Space-efficient randomized algorithms for K-SUM

Abstract

Recent results by Dinur et al. (2012) on random SubsetSum instances and by Austrin et al. (2013) on worst-case SubsetSum instances have improved the long-standing time-space tradeoff curve. We analyze a family of hash functions previously introduced by Dietzfelbinger (1996), 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 2) time Õ(√n) and space, a Monte Carlo algorithm solving k -Sum Õ(nk-√2k+1) in 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(nδ)sup space, for δ [0, 1] and k≥3. © 2014 Springer-Verlag Berlin Heidelberg.