## Abstract

We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include L_{p}-norms and additively weighted Euclidean distances. Our data structure supports general (convex, pairwise disjoint) sites that have constant description complexity (e.g., points, line segments, disks, etc.). Our structure uses O(nlog ^{3}n) storage, and requires polylogarithmic update and query time, improving an earlier data structure of Agarwal, Efrat, and Sharir which required O(n^{ε}) time for an update and O(log n) time for a query [SICOMP 1999]. Our data structure has numerous applications. In all of them, it gives faster algorithms, typically reducing an O(n^{ε}) factor in the previous bounds to polylogarithmic. In addition, we give here two new applications: an efficient construction of a spanner in a disk intersection graph, and a data structure for efficient connectivity queries in a dynamic disk graph. To obtain this data structure, we combine and extend various techniques from the literature. Along the way, we obtain several side results that are of independent interest. Our data structure depends on the existence and an efficient construction of “vertical” shallow cuttings in arrangements of bivariate algebraic functions. We prove that an appropriate level in an arrangement of a random sample of a suitable size provides such a cutting. To compute it efficiently, we develop a randomized incremental construction algorithm for computing the lowest k levels in an arrangement of bivariate algebraic functions (we mostly consider here collections of functions whose lower envelope has linear complexity, as is the case in the dynamic nearest-neighbor context, under both types of norm). To analyze this algorithm, we also improve a longstanding bound on the combinatorial complexity of the vertical decomposition of these levels. Finally, to obtain our structure, we combine our vertical shallow cutting construction with Chan’s algorithm for efficiently maintaining the lower envelope of a dynamic set of planes in R^{3}. Along the way, we also revisit Chan’s technique and present a variant that uses a single binary counter, with a simpler analysis and improved amortized deletion time (by a logarithmic factor; the insertion and query costs remain asymptotically the same).

Original language | English |
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Pages (from-to) | 838-904 |

Number of pages | 67 |

Journal | Discrete and Computational Geometry |

Volume | 64 |

Issue number | 3 |

DOIs | |

State | Published - 1 Oct 2020 |

### Bibliographical note

Publisher Copyright:© 2020, The Author(s).

### Funding

A preliminary version appeared as []. Work by Haim Kaplan, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth has been supported by Grants 1161/2011 and (with Micha Sharir) 1367/2016 from the German–Israeli Science Foundation. Work by Haim Kaplan has also been supported by Grants 822-10 and 1841-14 from the Israel Science Foundation, and by the Israeli Centers for Research Excellence (I-CORE) program (Center No. 4/11). Work by Wolfgang Mulzer and Paul Seiferth has also been supported by grant MU/3501/1 from Deutsche Forschungsgemeinschaft (DFG) and by ERC StG 757609. Work by Micha Sharir has been supported by Grant 2012/229 from the U.S.–Israel Binational Science Foundation, by Grants 892/13 and 260/18 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.

Funders | Funder number |
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German–Israeli Science Foundation | |

Horizon 2020 Framework Programme | 757609 |

European Commission | 2012/229 |

Deutsche Forschungsgemeinschaft | |

German-Israeli Foundation for Scientific Research and Development | 1367/2016, 1161/2011 |

United States-Israel Binational Science Foundation | 260/18 |

Israel Science Foundation | 822-10, 1841-14, 892/13 |

Tel Aviv University | |

Israeli Centers for Research Excellence | 4/11, MU/3501/1 |

## Keywords

- Dynamic structure
- General distance functions
- Voronoi diagram