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# Deterministic Time-Space Trade-Offs for k-SUM

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LIPIcs.ICALP.2016.58.pdf
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## Cite As

Andrea Lincoln, Virginia Vassilevska Williams, Joshua R. Wang, and R. Ryan Williams. Deterministic Time-Space Trade-Offs for k-SUM. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ICALP.2016.58

## Abstract

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
• algorithm

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