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(n2lglg(n)/lg(n)) and space
O(lglg(n)nlg(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