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Improved Space-Time Tradeoffs for kSUM

Authors Isaac Goldstein, Moshe Lewenstein, Ely Porat

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Author Details

Isaac Goldstein
  • Bar-Ilan University, Ramat Gan, Israel
Moshe Lewenstein
  • Bar-Ilan University, Ramat Gan, Israel
Ely Porat
  • Bar-Ilan University, Ramat Gan, Israel

Funding

  • Goldstein, Isaac: This research is supported by the Adams Foundation of the Israel Academy of Sciences and Humanities.
  • Lewenstein, Moshe: This work was partially supported by an ISF grant #1278/16, BSF grant 2010437 and a GIF grant 1147/2011.
  • Porat, Ely: This work was partially supported by an ISF grant #1278/16.

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Isaac Goldstein, Moshe Lewenstein, and Ely Porat. Improved Space-Time Tradeoffs for kSUM. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ESA.2018.37

Abstract

In the kSUM problem we are given an array of numbers a_1,a_2,...,a_n and we are required to determine if there are k different elements in this array such that their sum is 0. This problem is a parameterized version of the well-studied SUBSET-SUM problem, and a special case is the 3SUM problem that is extensively used for proving conditional hardness. Several works investigated the interplay between time and space in the context of SUBSET-SUM. Recently, improved time-space tradeoffs were proven for kSUM using both randomized and deterministic algorithms.
In this paper we obtain an improvement over the best known results for the time-space tradeoff for kSUM. A major ingredient in achieving these results is a general self-reduction from kSUM to mSUM where m<k, and several useful observations that enable this reduction and its implications. The main results we prove in this paper include the following: (i) The best known Las Vegas solution to kSUM running in approximately O(n^{k-delta sqrt{2k}}) time and using O(n^{delta}) space, for 0 <= delta <= 1. (ii) The best known deterministic solution to kSUM running in approximately O(n^{k-delta sqrt{k}}) time and using O(n^{delta}) space, for 0 <= delta <= 1. (iii) A space-time tradeoff for solving kSUM using O(n^{delta}) space, for delta>1. (iv) An algorithm for 6SUM running in O(n^4) time using just O(n^{2/3}) space. (v) A solution to 3SUM on random input using O(n^2) time and O(n^{1/3}) space, under the assumption of a random read-only access to random bits.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • kSUM
  • space-time tradeoff
  • self-reduction

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