Fast Low-Space Algorithms for Subset Sum

Abstract: We consider the canonical Subset Sum problem: given a list of positive integers $a_1,\ldots,a_n$ and a target integer $t$ with $t &gt; a_i$ for all $i$, determine if there is an $S \subseteq [n]$ such that $\sum_{i \in S} a_i = t$. The well-known pseudopolynomial-time dynamic programming algorithm [Bellman, 1957] solves Subset Sum in $O(nt)$ time, while requiring $\Omega(t)$ space. In this paper we present algorithms for Subset Sum with $\tilde O(nt)$ running time and much lower space requirements than Bellman's algorithm, as well as that of prior work. We show that Subset Sum can be solved in $\tilde O(nt)$ time and $O(\log(nt))$ space with access to $O(\log n \log \log n+\log t)$ random bits. This significantly improves upon the $\tilde O(n t<sup>{1+\varepsilon})$-time, $\tilde O(n\log t)$-space algorithm of Bringmann (SODA 2017). We also give an $\tilde O(n<sup>{1+\varepsilon}t)$-time, $O(\log(nt))$-space randomized algorithm, improving upon previous $(nt)<sup>{O(1)}$-time $O(\log(nt))$-space algorithms by Elberfeld, Jakoby, and Tantau (FOCS 2010), and Kane (2010). In addition, we also give a $\mathrm{poly} \log(nt)$-space, $\tilde O(n² t)$-time deterministic algorithm. We also study time-space trade-offs for Subset Sum. For parameter $1\le k\le \min{n,t}$, we present a randomized algorithm running in $\tilde O((n+t)\cdot k)$ time and $O((t/k) \mathrm{polylog} (nt))$ space. As an application of our results, we give an $$\tilde{O}(\min{n^{2/\varepsilon,} n/\varepsilon<sup>2})$$-time and $\mathrm{polylog}(nt)$-space algorithm for "weak" $\varepsilon$-approximations of Subset Sum.