Steinov algoritem za iskanje GCD
Preizkusite na GfG Practice 
Izhod
Izhod
Steinov algoritem oz binarni algoritem GCD je algoritem, ki izračuna največji skupni delitelj dveh nenegativnih celih števil. Steinov algoritem nadomešča deljenje z aritmetičnimi premiki, primerjavami in odštevanjem.
Primeri:
Vnos : a = 17 b = 34
Izhod : 17
Vnos : a = 50 b = 49
Izhod : 1
Algoritem za iskanje GCD z uporabo Steinovega algoritma gcd(a b)
Algoritem je v glavnem optimizacija standarda Evklidski algoritem za GCD
- Če sta a in b 0, je gcd nič gcd(0 0) = 0.
- gcd(a 0) = a in gcd(0 b) = b, ker vse deli 0.
- Če sta a in b soda, je gcd(a b) = 2*gcd(a/2 b/2), ker je 2 skupni delitelj. Množenje z 2 je mogoče izvesti z bitnim operaterjem premika.
- Če je a sodo in b liho, je gcd(a b) = gcd(a/2 b). Podobno je, če je a liho in b sodo
gcd(a b) = gcd(a b/2). To je zato, ker 2 ni skupni delitelj. - Če sta a in b liha, potem je gcd(a b) = gcd(|a-b|/2 b). Upoštevajte, da je razlika dveh lihih števil soda
- Ponavljajte korake 3–5, dokler ni a = b ali dokler ni a = 0. V obeh primerih je GCD potenca (2 k) * b, kjer je potenca (2 k) 2, povišana na potenco k, k pa je število skupnih faktorjev 2, najdenih v koraku 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 ?>
Izhod
Gcd of given numbers is 17
Časovna zapletenost: O(N*N)
Pomožni prostor: O(1)
[Pričakovan pristop 2] Rekurzivna implementacija - O(N*N) Čas in O(N*N) Vesolje
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 ?>
Izhod
Gcd of given numbers is 17
Časovna zapletenost : O(N*N), kjer je N število bitov v večjem številu.
Pomožni prostor: O(N*N), kjer je N število bitov v večjem številu.
Morda vam bo všeč tudi - Osnovni in razširjeni evklidski algoritem
Prednosti pred Evklidovim algoritmom GCD
- Steinov algoritem je optimizirana različica Evklidovega algoritma GCD.
- bolj učinkovito je z uporabo operatorja bitnega premika.
