Finding sub-array sum in an integer array

Question

Given an array of N positive integers. It can have n*(n+1)/2 sub-arrays including single element sub-arrays. Each sub-array has a sum S. Find S\'

Answer 1

When S is not too large, we can count the distinct sums with one (fast) polynomial multiplication. When S is larger, N is hopefully small enough to use a quadratic algorithm.

Let x_1, x_2, ..., x_n be the array elements. Let y_0 = 0 and y_i = x_1 + x_2 + ... + x_i. Let P(z) = z^{y_0} + z^{y_1} + ... + z^{y_n}. Compute the product of polynomials P(z) * P(z^{-1}); the coefficient of z^k with k > 0 is nonzero if and only if k is a sub-array sum, so we just have to read off the number of nonzero coefficients of positive powers. The powers of z, moreover, range from -S to S, so the multiplication takes time on the order of S log S.