Algorithme de Stein pour trouver GCD
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L'algorithme de Stein ou algorithme binaire GCD est un algorithme qui calcule le plus grand commun diviseur de deux entiers non négatifs. L’algorithme de Stein remplace la division par des comparaisons de décalages arithmétiques et des soustractions.
Exemples :
Saisir : une = 17 b = 34
Sortir : 17
Saisir : une = 50 b = 49
Sortir : 1
Algorithme pour trouver GCD en utilisant l'algorithme de Stein pgcd(a b)
L'algorithme est principalement une optimisation par rapport au standard Algorithme euclidien pour GCD
- Si a et b valent tous deux 0, pgcd est égal à zéro pgcd(0 0) = 0.
- pgcd(a 0) = a et pgcd(0 b) = b car tout divise 0.
- Si a et b sont tous deux pairs pgcd(a b) = 2*gcd(a/2 b/2) car 2 est un diviseur commun. La multiplication par 2 peut être effectuée avec un opérateur de décalage au niveau du bit.
- Si a est pair et b est impair pgcd(a b) = pgcd(a/2 b). De même, si a est impair et b est pair alors
pgcd(un b) = pgcd(un b/2). C’est parce que 2 n’est pas un diviseur commun. - Si a et b sont impairs alors pgcd(a b) = pgcd(|a-b|/2 b). Notez que la différence entre deux nombres impairs est paire
- Répétez les étapes 3 à 5 jusqu'à ce que a = b ou jusqu'à a = 0. Dans les deux cas, le PGCD est puissance (2 k) * b où puissance (2 k) est 2 élevée à la puissance k et k est le nombre de facteurs communs de 2 trouvés à l'étape 3.
// Iterative C++ program to // implement Stein's Algorithm #include using namespace std ; // Function to implement // Stein's Algorithm int gcd ( int a int b ) { /* GCD(0 b) == b; GCD(a 0) == a GCD(0 0) == 0 */ if ( a == 0 ) return b ; if ( b == 0 ) return a ; /*Finding K where K is the greatest power of 2 that divides both a and b. */ int k ; for ( k = 0 ; (( a | b ) & 1 ) == 0 ; ++ k ) { a >>= 1 ; b >>= 1 ; } /* Dividing a by 2 until a becomes odd */ while (( a & 1 ) == 0 ) a >>= 1 ; /* From here on 'a' is always odd. */ do { /* If b is even remove all factor of 2 in b */ while (( b & 1 ) == 0 ) b >>= 1 ; /* Now a and b are both odd. Swap if necessary so a <= b then set b = b - a (which is even).*/ if ( a > b ) swap ( a b ); // Swap u and v. b = ( b - a ); } while ( b != 0 ); /* restore common factors of 2 */ return a < < k ; } // Driver code int main () { int a = 34 b = 17 ; printf ( 'Gcd of given numbers is %d n ' gcd ( a b )); return 0 ; }
Java // Iterative Java program to // implement Stein's Algorithm import java.io.* ; class GFG { // Function to implement Stein's // Algorithm static int gcd ( int a int b ) { // GCD(0 b) == b; GCD(a 0) == a // GCD(0 0) == 0 if ( a == 0 ) return b ; if ( b == 0 ) return a ; // Finding K where K is the greatest // power of 2 that divides both a and b int k ; for ( k = 0 ; (( a | b ) & 1 ) == 0 ; ++ k ) { a >>= 1 ; b >>= 1 ; } // Dividing a by 2 until a becomes odd while (( a & 1 ) == 0 ) a >>= 1 ; // From here on 'a' is always odd. do { // If b is even remove // all factor of 2 in b while (( b & 1 ) == 0 ) b >>= 1 ; // Now a and b are both odd. Swap // if necessary so a <= b then set // b = b - a (which is even) if ( a > b ) { // Swap u and v. int temp = a ; a = b ; b = temp ; } b = ( b - a ); } while ( b != 0 ); // restore common factors of 2 return a < < k ; } // Driver code public static void main ( String args [] ) { int a = 34 b = 17 ; System . out . println ( 'Gcd of given ' + 'numbers is ' + gcd ( a b )); } } // This code is contributed by Nikita Tiwari
