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

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 $O(\log² n)$. 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 $O(\ell \log² n )$ 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.