🤖 AI Summary
This paper addresses the lack of high-probability bounds on depth and size for randomized incremental construction (RIC) search structures—such as DAGs used in planar point location, 2D nearest-neighbor search, and 3D convex hull extreme-point queries. We propose a unified probabilistic analysis framework that innovatively combines martingale difference sequences with exponential moment estimation. For the first time, we establish tight exponential tail bounds for three canonical geometric DAGs, rigorously proving that, with probability $1 - 2^{-Omega(n)}$, each achieves size $O(n)$, depth $O(log n)$, and construction time $O(n log n)$. This confirms long-standing conjectures regarding Delaunay and trapezoidal-map DAGs. Furthermore, our analysis yields a succinct Las Vegas verifier that guarantees optimal worst-case search performance.
📝 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].
