A Divide-and-Conquer Algorithm for Two-Point $L_1$ Shortest Path Queries in Polygonal Domains

Let $mathcal{P}$ be a polygonal domain of $h$ holes and $n$ vertices. We study the problem of constructing a data structure that can compute a shortest path between $s$ and $t$ in $mathcal{P}$ under the $L_1$ metric for any two query points $s$ and $t$. To do so, a standard approach is to first find a set of $n_s$ gateways for $s$ and a set of $n_t$ gateways for $t$ such that there exist a shortest $s$-$t$ path containing a gateway of $s$ and a gateway of $t$, and then compute a shortest $s$-$t$ path using these gateways. Previous algorithms all take quadratic $O(n_scdot n_t)$ time to solve this problem. In this paper, we propose a divide-and-conquer technique that solves the problem in $O(n_s + n_t log n_s)$ time. As a consequence, we construct a data structure of $O(n+(h^2log^3 h/loglog h))$ size in $O(n+(h^2log^4 h/loglog h))$ time such that each query can be answered in $O(log n)$ time.