Number of cells a queen can move with obstacles on the chessborad

Consider a N X N chessboard with a Queen and K obstacles. The Queen cannot pass through obstacles. Given the position (x, y) of Queen, the task is to find the number of cells the queen can move.

Examples:

Input : N = 8, x = 4, y = 4, 
        K = 0
Output : 27


Input : N = 8, x = 4, y = 4, 
        K = 1, kx1 = 3, ky1 = 5
Output : 24

Method 1:
The idea is to iterate over the cells the queen can attack and stop until there is an obstacle or end of the board. To do that, we need to iterate horizontally, vertically and diagonally. The moves from position (x, y) can be:
(x+1, y): one step horizontal move to the right.
(x-1, y): one step horizontal move to the left.
(x+1, y+1): one step diagonal move up-right.
(x-1, y-1): one step diagonal move down-left.
(x-1, y+1): one step diagonal move left-up.
(x+1, y-1): one step diagonal move right-down.
(x, y+1): one step downward.
(x, y-1): one step upward.

Below is C++ implementation of this approach:

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// C++ program to find number of cells a queen can move 
// with obstacles on the chessborad
#include<bits/stdc++.h>
using namespace std;
  
// Return if position is valid on chessboard
int range(int n, int x, int y)
{
  return (x <= n && x > 0 && y <= n && y > 0);
}
  
// Return the number of moves with a given direction
int check(int n, int x, int y, int xx, int yy, 
                  map <pair<int, int>, int> mp)
{
  int ans = 0;
    
  // Checking valid move of Queen in a direction.
  while (range(n, x, y) && ! mp[{x, y}])
  {
    x += xx;
    y += yy;
    ans++;
  }
    
  return ans;
}
  
// Return the number of position a Queen can move.
int numberofPosition(int n, int k, int x, int y, 
                  int obstPosx[], int obstPosy[])
{
  int x1, y1, ans = 0;
  map <pair<int, int>, int> mp;
    
  // Mapping each obstacle's position
  while(k--)
  {
    x1 = obstPosx[k];
    y1 = obstPosy[k];
      
    mp[{x1, y1}] = 1;
  }
    
  // Fetching number of position a queen can
  // move in each direction.
  ans += check(n, x + 1, y, 1, 0, mp);
  ans += check(n, x-1, y, -1, 0, mp);
  ans += check(n, x, y + 1, 0, 1, mp);
  ans += check(n, x, y-1, 0, -1, mp);
  ans += check(n, x + 1, y + 1, 1, 1, mp);
  ans += check(n, x + 1, y-1, 1, -1, mp);
  ans += check(n, x-1, y + 1, -1, 1, mp);
  ans += check(n, x-1, y-1, -1, -1, mp);
    
  return ans;
}
  
// Driven Program
int main()
{
  int n = 8;  // Chessboard size
  int k = 1;  // Number of obstacles
  int Qposx = 4; // Queen x position
  int Qposy = 4; // Queen y position
  int obstPosx[] = { 3 };  // x position of obstacles
  int obstPosy[] = { 5 };  // y position of obstacles
    
  cout << numberofPosition(n, k, Qposx, Qposy, 
                   obstPosx, obstPosy) << endl;
  return 0;
}

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Output:


24



 


Method 2:

The idea is to iterate over the obstacles and for those who are in the queen’s path, we calculate the free cells upto that obstacle. If there is no obstacle in the path we have to calculate the number of free cells upto end of board in that direction.

For any (x1, y1) and (x2, y2):


    

  • If they are horizontally at same level: abs(x1 – x2 – 1)
    • 

  • If they are vertically at same level: abs(y1 – y2 – 1) is the number of free cells between.
    • 

  • If they are diagonal: both abs(x1 – x2 – 1) or abs(y1 – y2 – 1) is the number of free cells between.
    • 



Below is the implementation of this approach:




C++















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// C++ program to find number of cells a queen can move 


// with obstacles on the chessborad 


#include <bits/stdc++.h> 


using namespace std; 


  


// Return the number of position a Queen can move. 


int numberofPosition(int n, int k, int x, int y, 


                    int obstPosx[], int obstPosy[]) 


{ 


    // d11, d12, d21, d22 are for diagnoal distances. 


