Space-Efficient Randomized Algorithms for K-SUM

Recent results by Dinur et al. (2012) and Austrin et al. (2013) have improved the time-space tradeoff curve for SubsetSum. We analyze a family of hash functions previously introduced by Dietzfelbinger, and apply it to decompose arbitrary k-Sum instances into smaller ones. This allows us to extend the aforementioned tradeoff curve to the k-Sum problem, which is SubsetSum restricted to sets of size k. Three consequences are: – a Las Vegas algorithm solving 3-Sum in O(n) time and Õ( √ n) space, – a Monte Carlo algorithm solving k-Sum in Õ(nk− √ ) time and Õ(n) space for k ≥ 3, and – a Monte Carlo algorithm solving k-Sum in Õ(nk−δ(k−1) + nk−1−δ( √ 2k−2)) time and Õ(n) space, for δ ∈ [0, 1] and k ≥ 3.