Add Linear Diophantine Equation

Signed-off-by: Bruno Freitas Tissei <bft15@inf.ufpr.br>

algorithms/math/extended_euclidean.cpp

 /// Extended Euclidean algorithm
 /// 
-/// Time: O(log n)
+/// Time: O(log min(a,b))
 /// Space: O(1)
 
 struct ExtendedEuclidean {

algorithms/math/linear_diophantine_equation.cpp

+/// Linear Diophantine Equation
+///
+/// Description:
+///   A Linear Diophantine Equation is an equation in the form $ax + by = c$.
+/// A solution of this equation is a pair $(x,y)$ that satisfies the equation. 
+/// The locus of (lattice) points whose coordinates $x$ and $y$ satisfy the 
+/// equation is a straigh line. \par
+///   The equation has a solution only if $gcd(a,b) | c$. In the case of 
+/// existing a solution for the provided $a, b, c$, the infinite set of
+/// coordinates $(x,y)$ can be obtained with get(t) for 
+/// $t=...,-2,-1,0,1,2,...$.
+///
+/// Time: O(log min(a,b))
+/// Space: O(1)
+
+struct Diophantine {
+  int a, b, c, d;
+  int x0, y0;
+
+  bool has_solution;
+
+  Diophantine(int a, int b, int c) :
+    a(a), b(b), c(c) 
+  { init(); }
+
+  void init() {
+    ExtendedEuclidean ext_gcd;
+
+    int w0, z0;
+    d = ext_gcd.run(a, b, w0, z0);
+    if (c % d == 0) {
+      x0 = w0 * (c / d);
+      y0 = z0 * (c / d);
+      has_solution = true;
+    } else {
+      has_solution = false;
+    }
+  }
+
+  ii get(int t) {
+    if (!has_solution) return ii(inf, inf);
+    return ii(x0 + t * (b / d), y0 - t * (a / d));
+  }
+};