Solving a Linear Diophantine Equation(see description for examples)

Question

Let me start off by clarifying that(before you guys dismiss me), this is not a homework problem and I\'m not a university student. :)

EDIT Thanks to

Answer 1

A brute force algorithm is as follows (3 variable case):

int sum = 25;
int a1 = 3;
int a2 = 4;
int a3 = 5;
for (int i = 0; i * a1 <= sum; i++) {
    for (int j = 0; i * a1 + j * a2 <= sum; j++) {
        for (int k = 0; i * a1 + j * a2 + k * a3 <= sum; k++) {
            if (i * a1 + j * a2 + k * a3 == sum) {
                System.out.println(i + "," + j + "," + k);
            }
        }
    }
}

To generalize this for the N variable case, you need to convert into a recursive form.

This algorithm is O(f(size, a)^N) for some function f.

• We can place bounds on f as follows: size / MaxValue(a) <= f(size, a) <= size / MinValue(a).
• In the worst case (where all of the a[i]s are 1) f(size, a) is size.

Either way, this is pretty horrendous for large values of N. So while the recursive N variable algorithm would be more elegant, it is probably not very practical.

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If you are willing to fork out 34 Euro's to Springer Verlag, here's a link to a paper which (according to the abstract) includes an algorithm for solving the N variable case.