Algorithm Summary-Euclidean Algorithm

Algorithm Summary-Euclidean Algorithm

1. Euclidean Algorithm

Euclidean algorithm, also known as the moving phase division, is used to calculate the maximum approximate number of two positive integers A and B.

The computation principle depends on the following theorem:

　　Gcd (a, B) = gcd (B, A mod B) (A> B and A mod B is not 0)

Code implementation:

1 int gcd(int a,int b)2 {3     return b==0?a:gcd(b,a%b);4 }

2. Extended Euclidean Algorithm

Basic Algorithms:

　　For non-negative integers a, B, gcd (A, B) with an incomplete value of 0, it indicates the maximum approximate number of A and B. There must be an integer pair X and Y, making gcd (A, B) = AX +.
Proof:
Let's assume that a positive integer of the equation AX + by = gcd (a, B) = D is interpreted as X1, Y1. Do not doubt that this equation must have a solution.

Then ax1 + by1 = gcd (A, B) (1)

For the equation bx + (a % B) y = gcd (B, A % B), there is a solution X2, Y2 (hypothesis)

Bx2 + (a % B) y2 = gcd (B, A % B) = gcd (A, B) (2)

And a % B = A-(A/B) * B;

Then the formula (2) is changed to bx2 + (a-(A/B) * B) y2 = gcd (A, B );

That is, ay2 + B (x2-(A/B) * Y2) = gcd (a, B) (3 );

Comparison (1) (3)

X1 = Y2; Y1 = x2-(A/B) * Y2

Therefore, the solution of AX + by = gcd (a, B) only needs to be in the equation bx + (a % B) y = gcd (B, A % B) on the basis of the original equation,

Because GCD always recurs When B = 0, the solution of the equation can be obtained through recursion.

Code implementation:

1 LL extended_gcd (ll a, LL B, ll & X, ll & Y) // The returned value is gcd (a, B) 2 {3 LL ret, TMP; 4 If (B = 0) 5 {6 x = 1, y = 0; 7 return a; 8} 9 ret = extended_gcd (B, A % B, X, y); 10 TMP = x; 11 x = y; 12 y = TMP-A/B * Y; 13 return ret; 14}

3. Some conclusions

1) The Equation AX + by = C meets the condition: c = K * gcd (A, B) (k is an integer), then the equation has an integer solution. Otherwise, there is no solution.

2) Set A, B, and C to any integer. If a group of integers in the equation AX + by = C is interpreted as (x0, y0), any integer solution of it can be written as (x0 + kb ', y0-ka '), here, A' = A/gcd (a, B), B '= B/gcd (a, B), and K are any integers.

3) if a group of Integer Solutions (x1, x2) is set to X0 = x1 % B 'Y0 = Y1 % B', x0 and Y0 are the closest to zero solutions.