
    {
        int min_deg_idx = -1, c,
            min_deg = (N + 1), nx, ny;
 
        // Try all N adjacent of (*x, *y) starting
        // from a random adjacent. Find the adjacent
        // with minimum degree.
        int start = ThreadLocalRandom.current().nextInt(1000) % N;
        for (int count = 0; count < N; ++count)
        {
            int i = (start + count) % N;
            nx = cell.x + cx[i];
            ny = cell.y + cy[i];
            if ((isempty(a, nx, ny)) &&
                (c = getDegree(a, nx, ny)) < min_deg)
            {
                min_deg_idx = i;
                min_deg = c;
            }
        }
 
        // IF we could not find a next cell
        if (min_deg_idx == -1)
            return null;
 
        // Store coordinates of next point
        nx = cell.x + cx[min_deg_idx];
        ny = cell.y + cy[min_deg_idx];
 
        // Mark next move
        a[ny * N + nx] = a[(cell.y) * N +
                           (cell.x)] + 1;
 
        // Update next point
        cell.x = nx;
        cell.y = ny;
 
        return cell;
    }
 
    /* displays the chessboard with all the
    legal knight's moves */
    void print(int a[])
    {
        for (int i = 0; i < N; ++i)
        {
            for (int j = 0; j < N; ++j)
                System.out.printf("%d\t", a[j * N + i]);
            System.out.printf("\n");
        }
    }
 
    /* checks its neighbouring squares */
    /* If the knight ends on a square that is one
    knight's move from the beginning square,
    then tour is closed */
    boolean neighbour(int x, int y, int xx, int yy)
    {
        for (int i = 0; i < N; ++i)
            if (((x + cx[i]) == xx) &&
                ((y + cy[i]) == yy))
                return true;
 
        return false;
    }
 
    /* Generates the legal moves using warnsdorff's
    heuristics. Returns false if not possible */
    boolean findClosedTour()
    {
        // Filling up the chessboard matrix with -1's
        int a[] = new int[N * N];
        for (int i = 0; i < N * N; ++i)
            a[i] = -1;
 
        // initial position
        int sx = 3;
        int sy = 2;
 
        // Current points are same as initial points
        Cell cell = new Cell(sx, sy);
 
        a[cell.y * N + cell.x] = 1; // Mark first move.
 
        // Keep picking next points using
        // Warnsdorff's heuristic
        Cell ret = null;
        for (int i = 0; i < N * N - 1; ++i)
        {
            ret = nextMove(a, cell);
            if (ret == null)
                return false;
        }
 
        // Check if tour is closed (Can end
        // at starting point)
        if (!neighbour(ret.x, ret.y, sx, sy))
            return false;
 
        print(a);
        return true;
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        // While we don't get a solution
        while (!new GFG().findClosedTour())
        {
            ;
        }
    }
}
 
class Cell
{
    int x;
    int y;
 
    public Cell(int x, int y)
    {
        this.x = x;
        this.y = y;
    }
}
 
// This code is contributed by SaeedZarinfam


Python3




# Python program to for Knight's tour problem using
# Warnsdorff's algorithm
import random
 
class Cell:
    def __init__(self, x, y):
        self.x = x
        self.y = y
 
N = 8
 
# Move pattern on basis of the change of
# x coordinates and y coordinates respectively
cx = [1, 1, 2, 2, -1, -1, -2, -2]
cy = [2, -2, 1, -1, 2, -2, 1, -1]
 
# function restricts the knight to remain within
# the 8x8 chessboard
def limits(x, y):
    return ((x >= 0 and y >= 0) and (x < N and y < N))
 
# Checks whether a square is valid and empty or not
def isempty(a, x, y):
    return (limits(x, y)) and (a[y * N + x] < 0)
 
# Returns the number of empty squares adjacent to (x, y)
def getDegree(a, x, y):
    count = 0
    for i in range(N):
        if isempty(a, (x + cx[i]), (y + cy[i])):
            count += 1
    return count
 
# Picks next point using Warnsdorff's heuristic.
# Returns false if it is not possible to pick
# next point.
def nextMove(a, cell):
    min_deg_idx = -1
    c = 0
    min_deg = (N + 1)
    nx = 0
    ny = 0
 
    # Try all N adjacent of (*x, *y) starting
    # from a random adjacent. Find the adjacent
    # with minimum degree.
    start = random.randint(0, 1000) % N
    for count in range(0, N):
        i = (start + count) % N
        nx = cell.x + cx[i]
        ny = cell.y + cy[i]
        c = getDegree(a, nx, ny)
        if ((isempty(a, nx, ny)) and c < min_deg):
            min_deg_idx = i
            min_deg = c
 
    # IF we could not find a next cell
    if (min_deg_idx == -1):
        return None
 
    # Store coordinates of next point
    nx = cell.x + cx[min_deg_idx]
    ny = cell.y + cy[min_deg_idx]
 
    # Mark next move
    a[ny * N + nx] = a[(cell.y) * N + (cell.x)] + 1
 
    # Update next point
    cell.x = nx
    cell.y = ny
 
    return cell
 
# displays the chessboard with all the legal knight's moves
def printA(a):
    for i in range(N):
        for j in range(N):
            print("%d\t" % a[j * N + i], end="")
        print()
 
