Check if a destination is reachable from source with two movements allowed | Set 2

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Given a pair of coordinates (X1, Y1)(source) and (X2, Y2)(destination), the task is to check if it is possible to reach the destination form the source by the following movements from any cell (X, Y):

(X + Y, Y)
(X, Y + X)

Note: All coordinates are positive and can be as large as 10¹⁸.

Examples:

Input: X1 = 2, Y1 = 10, X2 = 26, Y2 = 12
Output: Yes

Explanation: Possible path: (2, 10) ? (2, 12) ? (14, 12) ? (26, 12)
Therefore, a path exists between source and destination.

Input: X1 = 20, Y1 = 10, X2 = 6, Y2 = 12
Output: No

Naive Approach: The simplest approach to solve the problem is by using recursion. Refer to the article check if a destination is reachable from source with two movements allowed for the recursive approach.

Efficient Approach: The main idea is to check if a path from the destination coordinates (X2, Y2) to the source (X1, Y1) exists or not.

Follow the steps below to solve the problem:

Keep subtracting the smallest of (X2, Y2) from the largest of (X2, Y2) and stop if X2 becomes less than X1 or Y2 becomes less than Y1.
Now, compare (X1, Y1) and modified (X2, Y2). If X1 is equal to X2 and Y1 is equal to Y2, then print “Yes“.
If X1 is not equal to X2 or Y1 is equal, not Y2, then print “No“.

To optimize the complexity of the subtraction operation, the modulus operation can be used instead. Simply perform x2 = x2 % y2 and y2 = y2 % x2 and check for the necessary condition mentioned above.

C++

// C++ program to implement
// the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Check if (x2, y2) can be reached
// from (x1, y1)
bool isReachable(long long x1, long long y1,
                 long long x2, long long y2)
{
    while (x2 > x1 && y2 > y1) {
 
        // Reduce x2 by y2 until it is
        // less than or equal to x1
        if (x2 > y2)
            x2 %= y2;
 
        // Reduce y2 by x2 until it is
        // less than or equal to y1
        else
            y2 %= x2;
    }
 
    // If x2 is reduced to x1
    if (x2 == x1)
 
        // Check if y2 can be
        // reduced to y1 or not
        return (y2 - y1) >= 0
               && (y2 - y1) % x1 == 0;
 
    // If y2 is reduced to y1
    else if (y2 == y1)
 
        // Check if x2 can be
        // reduced to x1 or not
        return (x2 - x1) >= 0
               && (x2 - x1) % y1 == 0;
    else
        return 0;
}
 
// Driver Code
int main()
{
    long long source_x = 2, source_y = 10;
    long long dest_x = 26, dest_y = 12;
 
    if (isReachable(source_x, source_y,
                    dest_x, dest_y))
        cout << "Yes";
    else
        cout << "No";
    return 0;
}

Python3

# Python3 program to implement
# the above approach
 
# Check if (x2, y2) can be reached
# from (x1, y1)
def isReachable(x1, y1, x2, y2):
 
    while(x2 > x1 and y2 > y1):
 
        # Reduce x2 by y2 until it is
        # less than or equal to x1
        if(x2 > y2):
            x2 %= y2
 
        # Reduce y2 by x2 until it is
        # less than or equal to y1
        else:
            y2 %= x2
 
    # If x2 is reduced to x1
    if(x2 == x1):
 
        # Check if y2 can be
        # reduced to y1 or not
        return (y2 - y1) >= 0 and (
                y2 - y1) % x1 == 0
 
    # If y2 is reduced to y1
    elif(y2 == y1):
 
        # Check if x2 can be
        # reduced to x1 or not 
        return (x2 - x1) >= 0 and (
                x2 - x1) % y1 == 0
    else:
        return 0
 
# Driver Code
source_x = 2
source_y = 10
dest_x = 26
dest_y = 12
 
# Function call
if(isReachable(source_x, source_y,
                 dest_x, dest_y)):
    print("Yes")
else:
    print("No")
 