    // r1, r2 are for vertical distance. 


    // c1, c2 are for horizontal distance. 


    int d11, d12, d21, d22, r1, r2, c1, c2; 


  


    // Initialise the distance to end of the board. 


    d11 = min( x-1, y-1 ); 


    d12 = min( n-x, n-y ); 


    d21 = min( n-x, y-1 ); 


    d22 = min( x-1, n-y ); 


  


    r1 = y-1; 


    r2 = n-y; 


    c1 = x-1; 


    c2 = n-x; 


  


    // For each obstacle find the minimum distance. 


    // If obstacle is present in any direction, 


    // distance will be updated. 


    for (int i = 0; i < k; i++) 


    { 


        if ( x > obstPosx[i] && y > obstPosy[i] && 


                 x-obstPosx[i] == y-obstPosy[i] ) 


            d11 = min(d11, x-obstPosx[i]-1); 


  


        if ( obstPosx[i] > x && obstPosy[i] > y && 


                  obstPosx[i]-x == obstPosy[i]-y ) 


            d12 = min( d12, obstPosx[i]-x-1); 


  


        if ( obstPosx[i] > x && y > obstPosy[i] && 


                   obstPosx[i]-x == y-obstPosy[i] ) 


            d21 = min(d21, obstPosx[i]-x-1); 


  


        if ( x > obstPosx[i] && obstPosy[i] > y && 


                    x-obstPosx[i] == obstPosy[i]-y ) 


            d22 = min(d22, x-obstPosx[i]-1); 


  


        if ( x == obstPosx[i] && obstPosy[i] < y ) 


            r1 = min(r1, y-obstPosy[i]-1); 


  


        if ( x == obstPosx[i] && obstPosy[i] > y ) 


            r2 = min(r2, obstPosy[i]-y-1); 


  


        if ( y == obstPosy[i] && obstPosx[i] < x ) 


            c1 = min(c1, x-obstPosx[i]-1); 


  


        if ( y == obstPosy[i] && obstPosx[i] > x ) 


            c2 = min(c2, obstPosx[i]-x-1); 


    } 


  


    return d11 + d12 + d21 + d22 + r1 + r2 + c1 + c2; 


} 


  


// Driver code 


int main(void) 


{ 


    int n = 8;  // Chessboard size 


    int k = 1;  // number of obstacles 


    int Qposx = 4; // Queen x position 


    int Qposy = 4; // Queen y position 


    int obstPosx[] = { 3 };  // x position of obstacles 


    int obstPosy[] = { 5 };  // y position of obstacles 


  


    cout << numberofPosition(n, k, Qposx, Qposy, 


                             obstPosx, obstPosy); 


    return 0; 


} 



















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// Java program to find number of cells a  


// queen can move with obstacles on the  


// chessborad 


import java.io.*; 


  


class GFG { 


  


    // Return the number of position a Queen 


    // can move. 


    static int numberofPosition(int n, int k, int x, 


                 int y, int obstPosx[], int obstPosy[]) 


    { 


          


        // d11, d12, d21, d22 are for diagnoal distances. 


        // r1, r2 are for vertical distance. 


        // c1, c2 are for horizontal distance. 


        int d11, d12, d21, d22, r1, r2, c1, c2; 


      


        // Initialise the distance to end of the board. 


        d11 = Math.min( x-1, y-1 ); 


        d12 = Math.min( n-x, n-y ); 


        d21 = Math.min( n-x, y-1 ); 


        d22 = Math.min( x-1, n-y ); 


      


        r1 = y-1; 


        r2 = n-y; 


        c1 = x-1; 


        c2 = n-x; 


      


        // For each obstacle find the minimum distance. 


