Check if possible to move from given coordinate to desired coordinate

06 Jul, 2022
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++
Java
Python3
C#
PHP
Javascript
// 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 jint 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 Programint 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 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.
# Python program to check if it is possible to reach# (a, b) from (x, y).# Returns GCD of i and jdef 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 integerx, y = 35, 15a, b = 20, 25if(ispossible(x, y, a, b)):    print ("Yes")else:    print ("No")# Contributed by Afzal Ansari
// 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 program to check if it// is possible to reach// (a, b) from (x, y). // Returns GCD of i and jfunction 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.?>
<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>
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.
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