A Tail Estimate with Exponential Decay for the Randomized Incremental Construction of Search Structures

TL;DR

This work introduces a novel Pairwise-Events tail-analysis to study Randomized Incremental Construction (RIC) of search structures, yielding high-probability guarantees previously unavailable. The authors prove that trapezoidal search DAGs, as well as history DAGs for 2D Delaunay triangulations and 3D convex hulls, have size $O(n)$ and depth $O(\log n)$ with construction time $O(n\log n)$, enabling simple Las Vegas verifiers to achieve worst-case optimality. The results resolve longstanding conjectures about logarithmic query costs in these DAGs and extend to crossing-segment configurations, achieving robust space bounds and enabling efficient point-location bounds. Overall, the approach provides practical, provably optimal data-structure constructions with strong probabilistic guarantees, broadening the applicability of RIC methods in geometric data-structure design.

Abstract

The Randomized Incremental Construction (RIC) of search DAGs for point location in planar subdivisions, nearest-neighbor search in 2D points, and extreme point search in 3D convex hulls, are well known to take ${\cal O}(n \log n)$ expected time for structures of ${\cal O}(n)$ expected size. Moreover, searching takes w.h.p. ${\cal O}(\log n)$ comparisons in the first and w.h.p. ${\cal O}(\log^2 n)$ comparisons in the latter two DAGs. However, the expected depth of the DAGs and high probability bounds for their size are unknown. Using a novel analysis technique, we show that the three DAGs have w.h.p. i) a size of ${\cal O}(n)$, ii) a depth of ${\cal O}(\log n)$, and iii) a construction time of ${\cal O}(n \log n)$. One application of these new and improved results are \emph{remarkably simple} Las Vegas verifiers to obtain search DAGs with optimal worst-case bounds. This positively answers the conjectured logarithmic search cost in the DAG of Delaunay triangulations [Guibas et al.; ICALP 1990] and a conjecture on the depth of the DAG of Trapezoidal subdivisions [Hemmer et al.; ESA 2012].

Figures (5)

• Figure 1: Trapezoidations over the segments $S=\{a,b,c,d\}$, with $a=(a.l,a.r)$, $b=(b.l,b.r)$, $c=(c.l,c.r)$, and $d=(d.l,d.r)\}$, where $c.l=d.l$ is a common endpoint. $\mathcal{T}(\{a\})$, $\mathcal{T}(\{a,b\})$, $\mathcal{T}(\{a,b,c\})$, and $\mathcal{T}(\{a,b,c,d\})$ have $4$, $7$, $10$, and $13$ faces respectively (cf. Figure \ref{['fig:example-tsd']}).
• Figure 2: TSD for the history of trapezoidations under permutation $\pi = \bigl(abcd1234\bigr)$ from Figure \ref{['fig:example-trapezoidations']}. TSD node $v$ corresponds to the trapezoid $\Delta(v)$, which has the boundaries $\textit{top}(\Delta(v))=c$, $\textit{bot}(\Delta(v))=b$, $\textit{left}(\Delta(v))=a.r$, and $\textit{right}(\Delta(v))=b.r$ and the spatially empty $\Delta(u)$ is due to common endpoint left$(\Delta(u))=c.l=d.l=\textit{right}(\Delta(u))$. The path with heavy line width is not a search path, since $d.r$ is left of $a.r$.
• Figure 3: Insertion order for non-crossing segments that results in a TSD with depth $\Omega(n)$ and $\Omega(3^{n/2})$ root-to-leaf paths.
• Figure 4: Example of absent events (white) and occurring events (gray). The sequence $\sigma=(1,3,6,7)$ is feasible since $X_{1,3},X_{3,6}$, and $X_{6,7}$, occur (tiled circles). A total of six events occur on $\sigma$, i.e. $\sigma(\pi)=6$.
• Figure 5: Schema of a search DAG using the radial-search for 2D Delaunay triangulations (left) and 3D convex hulls (right).

Theorems & Definitions (14)

• Lemma 3.1
• Theorem 3.1
• Corollary 3.1
• Corollary 3.2
• Corollary 3.3
• Lemma 4.1
• Lemma 4.2
• Theorem 4.1
• Lemma 5.1
• Theorem 5.1