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

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 $O(n^{3\mathrm{/}2+\epsilon })$ preprocessing time and space, so that we can count the number of ranges containing a query point in $O(n^{1\mathrm{/}4+\epsilon })$ time, for an arbitrarily small constant $\epsilon >0$. 2. Ray shooting amid algebraic arcs. We present a data structure for $n$ algebraic arcs in 2D of constant description complexity with $O(n^{3\mathrm{/}2+\epsilon })$ preprocessing time and space, so that we can find the first arc hit by a query (straight-line) ray in $O(n^{1\mathrm{/}4+\epsilon })$ time. 3. Intersection counting amid algebraic arcs. We present a data structure for $n$ algebraic arcs in 2D of constant description complexity with $O(n^{3\mathrm{/}2+\epsilon })$ preprocessing time and space, so that we can count the number of intersection points with a query algebraic arc of constant description complexity in $O(n^{1\mathrm{/}2+\epsilon })$ time. In particular, this implies an $O(n^{3\mathrm{/}2+\epsilon })$-time algorithm for counting intersections between two sets of $n$ algebraic arcs in 2D.