### Abstract

The k-SUM problem is given n input real numbers to determine whether any k of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within P, and it is in particular open whether it admits an algorithm of complexity O(n^{c}) with c < ⌊k/2⌋. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth O(n^{3} log^{2} n) solving k-SUM. Furthermore, we show that there exists a randomized algorithm that runs in Õ(n⌊k/2⌋+8) time, and performs O(n^{3} log^{2} n) linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the +8) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of k. The O(n^{3} log^{2} n) bound on the number of linear queries is also a tighter bound than any known algorithm solving k-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-à-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-P. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist o(n)-linear decision trees of depth Õ(n^{3}) for the k-SUM problem.

Original language | English (US) |
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Title of host publication | 24th Annual European Symposium on Algorithms, ESA 2016 |

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

Volume | 57 |

ISBN (Electronic) | 9783959770156 |

DOIs | |

State | Published - Aug 1 2016 |

Event | 24th Annual European Symposium on Algorithms, ESA 2016 - Aarhus, Denmark Duration: Aug 22 2016 → Aug 24 2016 |

### Other

Other | 24th Annual European Symposium on Algorithms, ESA 2016 |
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Country | Denmark |

City | Aarhus |

Period | 8/22/16 → 8/24/16 |

### Fingerprint

### Keywords

- H-SUM problem
- Linear decision trees
- Point location
- ϵ-nets

### ASJC Scopus subject areas

- Software

### Cite this

*24th Annual European Symposium on Algorithms, ESA 2016*(Vol. 57). [25] Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ESA.2016.25

**Solving k-SUM using few linear queries.** / Cardinal, Jean; Iacono, John; Ooms, Aurélien.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*24th Annual European Symposium on Algorithms, ESA 2016.*vol. 57, 25, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 24th Annual European Symposium on Algorithms, ESA 2016, Aarhus, Denmark, 8/22/16. https://doi.org/10.4230/LIPIcs.ESA.2016.25

}

TY - GEN

T1 - Solving k-SUM using few linear queries

AU - Cardinal, Jean

AU - Iacono, John

AU - Ooms, Aurélien

PY - 2016/8/1

Y1 - 2016/8/1

N2 - The k-SUM problem is given n input real numbers to determine whether any k of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within P, and it is in particular open whether it admits an algorithm of complexity O(nc) with c < ⌊k/2⌋. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth O(n3 log2 n) solving k-SUM. Furthermore, we show that there exists a randomized algorithm that runs in Õ(n⌊k/2⌋+8) time, and performs O(n3 log2 n) linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the +8) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of k. The O(n3 log2 n) bound on the number of linear queries is also a tighter bound than any known algorithm solving k-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-à-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-P. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist o(n)-linear decision trees of depth Õ(n3) for the k-SUM problem.

AB - The k-SUM problem is given n input real numbers to determine whether any k of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within P, and it is in particular open whether it admits an algorithm of complexity O(nc) with c < ⌊k/2⌋. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth O(n3 log2 n) solving k-SUM. Furthermore, we show that there exists a randomized algorithm that runs in Õ(n⌊k/2⌋+8) time, and performs O(n3 log2 n) linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the +8) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of k. The O(n3 log2 n) bound on the number of linear queries is also a tighter bound than any known algorithm solving k-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-à-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-P. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist o(n)-linear decision trees of depth Õ(n3) for the k-SUM problem.

KW - H-SUM problem

KW - Linear decision trees

KW - Point location

KW - ϵ-nets

UR - http://www.scopus.com/inward/record.url?scp=85012977552&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85012977552&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2016.25

DO - 10.4230/LIPIcs.ESA.2016.25

M3 - Conference contribution

VL - 57

BT - 24th Annual European Symposium on Algorithms, ESA 2016

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -