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. 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. Our main result can be extended to include a space-time trade-off. Specifically, we devise data structures with $O(n^{10+\varepsilon}/\hspace{1pt} \ell^{5 + O(\varepsilon)})$ space complexity and $O(\ell \log n )$ query time for any integer $1 \leq \ell \leq n$. Furthermore, our main result can be improved if 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 can still lie anywhere in $P$, the space complexity becomes $O(n^{6+\varepsilon})$. When both query points are on the boundary, the space is decreased further to $O(n^{4+\varepsilon})$, thereby improving an earlier result of Bae and Okamoto.

翻译：我们设计了一种数据结构，用于在多边形域 $P$ 中回答两个查询点之间的最短路径查询，其中 $P$ 包含 $n$ 个顶点。对于任意 $\varepsilon > 0$，该数据结构的空间复杂度为 $O(n^{10+\varepsilon})$，且可在 $O(\log n)$ 时间内回答查询。这是对 Chiang 和 Mitchell 于 1999 年发表的会议论文的首次改进。他们提出的数据结构空间复杂度为 $O(n^{11})$。我们的主要结果可扩展以包含空间-时间权衡。具体而言，我们设计了空间复杂度为 $O(n^{10+\varepsilon}/\hspace{1pt} \ell^{5 + O(\varepsilon)})$、查询时间为 $O(\ell \log n)$ 的数据结构，适用于任意整数 $1 \leq \ell \leq n$。此外，如果限制一个（或两个）查询点位于 $P$ 的边界上，我们的主要结果可进一步改进。当其中一个查询点限制在边界上，而另一个查询点仍可位于 $P$ 中的任意位置时，空间复杂度降至 $O(n^{6+\varepsilon})$。当两个查询点均位于边界上时，空间复杂度进一步降至 $O(n^{4+\varepsilon})$，从而改进了 Bae 和 Okamoto 的先前结果。