Python # Iterative Python 3 program to # implement Stein's Algorithm # Function to implement # Stein's Algorithm def gcd ( a b ): # GCD(0 b) == b; GCD(a 0) == a # GCD(0 0) == 0 if ( a == 0 ): return b if ( b == 0 ): return a # Finding K where K is the # greatest power of 2 that # divides both a and b. k = 0 while ((( a | b ) & 1 ) == 0 ): a = a >> 1 b = b >> 1 k = k + 1 # Dividing a by 2 until a becomes odd while (( a & 1 ) == 0 ): a = a >> 1 # From here on 'a' is always odd. while ( b != 0 ): # If b is even remove all # factor of 2 in b while (( b & 1 ) == 0 ): b = b >> 1 # Now a and b are both odd. Swap if # necessary so a <= b then set # b = b - a (which is even). if ( a > b ): # Swap u and v. temp = a a = b b = temp b = ( b - a ) # restore common factors of 2 return ( a < < k ) # Driver code a = 34 b = 17 print ( 'Gcd of given numbers is ' gcd ( a b )) # This code is contributed by Nikita Tiwari.
C# // Iterative C# program to implement // Stein's Algorithm using System ; class GFG { // Function to implement Stein's // Algorithm static int gcd ( int a int b ) { // GCD(0 b) == b; GCD(a 0) == a // GCD(0 0) == 0 if ( a == 0 ) return b ; if ( b == 0 ) return a ; // Finding K where K is the greatest // power of 2 that divides both a and b int k ; for ( k = 0 ; (( a | b ) & 1 ) == 0 ; ++ k ) { a >>= 1 ; b >>= 1 ; } // Dividing a by 2 until a becomes odd while (( a & 1 ) == 0 ) a >>= 1 ; // From here on 'a' is always odd do { // If b is even remove // all factor of 2 in b while (( b & 1 ) == 0 ) b >>= 1 ; /* Now a and b are both odd. Swap if necessary so a <= b then set b = b - a (which is even).*/ if ( a > b ) { // Swap u and v. int temp = a ; a = b ; b = temp ; } b = ( b - a ); } while ( b != 0 ); /* restore common factors of 2 */ return a < < k ; } // Driver code public static void Main () { int a = 34 b = 17 ; Console . Write ( 'Gcd of given ' + 'numbers is ' + gcd ( a b )); } } // This code is contributed by nitin mittal
JavaScript < script > // Iterative JavaScript program to // implement Stein's Algorithm // Function to implement // Stein's Algorithm function gcd ( a b ) { /* GCD(0 b) == b; GCD(a 0) == a GCD(0 0) == 0 */ if ( a == 0 ) return b ; if ( b == 0 ) return a ; /*Finding K where K is the greatest power of 2 that divides both a and b. */ let k ; for ( k = 0 ; (( a | b ) & 1 ) == 0 ; ++ k ) { a >>= 1 ; b >>= 1 ; } /* Dividing a by 2 until a becomes odd */ while (( a & 1 ) == 0 ) a >>= 1 ; /* From here on 'a' is always odd. */ do { /* If b is even remove all factor of 2 in b */ while (( b & 1 ) == 0 ) b >>= 1 ; /* Now a and b are both odd. Swap if necessary so a <= b then set b = b - a (which is even).*/ if ( a > b ){ let t = a ; a = b ; b = t ; } b = ( b - a ); } while ( b != 0 ); /* restore common factors of 2 */ return a < < k ; } // Driver code let a = 34 b = 17 ; document . write ( 'Gcd of given numbers is ' + gcd ( a b )); // This code contributed by gauravrajput1 < /script>
