Sum of multiplication of all combination of m element from an array of n elements

Question

Suppose I have an array {1, 2, 5, 4} and m = 3. I need to find:

1*2*5 + 1*2*4 + 1*5*4 + 2*5*4

i.e Sum of multipli

Answer 1

I came up with the same problem. I found the solution via Vieta's Algorithm as well. I tweaked the algorithm not the calculate the coefficient of the polynomial which are not needed. The complexity is O(N*M). I looked into DFTs and FFTs but if M is small enough, this method will be even faster than Fast Fourier Transformation algorithms. This is java btw.

public BigInteger sumOfCombMult(Integer[] roots, int M)
{
    if (roots.length < M)
    {
        throw new IllegalArgumentException("size of roots cannot be smaller than M");
    }

    BigInteger[] R = new BigInteger[roots.length];

    for (int i = 0; i < roots.length; i++)
    {
        R[i] = BigInteger.valueOf(roots[i]);
    }

    BigInteger[] coeffs = new BigInteger[roots.length + 1];

    coeffs[0] = BigInteger.valueOf(roots[0]);

    int lb = 0;

    for (int i = 1; i < roots.length; i++)
    {
        lb = Math.max(i - M, 0);

        coeffs[i] = R[i].add(coeffs[i - 1]);

        for (int j = i - 1; j > lb; j--)
        {
            coeffs[j] = R[i].multiply(coeffs[j]).add(coeffs[j - 1]);

        }

        if (lb == 0)
        {
            coeffs[0] = coeffs[0].multiply(R[i]);
        }
    }

    return coeffs[roots.length - M];
}