# This code is contributed by Shivam Singh

Java

// Java program to implement
// the above approach
class GFG{
 
// Check if (x2, y2) can be reached
// from (x1, y1)
static boolean isReachable(long x1, long y1,
                           long x2, long y2)
{
    while (x2 > x1 && y2 > y1) 
    {
        // Reduce x2 by y2 until it is
        // less than or equal to x1
        if (x2 > y2)
            x2 %= y2;
 
        // Reduce y2 by x2 until it is
        // less than or equal to y1
        else
            y2 %= x2;
    }
 
    // If x2 is reduced to x1
    if (x2 == x1)
 
        // Check if y2 can be
        // reduced to y1 or not
        return (y2 - y1) >= 0 && 
               (y2 - y1) % x1 == 0;
 
    // If y2 is reduced to y1
    else if (y2 == y1)
 
        // Check if x2 can be
        // reduced to x1 or not
        return (x2 - x1) >= 0 && 
               (x2 - x1) % y1 == 0;
    else
        return false;
}
 
// Driver Code
public static void main(String[] args)
{
    long source_x = 2, source_y = 10;
    long dest_x = 26, dest_y = 12;
 
    if (isReachable(source_x, source_y,
                    dest_x, dest_y))
        System.out.print("Yes");
    else
        System.out.print("No");
}
}
 
// This code is contributed by shikhasingrajput

C#

// C# program to implement
// the above approach
using System;
class GFG{
 
// Check if (x2, y2) can be reached
// from (x1, y1)
static bool isReachable(long x1, long y1,
                        long x2, long y2)
{
  while (x2 > x1 && 
         y2 > y1) 
  {
    // Reduce x2 by y2 
    // until it is less 
    // than or equal to x1
    if (x2 > y2)
      x2 %= y2;
 
    // Reduce y2 by x2 
    // until it is less 
    // than or equal to y1
    else
      y2 %= x2;
  }
 
  // If x2 is reduced to x1
  if (x2 == x1)
 
    // Check if y2 can be
    // reduced to y1 or not
    return (y2 - y1) >= 0 && 
           (y2 - y1) % x1 == 0;
 
  // If y2 is reduced to y1
  else if (y2 == y1)
 
    // Check if x2 can be
    // reduced to x1 or not
    return (x2 - x1) >= 0 && 
           (x2 - x1) % y1 == 0;
  else
    return false;
}
 
// Driver Code
public static void Main(String[] args)
{
  long source_x = 2, source_y = 10;
  long dest_x = 26, dest_y = 12;
 
  if (isReachable(source_x, source_y,
                  dest_x, dest_y))
    Console.Write("Yes");
  else
    Console.Write("No");
}
}
 
// This code is contributed by shikhasingrajput

Javascript

<script>
 
// JavaScript  program for
//the above approach
 
// Check if (x2, y2) can be reached
// from (x1, y1)
function isReachable(x1, y1, x2, y2)
{
    while (x2 > x1 && y2 > y1)
    {
        // Reduce x2 by y2 until it is
        // less than or equal to x1
        if (x2 > y2)
            x2 %= y2;
  
        // Reduce y2 by x2 until it is
        // less than or equal to y1
        else
            y2 %= x2;
    }
  
    // If x2 is reduced to x1
    if (x2 == x1)
  
        // Check if y2 can be
        // reduced to y1 or not
        return (y2 - y1) >= 0 &&
               (y2 - y1) % x1 == 0;
  
    // If y2 is reduced to y1
    else if (y2 == y1)
  
        // Check if x2 can be
        // reduced to x1 or not
        return (x2 - x1) >= 0 &&
               (x2 - x1) % y1 == 0;
    else
        return false;
}
  
// Driver Code
 
    let source_x = 2, source_y = 10;
    let dest_x = 26, dest_y = 12;
  
    if (isReachable(source_x, source_y,
                    dest_x, dest_y))
        document.write("Yes");
    else
       document.write("No");
   
</script>

Output:

Yes

Time Complexity: O(1)
Auxiliary Space: O(1)

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