A Sublinear Algorithm for Path Feasibility Among Rectangular Obstacles

TL;DR

This work addresses rectilinear path feasibility for a rectangular robot amid axis-aligned rectangular obstacles that may overlap. It introduces a generalized Gabriel graph under the $L_\infty$ norm to capture tight gap constraints, and constructs a region-partitioning dual graph whose edges encode bottleneck widths. A persistent disjoint-set union enables online feasibility queries in $O(\log n)$ time after $O(n \log n)$ preprocessing and $O(n)$ space. The approach supports varying robot sizes and extends from circular to rectilinear obstacle configurations, with potential extensions via Minkowski-sum techniques for broader shapes and radii, offering practical tools for fast, size-varied motion planning in complex environments.

Abstract

The problem of finding a path between two points while avoiding obstacles is critical in robotic path planning. We focus on the feasibility problem: determining whether such a path exists. We model the robot as a query-specific rectangular object capable of moving parallel to its sides. The obstacles are axis-aligned, rectangular, and may overlap. Most previous works only consider nondisjoint rectangular objects and point-sized or statically sized robots. Our approach introduces a novel technique leveraging generalized Gabriel graphs and constructs a data structure to facilitate online queries regarding path feasibility with varying robot sizes in sublinear time. To efficiently handle feasibility queries, we propose an online algorithm utilizing sweep line to construct a generalized Gabriel graph under the $L_\infty$ norm, capturing key gap constraints between obstacles. We utilize a persistent disjoint-set union data structure to efficiently determine feasibility queries in $\mathcal{O}(\log n)$ time and $\mathcal{O}(n)$ total space.

Figures (11)

• Figure 1: An example where the red dots are the starting and ending positions, and the green line is our trajectory, indicating that there is a feasible path.
• Figure 2: Three cases for how a thin edge between obstacles is constructed.
• Figure 3: An example of a Gabriel graph on random points.
• Figure 4: Three cases of minimum pathways for obstacles, shown by the dashed lines.
• Figure 5: An example of several obstacles, thin edges, and two of the regions they form.

Theorems & Definitions (31)

• Definition 3.1
• Definition 3.2
• Theorem 3.1: Constraint Equivalence
• Lemma 4.1
• proof
• Theorem 4.1
• Lemma 4.2
• proof
• Lemma 4.3
• proof