# Towards Space Efficient Two-Point Shortest Path Queries in a Polygonal Domain

## TL;DR

This paper introduces new space-efficient data structures for two-point shortest path queries in polygonal domains, improving upon previous methods with faster query times and reduced space complexity, especially for boundary-restricted queries.

## Contribution

The authors develop the first data structures that significantly improve space complexity for two-point shortest path queries in polygonal domains, with flexible trade-offs and special case optimizations.

## Key findings

- Achieved $O(n^{10+ ext{ε}})$ space with $O( ext{log} n)$ query time.
- Reduced space to $O(n^{9+ ext{ε}})$ with $O( ext{log}^2 n)$ query time.
- Improved boundary-restricted query data structures with lower space complexity.

## Abstract

We devise a data structure that can answer shortest path queries for two query points in a polygonal domain $P$ on $n$ vertices. For any $\varepsilon > 0$, the space complexity of the data structure is $O(n^{10+\varepsilon })$ and queries can be answered in $O(\log n)$ time. Alternatively, we can achieve a space complexity of $O(n^{9+\varepsilon })$ by relaxing the query time to $O(\log^2 n)$. This is the first improvement upon a conference paper by Chiang and Mitchell from 1999. They present a data structure with $O(n^{11})$ space complexity and $O(\log n)$ query time. Our main result can be extended to include a space-time trade-off. Specifically, we devise data structures with $O(n^{9+\varepsilon}/\hspace{1pt} \ell^{4 + O(\varepsilon )})$ space complexity and $O(\ell \log^2 n )$ query time, for any integer $1 \leq \ell \leq n$. Furthermore, we present improved data structures with $O(\log n)$ query time for the special case where we restrict one (or both) of the query points to lie on the boundary of $P$. When one of the query points is restricted to lie on the boundary, and the other query point is unrestricted, the space complexity becomes $O(n^{6+\varepsilon})$. When both query points are on the boundary, the space complexity is decreased further to $O(n^{4+\varepsilon })$, thereby improving an earlier result of Bae and Okamoto.