Semialgebraic Range Stabbing, Ray Shooting, and Intersection Counting in the Plane

TL;DR

This paper presents a data structure for algebraic arcs in 2D of constant description complexity with preprocessing time and space, so that the number of intersection points with a query algebraic arc of constant description complexity in $O(n^{1/4+\varepsilon})$ time is counted.

Abstract

Polynomial partitioning techniques have recently led to improved geometric data structures for a variety of fundamental problems related to semialgebraic range searching and intersection searching in 3D and higher dimensions (e.g., see [Agarwal, Aronov, Ezra, and Zahl, SoCG 2019; Ezra and Sharir, SoCG 2021; Agarwal, Aronov, Ezra, Katz, and Sharir, SoCG 2022]). They have also led to improved algorithms for offline versions of semialgebraic range searching in 2D, via lens-cutting [Sharir and Zahl (2017)]. In this paper, we show that these techniques can yield new data structures for a number of other 2D problems even for online queries: 1. Semialgebraic range stabbing. We present a data structure for $n$ semialgebraic ranges in 2D of constant description complexity with $O(n^{3/2+\varepsilon})$ preprocessing time and space, so that we can count the number of ranges containing a query point in $O(n^{1/4+\varepsilon})$ time, for an arbitrarily small constant $\varepsilon>0$. 2. Ray shooting amid algebraic arcs. We present a data structure for $n$ algebraic arcs in 2D of constant description complexity with $O(n^{3/2+\varepsilon})$ preprocessing time and space, so that we can find the first arc hit by a query (straight-line) ray in $O(n^{1/4+\varepsilon})$ time. 3. Intersection counting amid algebraic arcs. We present a data structure for $n$ algebraic arcs in 2D of constant description complexity with $O(n^{3/2+\varepsilon})$ preprocessing time and space, so that we can count the number of intersection points with a query algebraic arc of constant description complexity in $O(n^{1/2+\varepsilon})$ time. In particular, this implies an $O(n^{3/2+\varepsilon})$-time algorithm for counting intersections between two sets of $n$ algebraic arcs in 2D.

Figures (3)

• Figure 1: The blue line shows the prior known trade-off curve for semialgebraic range stabbing, and the green curve shows the improved trade-off curve we obtain. The dotted red lines show the lower bounds of Afshani a13 for simplex stabbing and Afshani and Cheng ac21 for semialgebraic range stabbing, both of which apply.
• Figure 2: The line $\ell$ intersecting $\kappa$ intersects $\rho_v$ and $\gamma$.
• Figure 3: Three different ways a ray $\rho$ can intersect an arc $\gamma$. The left depicts Case A, the middle depicts Case B, and the right depicts Case C.

Theorems & Definitions (24)

• lemma 1: $(1/r)$-Cutting Lemma
• theorem 1: Matoušek's Partition Theorem m92a
• theorem 2: Lens cutting for algebraic curves
• theorem 3
• theorem 4
• proof
• proof : Proof of \ref{['thm:ps-counting']}
• corollary 1
• definition 1
• theorem 5