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

## TL;DR

This paper introduces a divide-and-conquer algorithm that significantly improves the efficiency of shortest path queries in polygonal domains under the $L_1$ metric, reducing query time and preprocessing complexity.

## Contribution

It presents a novel divide-and-conquer approach that reduces the query time for shortest path computations from quadratic to near-linear, with efficient data structure construction.

## Key findings

- Achieves $O(n_s + n_t \log n_s)$ query time for shortest paths.
- Constructs a compact data structure of size $O(n + (h^2 \log^3 h / \log\log h))$.
- Preprocessing time is improved to $O(n + (h^2 \log^4 h / \log\log h))$.

## Abstract

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 $O(n_s\cdot n_t)$ 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 $O(n+(h^2\log^3 h/\log\log h))$ size in $O(n+(h^2\log^4 h/\log\log h))$ time such that each query can be answered in $O(\log n)$ time.