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Given a multiset $S$ of $n$ positive integers and a target integer $t$, theSubset Sum problem asks to determine whether there exists a subset of $S$ thatsums up to $t$. The current best deterministic algorithm, by Koiliaris and Xu[SODA'17], runs in $\tilde O(\sqrt{n}t)$ time, where $\tilde O$ hidespoly-logarithm factors. Bringmann [SODA'17] later gave a randomized $\tilde O(n+ t)$ time algorithm using two-stage color-coding. The $\tilde O(n+t)$ runningtime is believed to be near-optimal.  In this paper, we present a simple and elegant randomized algorithm forSubset Sum in $\tilde O(n + t)$ time. Our new algorithm actually solves itscounting version modulo prime $p>t$, by manipulating generating functions usingFFT.

A Simple Near-Linear Pseudopolynomial Time Randomized Algorithm for Subset Sum