Approximate Euclidean shortest paths in polygonal domains

TL;DR

This paper introduces a new efficient method for approximating shortest paths in polygonal domains, using a compact sketch of obstacles that enables faster computation of near-optimal paths.

Contribution

It presents a novel obstacle sketch technique that is independent of the total number of vertices, improving the efficiency of approximate shortest path computations.

Findings

Achieves a $(1+psilon)$-approximate shortest path in near-linear time.

Develops a $(2+psilon)$-approximate distance query algorithm for convex obstacles.

Sketch size depends only on the number of obstacles, not on total vertices.

Abstract

Given a set $\mathcal{P}$ of $h$ pairwise disjoint simple polygonal obstacles in $\mathbb{R}^2$ defined with $n$ vertices, we compute a sketch $\Omega$ of $\mathcal{P}$ whose size is independent of $n$, depending only on $h$ and the input parameter $\epsilon$. We utilize $\Omega$ to compute a $(1+\epsilon)$-approximate geodesic shortest path between the two given points in $O(n + h((\lg{n}) + (\lg{h})^{1+\delta} + (\frac{1}{\epsilon}\lg{\frac{h}{\epsilon}})))$ time. Here, $\epsilon$ is a user parameter, and $\delta$ is a small positive constant (resulting from the time for triangulating the free space of $\cal P$ using the algorithm in \cite{journals/ijcga/Bar-YehudaC94}). Moreover, we devise a $(2+\epsilon)$-approximation algorithm to answer two-point Euclidean distance queries for the case of convex polygonal obstacles.