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(n^2*lg(lg(n))/lg(n)) and space O(sqrt(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(n^2) and space O(sqrt(n)), derandomizing an algorithm of Wang.
- A popular conjecture states that 3-SUM requires n^{2-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 n^{2-o(1)} time on the word-RAM.
- For k >= 4, k-SUM is in deterministic O(n^{k-2+2/k}) time and O(sqrt(n)) space.
3SUM
kSUM
time-space tradeoff
algorithm
58:1-58:14
Regular Paper
Andrea
Lincoln
Andrea Lincoln
Virginia
Vassilevska Williams
Virginia Vassilevska Williams
Joshua R.
Wang
Joshua R. Wang
R. Ryan
Williams
R. Ryan Williams
10.4230/LIPIcs.ICALP.2016.58
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