You are given a set S = {s1,s2,...,sn} of n distinct natural numbers such that 0 ≤ si ≤ 100n. Your task is to design an algorithm that takes as input S and a natural number N, and outputs True if the equation si + sj + sk = N has at least one solution, and return False otherwise. There is a simple solution that runs in O(n3) time, but I hope you improve that time using FFT. Example: For N = 6 and S = {1,2,3,5,10} your design should output True since 1+2+3 = 6. For N = 20 and the same set S the answer should be True again since 5+5+10 = 20 (yes, you can have si = sj = sk!) but for N = 19 the answer is False since no three numbers add up to 19.
You can use FFT as a black box. Explicitly write the polynomials you will use as input for FFT and explain how you use the output to answer your problem. State and justify the running time of your algorithm.