# checks its neighbouring squares
# If the knight ends on a square that is one knight's move from the beginning square,then tour is closed
def neighbour(x, y, xx, yy):
    for i in range(N):
        if ((x + cx[i]) == xx) and ((y + cy[i]) == yy):
            return True
    return False
 
#  Generates the legal moves using warnsdorff's heuristics. Returns false if not possible
def findClosedTour():
    # Filling up the chessboard matrix with -1's
    a = [-1] * N * N
 
    # initial position
    sx = 3
    sy = 2
 
    # Current points are same as initial points
    cell = Cell(sx, sy)
 
    a[cell.y * N + cell.x] = 1  # Mark first move.
 
    # Keep picking next points using Warnsdorff's heuristic
    ret = None
    for i in range(N * N - 1):
        ret = nextMove(a, cell)
        if ret == None:
            return False
 
    # Check if tour is closed (Can end at starting point)
    if not neighbour(ret.x, ret.y, sx, sy):
        return False
    printA(a)
    return True
 
 
# Driver Code
if __name__ == '__main__':
    # While we don't get a solution
    while not findClosedTour():
        pass
 
# This code is contributed by Tapesh(tapeshdua420)


C#




//C# program for Knight’s tour 
//problem using Warnsdorff’salgorithm
using System;
using System.Collections;
using System.Collections.Generic;
 
public class GFG{
     
    public static int N = 8;
  
    // Move pattern on basis of the change of
    // x coordinates and y coordinates respectively
    public int[] cx = new int[] {1, 1, 2, 2, -1, -1, -2, -2};
    public int[] cy = new int[] {2, -2, 1, -1, 2, -2, 1, -1};
  
    // function restricts the knight to remain within
    // the 8x8 chessboard
    bool limits(int x, int y)
    {
        return ((x >= 0 && y >= 0) && (x < N && y < N));
    }
  
    /* Checks whether a square is valid and
    empty or not */
    bool isempty(int[] a, int x, int y)
    {
        return ((limits(x, y)) && (a[y * N + x] < 0));
    }
 
    /* Returns the number of empty squares
    adjacent to (x, y) */
    int getDegree(int[] a, int x, int y)
    {
        int count = 0;
        for (int i = 0; i < N; ++i)
            if (isempty(a, (x + cx[i]),
                           (y + cy[i])))
                count++;
  
        return count;
    }
     
    // Picks next point using Warnsdorff's heuristic.
    // Returns false if it is not possible to pick
    // next point.
    Cell nextMove(int[] a, Cell cell)
    {
        int min_deg_idx = -1, c,
            min_deg = (N + 1), nx, ny;
  
        // Try all N adjacent of (*x, *y) starting
        // from a random adjacent. Find the adjacent
        // with minimum degree.
        Random random = new Random();
        int start=random.Next(0, 1000);
        for (int count = 0; count < N; ++count)
        {
            int i = (start + count) % N;
            nx = cell.x + cx[i];
            ny = cell.y + cy[i];
            if ((isempty(a, nx, ny)) &&
                (c = getDegree(a, nx, ny)) < min_deg)
            {
                min_deg_idx = i;
                min_deg = c;
            }
        }
  
        // IF we could not find a next cell
        if (min_deg_idx == -1)
            return null;
  
        // Store coordinates of next point
        nx = cell.x + cx[min_deg_idx];
        ny = cell.y + cy[min_deg_idx];
  
        // Mark next move
        a[ny * N + nx] = a[(cell.y) * N +
                           (cell.x)] + 1;
  
        // Update next point
        cell.x = nx;
        cell.y = ny;
  
        return cell;
    }
  
    /* displays the chessboard with all the
    legal knight's moves */
    void print(int[] a)
    {
        for (int i = 0; i < N; ++i)
        {
            for (int j = 0; j < N; ++j)
                Console.Write(a[j * N + i]+"\t");
            Console.Write("\n");
        }
    }
     
    /* checks its neighbouring squares */
    /* If the knight ends on a square that is one
    knight's move from the beginning square,
    then tour is closed */
    bool neighbour(int x, int y, int xx, int yy)
    {
        for (int i = 0; i < N; ++i)
            if (((x + cx[i]) == xx) &&
                ((y + cy[i]) == yy))
                return true;
  
        return false;
    }
  
    /* Generates the legal moves using warnsdorff's
    heuristics. Returns false if not possible */
    bool findClosedTour()
    {
        // Filling up the chessboard matrix with -1's
        int[] a = new int[N * N];
        for (int i = 0; i < N * N; ++i)
            a[i] = -1;
  
        // initial position
        int sx = 3;
        int sy = 2;
  
        // Current points are same as initial points
        Cell cell = new Cell(sx, sy);
  
        a[cell.y * N + cell.x] = 1; // Mark first move.
  
