
TL;DR
This paper presents improved dynamic data structures for unit-disk range reporting and emptiness queries in the plane, achieving faster query times while maintaining efficient update and space complexities.
Contribution
It introduces a new approach with a shallow cutting algorithm for circular arcs, significantly improving query times for dynamic unit-disk range reporting and emptiness queries.
Findings
Query time for range reporting improved to O(log n + k)
Supports dynamic updates with efficient amortized times
Maintains linear space complexity
Abstract
For a set of points in the plane and a value , the unit-disk range reporting problem is to construct a data structure so that given any query disk of radius , all points of in the disk can be reported efficiently. We consider the dynamic version of the problem where point insertions and deletions of are allowed. The previous best method provides a data structure of space that supports amortized insertion time, amortized deletion time, and query time, where is an arbitrarily small positive constant and is the output size. In this paper, we improve the query time to 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…
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