## Abstract

Let S⊆ R^{2} be a set of n sites in the plane, so that every site s∈ S has an associated radius r_{s}> 0 . Let D(S) be the disk intersection graph defined by S, i.e., the graph with vertex set S and an edge between two distinct sites s, t∈ S if and only if the disks with centers s, t and radii r_{s} , r_{t} intersect. Our goal is to design data structures that maintain the connectivity structure of D(S) as sites are inserted and/or deleted in S. First, we consider unit disk graphs, i.e., we fix r_{s}= 1 , for all sites s∈ S . For this case, we describe a data structure that has O(log ^{2}n) amortized update time and O(log n/ log log n) query time. Second, we look at disk graphs with bounded radius ratio Ψ , i.e., for all s∈ S , we have 1 ≤ r_{s}≤ Ψ , for a parameter Ψ that is known in advance. Here, we not only investigate the fully dynamic case, but also the incremental and the decremental scenario, where only insertions or only deletions of sites are allowed. In the fully dynamic case, we achieve amortized expected update time O(Ψ log ^{4}n) and query time O(log n/ log log n) . This improves the currently best update time by a factor of Ψ . In the incremental case, we achieve logarithmic dependency on Ψ , with a data structure that has O(α(n)) amortized query time and O(log Ψ log ^{4}n) amortized expected update time, where α(n) denotes the inverse Ackermann function. For the decremental setting, we first develop an efficient decremental disk revealing data structure: given two sets R and B of disks in the plane, we can delete disks from B, and upon each deletion, we receive a list of all disks in R that no longer intersect the union of B. Using this data structure, we get decremental data structures with a query time of O(log n/ log log n) that supports deletions in O(nlog Ψ log ^{4}n) overall expected time for disk graphs with bounded radius ratio Ψ and O(nlog ^{5}n) overall expected time for disk graphs with arbitrary radii, assuming that the deletion sequence is oblivious of the internal random choices of the data structures.

Original language | English |
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Pages (from-to) | 214-277 |

Number of pages | 64 |

Journal | Discrete and Computational Geometry |

Volume | 71 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2024 |

### Bibliographical note

Publisher Copyright:© 2024, The Author(s).

### Funding

Alexander Baumann: Supported in part by grant 1367/2016 from the German-Israeli Science Foundation (GIF), by the German Research Foundation within the collaborative DACH project Arrangements and Drawings as DFG Project MU 3501/3-1, and by ERC StG 757609. Wolfgang Mulzer: Supported in part by ERC StG 757609. Supported in part by grant 1367/2016 from the German-Israeli Science Foundation (GIF). Haim Kaplan: Partially supported by ISF grant 1595/19 and by the Blavatnik research foundation. Kristin Knorr: Supported by the German Science Foundation within the research training group ‘Facets of Complexity’ (GRK 2434).

Funders | Funder number |
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Blavatnik research foundation | |

German–Israeli Science Foundation | |

Blavatnik Family Foundation | |

European Research Council | 757609 |

Deutsche Forschungsgemeinschaft | MU 3501/3-1, GRK 2434 |

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

Israel Science Foundation | 1595/19 |

## Keywords

- Connectivity
- Disk graphs
- Lower envelopes