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 $L1$ metric for any two query points $s$ and $t$. To do so, a standard approach is to first find a set of $ns$ "gateways" for $s$ and a set of $nt$ "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 $O(ns\cdot nt)$ time to solve this problem. In this paper, we propose a divide-and-conquer technique that solves the problem in $O(ns + nt \log ns)$ time. As a consequence, we construct a data structure of $O(n+(h^2\log3 h/\log\log h))$ size in $O(n+(h^2\log4 h/\log\log h))$ time such that each query can be answered in $O(\log n)$ time.