## 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-δ√2k}) time and using O(n^{δ}) space, for 0 ≤ δ ≤ 1. (ii) The best known deterministic solution to kSUM running in approximately O(n^{k-δ√k}) time and using O(nδ) space, for 0 ≤ δ ≤ 1. (iii) A space-time tradeoff for solving kSUM using O(n^{δ}) space, for δ > 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.

Original language | English |
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Title of host publication | 26th European Symposium on Algorithms, ESA 2018 |

Editors | Hannah Bast, Grzegorz Herman, Yossi Azar |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Print) | 9783959770811 |

DOIs | |

State | Published - 1 Aug 2018 |

Event | 26th European Symposium on Algorithms, ESA 2018 - Helsinki, Finland Duration: 20 Aug 2018 → 22 Aug 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 112 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 26th European Symposium on Algorithms, ESA 2018 |
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Country/Territory | Finland |

City | Helsinki |

Period | 20/08/18 → 22/08/18 |

### Bibliographical note

Publisher Copyright:© Isaac Goldstein, Moshe Lewenstein, and Ely Porat.

### Funding

This research is supported by the Adams Foundation of the Israel Academy of Sciences and Humanities. 2 This work was partially supported by an ISF grant #1278/16, BSF grant 2010437 and a GIF grant 1147/2011. 3 This work was partially supported by an ISF grant #1278/16.

Funders | Funder number |
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German-Israeli Foundation for Scientific Research and Development | |

United States-Israel Binational Science Foundation | 2010437 |

Israel Academy of Sciences and Humanities | |

Israel Science Foundation | 1278/16 |

## Keywords

- KSUM
- Self-reduction
- Space-time tradeoff