        // If obstacle is present in any direction, 


        // distance will be updated. 


        for (int i = 0; i < k; i++) 


        { 


            if ( x > obstPosx[i] && y > obstPosy[i] && 


                    x-obstPosx[i] == y-obstPosy[i] ) 


                d11 = Math.min(d11, x-obstPosx[i]-1); 


      


            if ( obstPosx[i] > x && obstPosy[i] > y && 


                    obstPosx[i]-x == obstPosy[i]-y ) 


                d12 = Math.min( d12, obstPosx[i]-x-1); 


      


            if ( obstPosx[i] > x && y > obstPosy[i] && 


                    obstPosx[i]-x == y-obstPosy[i] ) 


                d21 = Math.min(d21, obstPosx[i]-x-1); 


      


            if ( x > obstPosx[i] && obstPosy[i] > y && 


                        x-obstPosx[i] == obstPosy[i]-y ) 


                d22 = Math.min(d22, x-obstPosx[i]-1); 


      


            if ( x == obstPosx[i] && obstPosy[i] < y ) 


                r1 = Math.min(r1, y-obstPosy[i]-1); 


      


            if ( x == obstPosx[i] && obstPosy[i] > y ) 


                r2 = Math.min(r2, obstPosy[i]-y-1); 


      


            if ( y == obstPosy[i] && obstPosx[i] < x ) 


                c1 = Math.min(c1, x-obstPosx[i]-1); 


      


            if ( y == obstPosy[i] && obstPosx[i] > x ) 


                c2 = Math.min(c2, obstPosx[i]-x-1); 


        } 


      


        return d11 + d12 + d21 + d22 + r1 + r2 + c1 + c2; 


    } 


      


    // Driver code 


    public static void main (String[] args) { 


    int n = 8; // Chessboard size 


    int k = 1; // number of obstacles 


    int Qposx = 4; // Queen x position 


    int Qposy = 4; // Queen y position 


    int obstPosx[] = { 3 }; // x position of obstacles 


    int obstPosy[] = { 5 }; // y position of obstacles 


  


    System.out.println(numberofPosition(n, k, Qposx, 


                          Qposy, obstPosx, obstPosy)); 


    } 


} 


  


// This code is contributed by anuj_67. 



















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// C# program to find number of cells a  


// queen can move with obstacles on the  


// chessborad 


using System; 


  


class GFG { 


  


    // Return the number of position a Queen 


    // can move. 


    static int numberofPosition(int n, int k, int x, 


                int y, int []obstPosx, int []obstPosy) 


    { 


          


        // d11, d12, d21, d22 are for diagnoal distances. 


        // r1, r2 are for vertical distance. 


        // c1, c2 are for horizontal distance. 


        int d11, d12, d21, d22, r1, r2, c1, c2; 


      


        // Initialise the distance to end of the board. 


        d11 = Math.Min( x-1, y-1 ); 


        d12 = Math.Min( n-x, n-y ); 


        d21 = Math.Min( n-x, y-1 ); 


        d22 = Math.Min( x-1, n-y ); 


      


        r1 = y-1; 


        r2 = n-y; 


        c1 = x-1; 


        c2 = n-x; 


      


        // For each obstacle find the Minimum distance. 


        // If obstacle is present in any direction, 


        // distance will be updated. 


        for (int i = 0; i < k; i++) 


        { 


            if ( x > obstPosx[i] && y > obstPosy[i] && 


                    x-obstPosx[i] == y-obstPosy[i] ) 


                d11 = Math.Min(d11, x-obstPosx[i]-1); 


      


            if ( obstPosx[i] > x && obstPosy[i] > y && 


                    obstPosx[i]-x == obstPosy[i]-y ) 


                d12 = Math.Min( d12, obstPosx[i]-x-1); 


      


            if ( obstPosx[i] > x && y > obstPosy[i] && 


                    obstPosx[i]-x == y-obstPosy[i] ) 


                d21 = Math.Min(d21, obstPosx[i]-x-1); 


      


            if ( x > obstPosx[i] && obstPosy[i] > y && 


                        x-obstPosx[i] == obstPosy[i]-y) 


                d22 = Math.Min(d22, x-obstPosx[i]-1); 


      


            if ( x == obstPosx[i] && obstPosy[i] < y ) 


                r1 = Math.Min(r1, y-obstPosy[i]-1); 


      


            if ( x == obstPosx[i] && obstPosy[i] > y ) 


                r2 = Math.Min(r2, obstPosy[i]-y-1); 


      


            if ( y == obstPosy[i] && obstPosx[i] < x ) 


                c1 = Math.Min(c1, x-obstPosx[i]-1); 


