For a set P of n points in the plane and a value r > 0, the unit-disk range reporting problem is to construct a data structure so that given any query disk of radius r, all points of P in the disk can be reported efficiently. We consider the dynamic version of the problem where point insertions and deletions of P are allowed. The previous best method provides a data structure of O(n log n) space that supports O(log^{3+ε} n) amortized insertion time, O(log^{5+ε} n) amortized deletion time, and O(log² n/log log n+k) query time, where ε is an arbitrarily small positive constant and k is the output size. In this paper, we improve the query time to O(log n+k) while keeping other complexities the same as before. A key ingredient of our approach is a shallow cutting algorithm for circular arcs, which may be interesting in its own right. A related problem that can also be solved by our techniques is the dynamic unit-disk range emptiness queries: Given a query unit disk, we wish to determine whether the disk contains a point of P. The best previous work can maintain P in a data structure of O(n) space that supports O(log² n) amortized insertion time, O(log⁴n) amortized deletion time, and O(log² n) query time. Our new data structure also uses O(n) space but can support each update in O(log^{1+ε} n) amortized time and support each query in O(log n) time.
@InProceedings{wang_et_al:LIPIcs.STACS.2025.76, author = {Wang, Haitao and Zhao, Yiming}, title = {{Dynamic Unit-Disk Range Reporting}}, booktitle = {42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)}, pages = {76:1--76:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-365-2}, ISSN = {1868-8969}, year = {2025}, volume = {327}, editor = {Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.76}, URN = {urn:nbn:de:0030-drops-229019}, doi = {10.4230/LIPIcs.STACS.2025.76}, annote = {Keywords: Unit disks, range reporting, range emptiness, alpha-hulls, dynamic data structures, shallow cuttings} }
Feedback for Dagstuhl Publishing