Subset Sum in Near-Linear Pseudopolynomial Time and Polynomial Space

Abstract: Given a multiset $A = {a_1, \dots, a_n}$ of positive integers and a target integer $t$, the Subset Sum problem asks if there is a subset of $A$ that sums to $t$. Bellman's [1957] classical dynamic programming algorithm runs in $O(nt)$ time and $O(t)$ space. Since then, there have been multiple improvements in both time and space complexity. Notably, Bringmann [SODA 2017] uses a two-step color-coding technique to obtain a randomized algorithm that runs in $\tilde{O}(n+t)$ time and $\tilde{O}(t)$ space. On the other hand, there are polynomial space algorithms -- for example, Jin, Vyas and Williams [SODA 2021] build upon the algorithm given by Bringmann, using a clever algebraic trick first seen in Kane's Logspace algorithm, to obtain an $\tilde{O}(nt)$ time and $\tilde{O}(\log(nt))$ space algorithm. A natural question, asked by Jin et al. is if there is an $\tilde{O}(n+t)$ time algorithm running in poly$(n, \log t)$ space. Another natural question is whether it is possible to construct a deterministic polynomial space algorithm with time complexity comparable to that of Bellman's. In this paper, we answer both questions affirmatively. We build on the framework given by Jin et al., using a multipoint evaluation-based approach to speed up a bottleneck step in their algorithm. We construct a deterministic algorithm that runs in $\tilde{O}(nt)$ time and $\tilde{O}(n \log^2 t)$ space and a randomized algorithm that runs in $\tilde{O}(n+t)$ time and $\tilde{O}(n^2 + n \log^2 t)$ space.