        // Keep picking next points using
        // Warnsdorff's heuristic
        Cell ret = null;
        for (int i = 0; i < N * N - 1; ++i)
        {
            ret = nextMove(a, cell);
            if (ret == null)
                return false;
        }
  
        // Check if tour is closed (Can end
        // at starting point)
        if (!neighbour(ret.x, ret.y, sx, sy))
            return false;
  
        print(a);
        return true;
    }
     
    static public void Main (){
        // While we don't get a solution
        while (!new GFG().findClosedTour())
        {
            ;
        }
    }
}
 
class Cell
{
    public int x;
    public int y;
  
    public Cell(int x, int y)
    {
        this.x = x;
        this.y = y;
    }
}
 
//This code is contributed by shruti456rawal


Javascript




// JavaScript program to for Knight's tour problem using
// Warnsdorff's algorithm
class Cell {
    constructor(x, y) {
        this.x = x;
        this.y = y;
    }
}
 
const N = 8;
 
// Move pattern on basis of the change of
// x coordinates and y coordinates respectively
const cx = [1, 1, 2, 2, -1, -1, -2, -2];
const cy = [2, -2, 1, -1, 2, -2, 1, -1];
 
// function restricts the knight to remain within
// the 8x8 chessboard
function limits(x, y) {
    return ((x >= 0 && y >= 0) && (x < N && y < N));
}
// Checks whether a square is valid and empty or not
function isempty(a, x, y) {
    return (limits(x, y)) && (a[y * N + x] < 0);
}
 
// Returns the number of empty squares adjacent to (x, y)
function getDegree(a, x, y) {
    let count = 0;
    for (let i = 0; i < N; i++) {
        if (isempty(a, (x + cx[i]), (y + cy[i]))) {
            count += 1;
        }
    }
    return count;
}
 
// Picks next point using Warnsdorff's heuristic.
// Returns false if it is not possible to pick
// next point.
function nextMove(a, cell) {
    var min_deg_idx = -1;
    var c = 0;
    var min_deg = (N + 1);
    var nx = 0;
    var ny = 0;
     
    // Try all N adjacent of (*x, *y) starting
    // from a random adjacent. Find the adjacent
    // with minimum degree.
    var start = Math.floor(Math.random() * 1000) % N;
    for (var count = 0; count < N; count++) {
        i = (start + count) % N;
        nx = cell.x + cx[i];
        ny = cell.y + cy[i];
        c = getDegree(a, nx, ny);
        if ((isempty(a, nx, ny)) && c < min_deg) {
            min_deg_idx = i;
            min_deg = c;
        }
    }
      
    // IF we could not find a next cell 
    if (min_deg_idx == -1) {
        return null
    }   
     
    // Store coordinates of next point
    nx = cell.x + cx[min_deg_idx];
    ny = cell.y + cy[min_deg_idx];
     
    // Mark next move
    a[ny * N + nx] = a[(cell.y) * N + (cell.x)] + 1; 
 
    // Update next point
    cell.x = nx;
    cell.y = ny;
     
    return cell;
 
}
 
// checks its neighbouring squares
// If the knight ends on a square that is one knight's move from the beginning square,then tour is closed
function neighbour(x, y, xx, yy) {
    for (var i = 0; i < N; i++) {
        if (((x + cx[i]) == xx) && ((y + cy[i]) == yy)){
            return true
        } else {
            return false
        }
    }   
}
 
//  Generates the legal moves using warnsdorff's heuristics. Returns false if not possible
function findClosedTour() {
    // Filling up the chessboard matrix with -1's
    var a = new Array(N * N).fill(-1);
     
    // initial position
    var sx = 3;
    var sy = 2;
     
    // Current points are same as initial points
    var cell = new Cell(sx, sy);
     
    a[cell.y * N + cell.x] = 1; // Mark first move.
     
    // Keep picking next points using Warnsdorff's heuristic
    var ret = null;
    for (var i = 0; i < N * N - 1; i++) {
        ret = nextMove(a, cell);
        if (ret == null) {
            return false;
        }
    }
     
    // Check if tour is closed (Can end at starting point)
    if (!neighbour(ret.x, ret.y, sx, sy)) {
        return false;
    }
    console.log(a);
    return true;
}
 
// Driver Code
// While we don't get a solution
while(!findClosedTour()){
}
 
// This code is contributed by Tapesh(tapeshdua420)


Output: 

59    14    63    32    1    16    19    34    
62    31    60    15    56    33    2    17    
13    58    55    64    49    18    35    20    
30    61    42    57    54    51    40    3    
43    12    53    50    41    48    21    36    
26    29    44    47    52    39    4    7    
11    46    27    24    9    6    37    22    
28    25    10    45    38    23    8    5    

The Hamiltonian path problem is NP-hard in general. In practice, Warnsdorff’s heuristic successfully finds a solution in linear time.

Do you know? 
“On an 8 × 8 board, there are exactly 26,534,728,821,064 directed closed tours (i.e. two tours along the same path that travel in opposite directions are counted separately, as are rotations and reflections). The number of undirected closed tours is half this number, since every tour can be traced in reverse!”

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