Time-Dependent Shortest Path Queries Among Growing Discs

TL;DR

This paper introduces a $(1+psilon)$-approximation algorithm for computing time-dependent shortest paths among growing discs, enabling efficient queries for minimum arrival times with preprocessing based on multiple shortest path computations.

Contribution

It presents a novel approximation algorithm for time-dependent shortest paths in dynamic environments with growing obstacles, generalizing previous fixed departure time models.

Findings

Preprocessing involves $O(rac{1}{psilon}\, ext{log}(rac{ ext{max speed}}{ ext{min growth rate}}))$ shortest path computations.

Query time for approximate minimum arrival is $O( ext{log}(rac{1}{psilon}) + ext{loglog}(rac{ ext{max speed}}{ ext{min growth rate}}))$.

The approach is adaptable by plugging in different shortest path algorithms as black-box functions.

Abstract

The determination of time-dependent collision-free shortest paths has received a fair amount of attention. Here, we study the problem of computing a time-dependent shortest path among growing discs which has been previously studied for the instance where the departure times are fixed. We address a more general setting: For two given points $s$ and $d$, we wish to determine the function $\mathcal{A}(t)$ which is the minimum arrival time at $d$ for any departure time $t$ at $s$. We present a $(1+\epsilon)$-approximation algorithm for computing $\mathcal{A}(t)$. As part of preprocessing, we execute $O({1 \over \epsilon} \log({\mathcal{V}_{r} \over \mathcal{V}_{c}}))$ shortest path computations for fixed departure times, where $\mathcal{V}_{r}$ is the maximum speed of the robot and $\mathcal{V}_{c}$ is the minimum growth rate of the discs. For any query departure time $t \geq 0$ from $s$, we can approximate the minimum arrival time at the destination in $O(\log ({1 \over \epsilon}) + \log\log({\mathcal{V}_{r} \over \mathcal{V}_{c}}))$ time, within a factor of $1+\epsilon$ of optimal. Since we treat the shortest path computations as black-box functions, for different settings of growing discs, we can plug-in different shortest path algorithms. Thus, the exact time complexity of our algorithm is determined by the running time of the shortest path computations.