Truly Low-Space Element Distinctness and Subset Sum via Pseudorandom Hash Functions

TL;DR

This paper develops low-space randomized algorithms for Element Distinctness and Subset Sum problems, removing the need for random oracles by using pseudorandom hash functions, achieving near-optimal time with minimal space.

Contribution

It introduces a pseudorandom hash family that replaces random oracles in low-space algorithms for Element Distinctness and Subset Sum, improving their practicality.

Findings

Achieves $ ilde O(n^{1.5})$-time algorithm with $O( ext{polylog}(n))$ space for Element Distinctness.

Removes the need for random oracles in existing algorithms, making them more feasible.

Provides a $ ext{poly}(n)$-space, $O^*(2^{0.86n})$-time algorithm for Subset Sum.

Abstract

We consider low-space algorithms for the classic Element Distinctness problem: given an array of $n$ input integers with $O(\log n)$ bit-length, decide whether or not all elements are pairwise distinct. Beame, Clifford, and Machmouchi [FOCS 2013] gave an $\tilde O(n^{1.5})$-time randomized algorithm for Element Distinctness using only $O(\log n)$ bits of working space. However, their algorithm assumes a random oracle (in particular, read-only random access to polynomially many random bits), and it was asked as an open question whether this assumption can be removed. In this paper, we positively answer this question by giving an $\tilde O(n^{1.5})$-time randomized algorithm using $O(\log ^3 n\log \log n)$ bits of space, with one-way access to random bits. As a corollary, we also obtain a $\operatorname{\mathrm{poly}}(n)$-space $O^*(2^{0.86n})$-time randomized algorithm for the Subset Sum problem, removing the random oracles required in the algorithm of Bansal, Garg, Nederlof, and Vyas [STOC 2017]. The main technique underlying our results is a pseudorandom hash family based on iterative restrictions, which can fool the cycle-finding procedure in the algorithms of Beame et al. and Bansal et al.