Deterministic Time-Space Trade-Offs for k-SUM

Given a set of numbers, the k-SUM problem asks for a subset of k numbers that sums to zero. When the numbers are integers, the time and space complexity of k-SUM is generally studied in the word-RAM model; when the numbers are reals, the complexity is studied in the real-RAM model, and space is measured by the number of reals held in memory at any point. We present a time and space efficient deterministic self-reduction for the k-SUM problem which holds for both models, and has many interesting consequences. To illustrate: 3-SUM is in deterministic time O(n2 lg lg(n)/ lg(n)) and space O (√ n lg(n) lg lg(n) ) . In general, any polylogarithmic-time improvement over quadratic time for 3-SUM can be converted into an algorithm with an identical time improvement but low space complexity as well. 3-SUM is in deterministic time O(n2) and space O( √ n), derandomizing an algorithm of Wang. A popular conjecture states that 3-SUM requires n2−o(1) time on the word-RAM. We show that the 3-SUM Conjecture is in fact equivalent to the (seemingly weaker) conjecture that every O(n.51)-space algorithm for 3-SUM requires at least n2−o(1) time on the word-RAM. For k ≥ 4, k-SUM is in deterministic O(nk−2+2/k) time and O( √ n) space. 1998 ACM Subject Classification F.2.1 Numerical Algorithms and Problems