Improved space-time tradeoffs for kSUM

In the kSUM problem we are given an array of numbers a₁,a₂,⋯,a_n and we are required to determine if there are k different elements in this array such that their sum is 0. This problem is a parameterized version of the well-studied SUBSET-SUM problem, and a special case is the 3SUM problem that is extensively used for proving conditional hardness. Several works investigated the interplay between time and space in the context of SUBSET-SUM. Recently, improved time-space tradeoffs were proven for kSUM using both randomized and deterministic algorithms. In this paper we obtain an improvement over the best known results for the time-space tradeoff for kSUM. A major ingredient in achieving these results is a general self-reduction from kSUM to mSUM where m < k, and several useful observations that enable this reduction and its implications. The main results we prove in this paper include the following: (i) The best known Las Vegas solution to kSUM running in approximately O(n^k-δ√2k) time and using O(n^δ) space, for 0 ≤ δ ≤ 1. (ii) The best known deterministic solution to kSUM running in approximately O(n^k-δ√k) time and using O(nδ) space, for 0 ≤ δ ≤ 1. (iii) A space-time tradeoff for solving kSUM using O(n^δ) space, for δ > 1. (iv) An algorithm for 6SUM running in O(n⁴) time using just O(n²/³) space. (v) A solution to 3SUM on random input using O(n²) time and O(n¹/³) space, under the assumption of a random read-only access to random bits.