스타인의 알고리즘 또는 이진 GCD 알고리즘 음이 아닌 두 정수의 최대 공약수를 계산하는 알고리즘입니다. Stein의 알고리즘은 나눗셈을 산술 이동 비교 및 뺄셈으로 대체합니다.
예:
입력 : a = 17 b = 34
산출 : 17
입력 : a = 50 b = 49
산출 : 1
Stein의 알고리즘 gcd(a b)를 사용하여 GCD를 찾는 알고리즘
알고리즘은 주로 표준에 대한 최적화입니다. GCD에 대한 유클리드 알고리즘
- a와 b가 모두 0이면 gcd는 0입니다. gcd(0 0) = 0입니다.
- gcd(a 0) = a 및 gcd(0 b) = b 왜냐하면 모든 것이 0으로 나누어지기 때문입니다.
- a와 b가 모두 짝수인 경우 gcd(a b) = 2*gcd(a/2 b/2)는 2가 공약수이기 때문입니다. 2와의 곱셈은 비트 시프트 연산자를 사용하여 수행할 수 있습니다.
- a가 짝수이고 b가 홀수이면 gcd(a b) = gcd(a/2 b). 마찬가지로 a가 홀수이고 b가 짝수이면
gcd(ab) = gcd(ab/2). 2는 공약수가 아니기 때문이다. - a와 b가 모두 홀수이면 gcd(a b) = gcd(|a-b|/2 b)입니다. 두 홀수의 차이는 짝수라는 점에 유의하세요.
- a = b 또는 a = 0이 될 때까지 3~5단계를 반복합니다. 두 경우 모두 GCD는 power(2 k) * b입니다. 여기서 power(2 k)는 2의 k 거듭제곱이고 k는 3단계에서 찾은 2의 공통 인자 수입니다.
// Iterative C++ program to // implement Stein's Algorithm #include
// 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
# 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.
// 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
<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>
// 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 ?> 산출
Gcd of given numbers is 17
시간 복잡도: O(N*N)
보조 공간: 오(1)
[예상 접근방식 2] 재귀적 구현 - O(N*N) 시간과 O(N*N) 공간
C++// Recursive C++ program to // implement Stein's Algorithm #include
// 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
# 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.
// 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.
<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>
// 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 ?> 산출
Gcd of given numbers is 17
시간 복잡도 : O(N*N) 여기서 N은 더 큰 숫자의 비트 수입니다.
보조 공간: O(N*N) 여기서 N은 더 큰 숫자의 비트 수입니다.
당신은 또한 좋아할 수도 있습니다 - 기본 및 확장 유클리드 알고리즘
유클리드의 GCD 알고리즘에 비해 장점
- Stein의 알고리즘은 Euclid의 GCD 알고리즘의 최적화된 버전입니다.
- 비트 시프트 연산자를 사용하면 더 효율적입니다.