Euclid’s Algorithm when % and / operations are costly
Last Updated :
27 Feb, 2023
Euclid’s algorithm is used to find GCD of two numbers.
There are mainly two versions of algorithm.
Version 1 (Using subtraction)
C++
int gcd( int a, int b)
{
if (a == b)
return a;
return (a > b) ? gcd(a - b, b) : gcd(a, b - a);
}
|
C
int gcd( int a, int b)
{
if (a == b)
return a;
return (a > b)? gcd(a-b, b): gcd(a, b-a);
}
|
Java
static int gcd( int a, int b)
{
if (a == b)
return a;
return (a > b) ? gcd(a - b, b) : gcd(a, b - a);
}
|
Python3
def gcd(a, b):
if (a = = b):
return a
return gcd(a - b, b) if (a > b) else gcd(a, b - a)
|
C#
static int gcd( int a, int b)
{
if (a == b)
return a;
return (a > b) ? gcd(a - b, b) : gcd(a, b - a);
}
|
Javascript
function gcd(a, b)
{
if (a === b)
return a;
return (a > b)? gcd(a-b, b): gcd(a, b-a);
}
|
Time Complexity: O(max(a, b))
Auxiliary Space: O(1)
Version 2 (Using modulo operator)
C++
static int gcd( int a, int b)
{
if (a == 0)
return b;
return gcd(b % a, a);
}
|
C
int gcd( int a, int b)
{
if (a == 0)
return b;
return gcd(b%a, a);
}
|
Java
static int gcd( int a, int b)
{
if (a == 0 )
return b;
return gcd(b%a, a);
}
|
C#
static int gcd( int a, int b)
{
if (a == 0)
return b;
return gcd(b%a, a);
}
|
Javascript
function gcd(a, b)
{
if (a === 0)
return b;
return gcd(b%a, a);
}
|
Python3
def gcd(a, b):
if (a = = 0 ):
return b
return gcd(b % a, a)
|
Time Complexity: O(log(max(a, b)))
Auxiliary Space: O(1)
Which of the above two is more efficient?
Version 1 can take linear time to find the GCD, consider the situation when one of the given numbers is much bigger than the other. Version 2 is obviously more efficient as there are less recursive calls and takes logarithmic time.
Consider a situation where modulo operator is not allowed, can we optimize version 1 to work faster?
Below are some important observations. The idea is to use bitwise operators. We can find x/2 using x>>1. We can check whether x is odd or even using x&1.
gcd(a, b) = 2*gcd(a/2, b/2) if both a and b are even.
gcd(a, b) = gcd(a/2, b) if a is even and b is odd.
gcd(a, b) = gcd(a, b/2) if a is odd and b is even.
Below is C++ implementation.
C++
int gcd( int a, int b)
{
if (b == 0 || a == b)
return a;
if (a == 0)
return b;
if ((a & 1) == 0 && (b & 1) == 0)
return gcd(a >> 1, b >> 1) << 1;
if ((a & 1) == 0 && (b & 1) != 0)
return gcd(a >> 1, b);
if ((a & 1) != 0 && (b & 1) == 0)
return gcd(a, b >> 1);
return (a > b) ? gcd(a - b, b) : gcd(a, b - a);
}
|
C
int gcd( int a, int b)
{
if (b == 0 || a == b) return a;
if (a == 0) return b;
if ( (a & 1) == 0 && (b & 1) == 0 )
return gcd(a>>1, b>>1) << 1;
if ( (a & 1) == 0 && (b & 1) != 0 )
return gcd(a>>1, b);
if ( (a & 1) != 0 && (b & 1) == 0 )
return gcd(a, b>>1);
return (a > b)? gcd(a-b, b): gcd(a, b-a);
}
|
Java
import java.util.*;
class GFG {
static int gcd( int a, int b)
{
if (b == 0 || a == b)
return a;
if (a == 0 )
return b;
if ((a & 1 ) == 0 && (b & 1 ) == 0 )
return gcd(a >> 1 , b >> 1 ) << 1 ;
if ((a & 1 ) == 0 && (b & 1 ) != 0 )
return gcd(a >> 1 , b);
if ((a & 1 ) != 0 && (b & 1 ) == 0 )
return gcd(a, b >> 1 );
return (a > b) ? gcd(a - b, b) : gcd(a, b - a);
}
public static void main(String[] args)
{
System.out.println(gcd( 54 , 36 ));
}
}
|
Python3
def gcd(a, b):
if (b = = 0 or a = = b):
return a
if (a = = 0 ):
return b
if ((a & 1 ) = = 0 and (b & 1 ) = = 0 ):
return gcd(a >> 1 , b >> 1 ) * 2
if ((a & 1 ) = = 0 and (b & 1 ) ! = 0 ):
return gcd(a >> 1 , b)
if ((a & 1 ) ! = 0 and (b & 1 ) = = 0 ):
return gcd(a, b >> 1 )
return gcd(a - b, b) if a > b else gcd(a, b - a)
|
C#
using System;
class GFG
{
int gcd( int a, int b)
{
if (b == 0 || a == b) return a;
if (a == 0) return b;
if ( (a & 1) == 0 && (b & 1) == 0 )
return gcd(a>>1, b>>1) << 1;
if ( (a & 1) == 0 && (b & 1) != 0 )
return gcd(a>>1, b);
if ( (a & 1) != 0 && (b & 1) == 0 )
return gcd(a, b>>1);
return (a > b)? gcd(a-b, b): gcd(a, b-a);
}
}
|
Javascript
function gcd(a, b)
{
if (b == 0 || a == b) return a;
if (a == 0) return b;
if ( (a & 1) == 0 && (b & 1) == 0 )
return gcd(a>>1, b>>1) << 1;
if ( (a & 1) == 0 && (b & 1) != 0 )
return gcd(a>>1, b);
if ( (a & 1) != 0 && (b & 1) == 0 )
return gcd(a, b>>1);
return (a > b)? gcd(a-b, b): gcd(a, b-a);
}
|
Time Complexity: O(log(max(a, b)))
Auxiliary Space: O(1)
This article is compiled by Shivam Agrawal.
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