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Check if possible to move from given coordinate to desired coordinate

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  • Difficulty Level : Medium
  • Last Updated : 06 Jul, 2022
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Given two coordinates (x, y) and (a, b). Find if it is possible to reach (x, y) from (a, b). 

Only possible moves from any coordinate (i, j) are 

  • (i-j, j)
  • (i, i-j)
  • (i+j, j)
  • (i, i+j)

Given x, y, a, b can be negative.

Examples: 

Input : (x, y) = (1, 1) and  (a, b) = (2, 3).
Output : Yes.
(1, 1) -> (2, 1) -> (2, 3).

Input : (x, y) = (2, 1) and  (a, b) = (2, 3).
Output : Yes.

Input : (x, y) = (35, 15) and  (a, b) = (20, 25).
Output : Yes.
(35, 15) -> (20, 15) -> (5, 15) -> (5, 10) -> (5, 5) ->
(10, 5) -> (15, 5) -> (20, 5) -> (20, 25)

If we take a closer look at the problem, we can notice that the moves are similar steps of Euclidean algorithm for finding GCD. So, it is only possible to reach coordinate (a, b) from (x, y) if GCD of x, y is equal to GCD of a, b. Otherwise, it is not possible.

Let GCD of (x, y) be gcd. From (x, y), we can reach (gcd, gcd) and from this point, we can reach to (a, b) if and only if GCD of ‘a’ and ‘b’ is also gcd.

Below is the implementation of this approach: 

C++




// C++ program to check if it is possible to reach
// (a, b) from (x, y).
#include <bits/stdc++.h>
using namespace std;
 
// Returns GCD of i and j
int gcd(int i, int j)
{
    if (i == j)
        return i;
 
    if (i > j)
        return gcd(i - j, j);
    return gcd(i, j - i);
}
 
// Returns true if it is possible to go to (a, b)
// from (x, y)
bool ispossible(int x, int y, int a, int b)
{
    // Find absolute values of all as sign doesn't
    // matter.
    x = abs(x), y = abs(y), a = abs(a), b = abs(b);
 
    // If gcd is equal then it is possible to reach.
    // Else not possible.
    return (gcd(x, y) == gcd(a, b));
}
 
// Driven Program
int main()
{
    // Converting coordinate into positive integer
    int x = 35, y = 15;
    int a = 20, b = 25;
    (ispossible(x, y, a, b)) ? (cout << "Yes") : (cout << "No");
    return 0;
}

Java




// Java program to check if it is possible
// to reach (a, b) from (x, y).
class GFG {
 
    // Returns GCD of i and j
    static int gcd(int i, int j)
    {
        if (i == j)
            return i;
 
        if (i > j)
            return gcd(i - j, j);
        return gcd(i, j - i);
    }
 
    // Returns true if it is possible to go to (a, b)
    // from (x, y)
    static boolean ispossible(int x, int y, int a, int b)
    {
 
        // Find absolute values of all as
        // sign doesn't matter.
        x = Math.abs(x);
        y = Math.abs(y);
        a = Math.abs(a);
        b = Math.abs(b);
 
        // If gcd is equal then it is possible to reach.
        // Else not possible.
        return (gcd(x, y) == gcd(a, b));
    }
 
    // Driver code
    public static void main(String[] args)
    {
 
        // Converting coordinate into positive integer
        int x = 35, y = 15;
        int a = 20, b = 25;
        if (ispossible(x, y, a, b))
            System.out.print("Yes");
        else
            System.out.print("No");
    }
}
// This code is contributed by Anant Agarwal.

