# Quickest Visibility Queries in Polygonal Domains

## TL;DR

This paper introduces efficient data structures for quickest visibility and shortest-path-to-segment queries in polygonal domains with holes, improving query times and sizes especially when the number of holes is small.

## Contribution

The authors develop new data structures that significantly improve query efficiency for visibility and shortest-path problems in polygonal domains with few holes, matching optimal results in simple cases.

## Key findings

- New data structure of size O(n log h + h^2) for visibility queries
- Query time of O(h log h log n) for visibility, better for small h
- Optimal data structure for shortest-path-to-segment queries when h=O(1)

## 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 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 $\Theta(n)$ 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 $O(h\log \frac{n}{h})$, which also favors small values of $h$ and is optimal when $h=O(1)$.