PHP // Iterative php program to // implement Stein's Algorithm // Function to implement // Stein's Algorithm function gcd ( $a $b ) { // GCD(0 b) == b; GCD(a 0) == a // GCD(0 0) == 0 if ( $a == 0 ) return $b ; if ( $b == 0 ) return $a ; // Finding K where K is the greatest // power of 2 that divides both a and b. $k ; for ( $k = 0 ; (( $a | $b ) & 1 ) == 0 ; ++ $k ) { $a >>= 1 ; $b >>= 1 ; } // Dividing a by 2 until a becomes odd while (( $a & 1 ) == 0 ) $a >>= 1 ; // From here on 'a' is always odd. do { // If b is even remove // all factor of 2 in b while (( $b & 1 ) == 0 ) $b >>= 1 ; // Now a and b are both odd. Swap // if necessary so a <= b then set // b = b - a (which is even) if ( $a > $b ) swap ( $a $b ); // Swap u and v. $b = ( $b - $a ); } while ( $b != 0 ); // restore common factors of 2 return $a < < $k ; } // Driver code $a = 34 ; $b = 17 ; echo 'Gcd of given numbers is ' . gcd ( $a $b ); // This code is contributed by ajit ?>
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Gcd of given numbers is 17
Complexité temporelle : O(N*N)
Espace auxiliaire : O(1)
[Approche attendue 2] Implémentation récursive - O(N*N) Le temps et O(N*N) Espace
C++ // Recursive C++ program to // implement Stein's Algorithm #include using namespace std ; // Function to implement // Stein's Algorithm int gcd ( int a int b ) { if ( a == b ) return a ; // GCD(0 b) == b; GCD(a 0) == a // GCD(0 0) == 0 if ( a == 0 ) return b ; if ( b == 0 ) return a ; // look for factors of 2 if ( ~ a & 1 ) // a is even { if ( b & 1 ) // b is odd return gcd ( a >> 1 b ); else // both a and b are even return gcd ( a >> 1 b >> 1 ) < < 1 ; } if ( ~ b & 1 ) // a is odd b is even return gcd ( a b >> 1 ); // reduce larger number if ( a > b ) return gcd (( a - b ) >> 1 b ); return gcd (( b - a ) >> 1 a ); } // Driver code int main () { int a = 34 b = 17 ; printf ( 'Gcd of given numbers is %d n ' gcd ( a b )); return 0 ; }
Java // Recursive Java program to // implement Stein's Algorithm import java.io.* ; class GFG { // Function to implement // Stein's Algorithm static int gcd ( int a int b ) { if ( a == b ) return a ; // GCD(0 b) == b; GCD(a 0) == a // GCD(0 0) == 0 if ( a == 0 ) return b ; if ( b == 0 ) return a ; // look for factors of 2 if (( ~ a & 1 ) == 1 ) // a is even { if (( b & 1 ) == 1 ) // b is odd return gcd ( a >> 1 b ); else // both a and b are even return gcd ( a >> 1 b >> 1 ) < < 1 ; } // a is odd b is even if (( ~ b & 1 ) == 1 ) return gcd ( a b >> 1 ); // reduce larger number if ( a > b ) return gcd (( a - b ) >> 1 b ); return gcd (( b - a ) >> 1 a ); } // Driver code public static void main ( String args [] ) { int a = 34 b = 17 ; System . out . println ( 'Gcd of given' + 'numbers is ' + gcd ( a b )); } } // This code is contributed by Nikita Tiwari
Python # Recursive Python 3 program to # implement Stein's Algorithm # Function to implement # Stein's Algorithm def gcd ( a b ): if ( a == b ): return a # GCD(0 b) == b; GCD(a 0) == a # GCD(0 0) == 0 if ( a == 0 ): return b if ( b == 0 ): return a # look for factors of 2 # a is even if (( ~ a & 1 ) == 1 ): # b is odd if (( b & 1 ) == 1 ): return gcd ( a >> 1 b ) else : # both a and b are even return ( gcd ( a >> 1 b >> 1 ) < < 1 ) # a is odd b is even if (( ~ b & 1 ) == 1 ): return gcd ( a b >> 1 ) # reduce larger number if ( a > b ): return gcd (( a - b ) >> 1 b ) return gcd (( b - a ) >> 1 a ) # Driver code a b = 34 17 print ( 'Gcd of given numbers is ' gcd ( a b )) # This code is contributed # by Nikita Tiwari.
