## Abstract

We show that the k-SUM problem can be solved by a linear decision tree of depth O(n^{2} log^{2} n), improving the recent bound O(n^{3} log^{3} n) of Cardinal et al. [7]. Our bound depends linearly on k, and allows us to conclude that the number of linear queries required to decide the n-dimensional KNAPSACK or SUBSETSUM problems is only O(n^{3} log n), improving the currently best known bounds by a factor of n [28, 29]. Our algorithm extends to the RAM model, showing that the k-SUM problem can be solved in expected polynomial time, for any fixed k, with the above bound on the number of linear queries. Our approach relies on a new point-location mechanism, exploiting "ϵ-cuttings" that are based on vertical decompositions in hyperplane arrangements in high dimensions. A major side result of the analysis in this paper is a sharper bound on the complexity of the vertical decomposition of such an arrangement (in terms of its dependence on the dimension). We hope that this study will reveal further structural properties of vertical decompositions in hyperplane arrangements.

Original language | English |
---|---|

Title of host publication | 33rd International Symposium on Computational Geometry, SoCG 2017 |

Editors | Matthew J. Katz, Boris Aronov |

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

Pages | 411-4115 |

Number of pages | 3705 |

ISBN (Electronic) | 9783959770385 |

DOIs | |

State | Published - 1 Jun 2017 |

Externally published | Yes |

Event | 33rd International Symposium on Computational Geometry, SoCG 2017 - Brisbane, Australia Duration: 4 Jul 2017 → 7 Jul 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 77 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 33rd International Symposium on Computational Geometry, SoCG 2017 |
---|---|

Country/Territory | Australia |

City | Brisbane |

Period | 4/07/17 → 7/07/17 |

### Bibliographical note

Publisher Copyright:© Esther Ezra and Micha Sharir.

## Keywords

- Hyperplane arrangements
- K-SUM and k-LDT
- Linear decision tree
- Point-location
- Vertical decompositions
- ϵ-cuttings