      


            if ( y == obstPosy[i] && obstPosx[i] > x ) 


                c2 = Math.Min(c2, obstPosx[i]-x-1); 


        } 


      


        return d11 + d12 + d21 + d22 + r1 + r2 + c1 + c2; 


    } 


      


    // Driver code 


    public static void Main ()  


    { 


        int n = 8; // Chessboard size 


        int k = 1; // number of obstacles 


        int Qposx = 4; // Queen x position 


        int Qposy = 4; // Queen y position 


        int []obstPosx = { 3 }; // x position of obstacles 


        int []obstPosy = { 5 }; // y position of obstacles 


      


        Console.WriteLine(numberofPosition(n, k, Qposx, 


                            Qposy, obstPosx, obstPosy)); 


    } 


} 


  


// This code is contributed by anuj_67. 



















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<?php 


//PHP program to find number of cells a queen can move 


// with obstacles on the chessborad 


  


// Return the number of position a Queen can move. 


function numberofPosition($n, $k, $x, $y, 


                    $obstPosx, $obstPosy) 


{ 


    // d11, d12, d21, d22 are for diagnoal distances. 


    // r1, r2 are for vertical distance. 


    // c1, c2 are for horizontal distance. 


    $d11; 


    $d12; 


    $d21; 


    $d22; 


    $r1; 


    $r2; 


    $c1; 


    $c2; 


  


    // Initialise the distance to end of the board. 


    $d11 = min( $x-1, $y-1 ); 


    $d12 = min( $n-$x, $n-$y ); 


    $d21 = min( $n-$x, $y-1 ); 


    $d22 = min( $x-1, $n-$y ); 


  


    $r1 = $y-1; 


    $r2 = $n-$y; 


    $c1 = $x-1; 


    $c2 = $n-$x; 


  


    // For each obstacle find the minimum distance. 


    // If obstacle is present in any direction, 


    // distance will be updated. 


    for ($i = 0; $i < $k; $i++) 


    { 


        if ( $x > $obstPosx[$i] && $y > $obstPosy[$i] && 


                $x-$obstPosx[$i] == $y-$obstPosy[$i] ) 


            $d11 = min($d11, $x-$obstPosx[$i]-1); 


  


        if ( $obstPosx[$i] > $x && $obstPosy[$i] > $y && 


                $obstPosx[$i]-$x == $obstPosy[$i]-$y ) 


            $d12 = min( $d12, $obstPosx[$i]-$x-1); 


  


        if ( $obstPosx[$i] > $x && $y > $obstPosy[$i] && 


                $obstPosx[$i]-$x == $y-$obstPosy[$i] ) 


            $d21 = min($d21, $obstPosx[$i]-$x-1); 


  


        if ( $x > $obstPosx[$i] && $obstPosy[$i] > $y && 


                    $x-$obstPosx[$i] == $obstPosy[$i]-$y ) 


            $d22 = min($d22, $x-$obstPosx[$i]-1); 


  


        if ( $x == $obstPosx[$i] && $obstPosy[$i] < $y ) 


            $r1 = min($r1, $y-$obstPosy[$i]-1); 


  


        if ( $x == $obstPosx[$i] && $obstPosy[$i] > $y ) 


            $r2 = min($r2, $obstPosy[$i]-$y-1); 


  


        if ( $y == $obstPosy[$i] && $obstPosx[$i] < $x ) 


            $c1 = min($c1, $x-$obstPosx[$i]-1); 


  


        if ( $y == $obstPosy[$i] && $obstPosx[$i] > $x ) 


            $c2 = min($c2, $obstPosx[$i]-$x-1); 


    } 


  


    return $d11 + $d12 + $d21 + $d22 + $r1 + $r2 + $c1 + $c2; 


} 


  


// Driver code 


  


$n = 8; // Chessboard size 


$k = 1; // number of obstacles 


$Qposx = 4; // Queen x position 


$Qposy = 4; // Queen y position 


$obstPosx = array(3 ); // x position of obstacles 


$obstPosy = array(5 ); // y position of obstacles 


  


echo numberofPosition($n, $k, $Qposx, $Qposy, 


                        $obstPosx, $obstPosy); 


  


// This code is contributed by ajit. 


?> 



















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