Computing $k$-Crossing Visibility through $k$-levels

Abstract

Let $\mathcal{A}$ be a set of straight lines in the plane (or planes in $\mathbb{R}^3$). The $k$-crossing visibility of a point $p$ on $\mathcal{A}$ is the set $Q$ of points in the elements of $\mathcal{A}$ such that the segment $pq$, where $q\in Q$, intersects at most $k$ elements of $\mathcal{A}$. In this paper, we present algorithms for computing the $k$-crossing visibility. Specifically, we provide $O(n\log n + kn)$ and $O(n\log n + k^2n)$ time algorithms for sets of $n$ lines in the plane and arrangements of $n$ planes in $\mathbb{R}^3$, which are optimal for $k=\Omega(\log n)$ and $k=\Omega(\sqrt{\log n})$, respectively. We also introduce an algorithm for computing $k$-crossing visibilities on polygons, which achieves the same asymptotic time complexity as the one presented by Bahoo et al. The techniques proposed in this paper can be easily adapted for computing $k$-crossing visibilities on other instances where the $(\leq k)$-level is known.