Quickest Visibility Queries in Polygonal Domains

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 see q 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 $$h=O(1)$$.