Dynamic Unit-Disk Range Reporting

TL;DR

This work advances dynamic unit-disk range reporting by achieving $O(\log n+k)$ query time while preserving $O(n\log n)$ space and $O(\log^{3+\varepsilon} n)$ insertions and $O(\log^{5+\varepsilon} n)$ deletions, a bound that matches the implicit lower bounds for static problems and drives practical dynamic queries. The approach centers on maintaining a conforming coverage of the point set via a grid of small axis-aligned cells and employing shallow cuttings for circular arcs to support arc reporting; these arcs underpin a dynamic line-separable UDRR subproblem and enable efficient fully dynamic reporting. A key technical contribution is a vertex-segment form shallow cutting for arc sets, which the authors leverage to construct fast, space-efficient data structures, including a dynamic unit-disk emptiness structure with $O(n)$ space and $O(\log n)$ query time plus $O(\log^{1+\varepsilon} n)$ amortized updates. The static problem is also addressed using simpler, self-contained methods based on line-separable reporting and lower envelopes, yielding results that match prior work but with greater simplicity. Overall, the paper introduces practical dynamic machinery for unit-disk range queries and related problems, with potential impact on related geometric data-structure tasks and dynamic nearest-neighbor settings.

Abstract

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 $O(\log^2 n/\log\log n+k)$ 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 $O(\log^2 n)$ amortized insertion time, $O(\log^4n)$ amortized deletion time, and $O(\log^2 n)$ 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.

Figures (26)

• Figure 1: The point-zones lie in the grey area. The black dots are points of $P$.
• Figure 2: Illustrating the case where $C^*$ is a regular cell. The red box is $\boxplus_{C^*}$. The two blue vertical lines are vertical point-zone bounding lines; the left one is $\ell^*$. All grey cells are in vertical point-zones while cells are in vertical gap zones.
• Figure 4: Illustrating the case where $C^*$ is in a vertical gap-zone. The red box is $\boxplus_{C^*}$. All grey cells are in vertical point-zones while white cells are in vertical gap zones. $\ell_1$ and $\ell_2$ are respectively the left and right bounding lines of the vertical gap-zone containing $C^*$.
• Figure 5: Illustration the boundary of $H_{\ell}(Q)$, where $Q$ is the set of points below the $x$-axis $\ell$. It consists of three (blue) dashed horizontal line segments of $y$-coordinates $-1$, four (red) dotted $\mathbb{R}^+$-constrained arcs with centers on $\ell$, and four other solid $\mathbb{R}^+$-constrained arcs. The region below the boundary is $H_{\ell}(Q)$.
• Figure 6: Illustration the wings of the a point $q$. The two (red) dotted curves are wing arcs and the two (blue) dashed segments are wing half-lines. $p$ and $p'$ are the left and right wing vertices, respectively.

Theorems & Definitions (26)

• Definition 1
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• Lemma 4
• Theorem 2
• Lemma 5
• Lemma 6
• Lemma 7