Rectilinear Shortest Paths Among Transient Obstacles

TL;DR

This paper introduces an optimal algorithm for finding time-minimal rectilinear paths among transient obstacles, accounting for waiting times, with applications in robot navigation and efficient query handling.

Contribution

It presents the first optimal $ heta(n ext{log} n)$ algorithm for rectilinear shortest paths among transient obstacles and a query method for fast minimum arrival time reporting.

Findings

Achieves optimal time complexity for the problem.

Provides a query structure for rapid path time estimation.

Handles obstacle appearance and disappearance in path planning.

Abstract

This paper presents an optimal $\Theta(n \log n)$ algorithm for determining time-minimal rectilinear paths among $n$ transient rectilinear obstacles. An obstacle is transient if it exists in the scene only for a specific time interval, i.e., it appears and then disappears at specific times. Given a point robot moving with bounded speed among transient rectilinear obstacles and a pair of points $s$, $d$, we determine a time-minimal, obstacle-avoiding path from $s$ to $d$. The main challenge in solving this problem arises as the robot may be required to wait for an obstacle to disappear, before it can continue moving toward the destination. Our algorithm builds on the continuous Dijkstra paradigm, which simulates propagating a wavefront from the source point. We also solve a query version of this problem. For this, we build a planar subdivision with respect to a fixed source point, so that minimum arrival time to any query point can be reported in $O(\log n)$ time, using point location for the query point in this subdivision.