C# // Recursive C# program to // implement Stein's Algorithm using System ; class GFG { // Function to implement // Stein's Algorithm static int gcd ( int a int b ) { if ( a == b ) return a ; // GCD(0 b) == b; // GCD(a 0) == a // GCD(0 0) == 0 if ( a == 0 ) return b ; if ( b == 0 ) return a ; // look for factors of 2 // a is even if (( ~ a & 1 ) == 1 ) { // b is odd if (( b & 1 ) == 1 ) return gcd ( a >> 1 b ); else // both a and b are even return gcd ( a >> 1 b >> 1 ) < < 1 ; } // a is odd b is even if (( ~ b & 1 ) == 1 ) return gcd ( a b >> 1 ); // reduce larger number if ( a > b ) return gcd (( a - b ) >> 1 b ); return gcd (( b - a ) >> 1 a ); } // Driver code public static void Main () { int a = 34 b = 17 ; Console . Write ( 'Gcd of given' + 'numbers is ' + gcd ( a b )); } } // This code is contributed by nitin mittal.
JavaScript < script > // JavaScript program to // implement Stein's Algorithm // Function to implement // Stein's Algorithm function gcd ( a b ) { if ( a == b ) return a ; // GCD(0 b) == b; GCD(a 0) == a // GCD(0 0) == 0 if ( a == 0 ) return b ; if ( b == 0 ) return a ; // look for factors of 2 if (( ~ a & 1 ) == 1 ) // a is even { if (( b & 1 ) == 1 ) // b is odd return gcd ( a >> 1 b ); else // both a and b are even return gcd ( a >> 1 b >> 1 ) < < 1 ; } // a is odd b is even if (( ~ b & 1 ) == 1 ) return gcd ( a b >> 1 ); // reduce larger number if ( a > b ) return gcd (( a - b ) >> 1 b ); return gcd (( b - a ) >> 1 a ); } // Driver Code let a = 34 b = 17 ; document . write ( 'Gcd of given ' + 'numbers is ' + gcd ( a b )); < /script>
PHP // Recursive PHP program to // implement Stein's Algorithm // Function to implement // Stein's Algorithm function gcd ( $a $b ) { if ( $a == $b ) return $a ; /* GCD(0 b) == b; GCD(a 0) == a GCD(0 0) == 0 */ if ( $a == 0 ) return $b ; if ( $b == 0 ) return $a ; // look for factors of 2 if ( ~ $a & 1 ) // a is even { if ( $b & 1 ) // b is odd return gcd ( $a >> 1 $b ); else // both a and b are even return gcd ( $a >> 1 $b >> 1 ) < < 1 ; } if ( ~ $b & 1 ) // a is odd b is even return gcd ( $a $b >> 1 ); // reduce larger number if ( $a > $b ) return gcd (( $a - $b ) >> 1 $b ); return gcd (( $b - $a ) >> 1 $a ); } // Driver code $a = 34 ; $b = 17 ; echo 'Gcd of given numbers is: ' gcd ( $a $b ); // This code is contributed by aj_36 ?>
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Gcd of given numbers is 17
Complexité temporelle : O(N*N) où N est le nombre de bits dans le plus grand nombre.
Espace auxiliaire : O(N*N) où N est le nombre de bits dans le plus grand nombre.
Vous aimerez peut-être aussi - Algorithme euclidien de base et étendu
Avantages par rapport à l'algorithme GCD d'Euclide
- L'algorithme de Stein est une version optimisée de l'algorithme GCD d'Euclide.
- il est plus efficace en utilisant l'opérateur de décalage au niveau du bit.
