Quickest Visibility Queries in Polygonal Domains

Abstract

Let s be a point in a polygonal domain ${\mathcal {P}}$ of $h-1$ holes and n vertices. We consider a quickest visibility query problem. Given a query point q in ${\mathcal {P}}$, the goal is to find a shortest path in ${\mathcal {P}}$ to move from s to seeq as quickly as possible. Previously, Arkin et al. (SoCG 2015) built a data structure of size $O(n^22^{\alpha (n)}\log n)$ that can answer each query in $O(K\log ^2 n)$ time, where $\alpha (n)$ is the inverse Ackermann function and K is the size of the visibility polygon of q in ${\mathcal {P}}$ (and K can be $\varTheta (n)$ in the worst case). In this paper, we present a new data structure of size $O(n\log h + h^2)$ that can answer each query in $O(h\log h\log n)$ time. Our result improves the previous work when h is relatively small. In particular, if h is a constant, then our result even matches the best result for the simple polygon case (i.e., $h=1$), which is optimal. As a by-product, we also have a new algorithm for a shortest-path-to-segment query problem. Given a query line segment $\tau$ in ${\mathcal {P}}$, the query seeks a shortest path from s to all points of $\tau$. Previously, Arkin et al. gave a data structure of size $O(n^22^{\alpha (n)}\log n)$ that can answer each query in $O(\log ^2 n)$ time, and another data structure of size $O(n^3\log n)$ with $O(\log n)$ query time. We present a data structure of size O(n) with query time $O\big (h\log \frac{n}{h}\big )$, which also favors small values of h and is optimal when $h=O(1)$.