Dynamic Connectivity in Disk Graphs

Alexander Baumann, Haim Kaplan, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, Paul Seiferth

Research output: Contribution to journalArticlepeer-review

Abstract

Let S⊆ R2 be a set of n sites in the plane, so that every site s∈ S has an associated radius rs> 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 rs , rt 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 rs= 1 , for all sites s∈ S . For this case, we describe a data structure that has O(log 2n) 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 ≤ rs≤ 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 4n) 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 4n) 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 4n) overall expected time for disk graphs with bounded radius ratio Ψ and O(nlog 5n) 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 languageEnglish
Pages (from-to)214-277
Number of pages64
JournalDiscrete and Computational Geometry
Volume71
Issue number1
DOIs
StatePublished - 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).

FundersFunder number
Blavatnik research foundation
German–Israeli Science Foundation
Blavatnik Family Foundation
European Research Council757609
Deutsche ForschungsgemeinschaftMU 3501/3-1, GRK 2434
German-Israeli Foundation for Scientific Research and Development1367/2016
Israel Science Foundation1595/19

    Keywords

    • Connectivity
    • Disk graphs
    • Lower envelopes

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