Python3




# Python program to check if it is possible to reach
# (a, b) from (x, y).
# Returns GCD of i and j
def gcd(i, j):
    if (i == j):
        return i
  
    if (i > j):
        return gcd(i - j, j)
    return gcd(i, j - i)
  
# Returns true if it is possible to go to (a, b)
# from (x, y)
def ispossible(x, y, a, b):
    # Find absolute values of all as sign doesn't
    # matter.
    x, y, a, b = abs(x), abs(y), abs(a), abs(b)
  
    # If gcd is equal then it is possible to reach.
    # Else not possible.
    return (gcd(x, y) == gcd(a, b))
  
# Driven Program
    # Converting coordinate into positive integer
x, y = 35, 15
a, b = 20, 25
if(ispossible(x, y, a, b)):
    print ("Yes")
else:
    print ("No")
# Contributed by Afzal Ansari

C#




// C# program to check if it is possible
// to reach (a, b) from (x, y).
using System;
 
class GFG {
 
    // Returns GCD of i and j
    static int gcd(int i, int j)
    {
        if (i == j)
            return i;
 
        if (i > j)
            return gcd(i - j, j);
        return gcd(i, j - i);
    }
 
    // Returns true if it is possible
    // to go to (a, b) from (x, y)
    static bool ispossible(int x, int y,
                           int a, int b)
    {
 
        // Find absolute values of all as
        // sign doesn't matter.
        x = Math.Abs(x);
        y = Math.Abs(y);
        a = Math.Abs(a);
        b = Math.Abs(b);
 
        // If gcd is equal then it is possible
        // to reach. Else not possible.
        return (gcd(x, y) == gcd(a, b));
    }
 
    // Driver code
    public static void Main()
    {
 
        // Converting coordinate
        // into positive integer
        int x = 35, y = 15;
        int a = 20, b = 25;
         
        if (ispossible(x, y, a, b))
            Console.Write("Yes");
        else
            Console.Write("No");
    }
}
 
// This code is contributed by nitin mittal.

PHP




<?php
// PHP program to check if it
// is possible to reach
// (a, b) from (x, y).
 
// Returns GCD of i and j
function gcd($i, $j)
{
    if ($i == $j)
        return $i;
 
    if ($i > $j)
        return gcd($i - $j, $j);
    return gcd($i, $j - $i);
}
 
// Returns true if it is
// possible to go to (a, b)
// from (x, y)
function ispossible($x, $y, $a, $b)
{
     
    // Find absolute values
    // of all as sign doesn't
    // matter.
    $x = abs($x);
    $y = abs($y);
    $a = abs($a);
    $b = abs($b);
 
    // If gcd is equal then
    // it is possible to reach.
    // Else not possible.
    return (gcd($x, $y) == gcd($a, $b));
}
 
// Driver Code
{
     
    // Converting coordinate
    // into positive integer
    $x = 35; $y = 15;
    $a = 20; $b = 25;
    if (ispossible($x, $y, $a, $b))
        echo( "Yes");
    else
        echo( "No");
    return 0;
}
 
// This code is contributed by nitin mittal.
?>

Javascript




<script>
// javascript program to check if it is possible
// to reach (a, b) from (x, y).   
// Returns GCD of i and j
    function gcd(i , j) {
        if (i == j)
            return i;
 
        if (i > j)
            return gcd(i - j, j);
        return gcd(i, j - i);
    }
 
    // Returns true if it is possible to go to (a, b)
    // from (x, y)
    function ispossible(x , y , a , b)
    {
 
        // Find absolute values of all as
        // sign doesn't matter.
        x = Math.abs(x);
        y = Math.abs(y);
        a = Math.abs(a);
        b = Math.abs(b);
 
        // If gcd is equal then it is possible to reach.
        // Else not possible.
        return (gcd(x, y) == gcd(a, b));
    }
 
    // Driver code
     
        // Converting coordinate into positive integer
        var x = 35, y = 15;
        var a = 20, b = 25;
        if (ispossible(x, y, a, b))
            document.write("Yes");
        else
            document.write("No");
 
// This code is contributed by todaysgaurav
</script>

Output

Yes

Time Complexity: O(min(x, y) + min(a, b)), where x, y, a and b are the given integers.
Auxiliary Space: O(min(x, y) + min(a, b)), space required due to the recursion stack.

This article is contributed by Anuj Chauhan(anuj0503). If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks. 


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