Spanner for the $0/1/\infty$ weighted region problem

TL;DR

This work addresses the 0/1/∞ weighted region problem by introducing a data structure that enables near-linear time construction and $(1+\varepsilon)$-approximate weighted shortest paths in planar subdivisions with convex zero-cost regions and convex obstacles. The core method combines trapezoidal maps for local region adjacency with a Θ-graph to guarantee good long-range connectivity, resulting in rigorous approximation guarantees and query times that scale polylogarithmically with input size. The framework is extended to handle convex obstacles (weight $\infty$) with propagated and tangent sample points, preserving the same favorable complexity while enabling paths amidst both zeros and obstacles; in polygonal cases, the data structure yields a $(1+\varepsilon)$-spanner. A notable application is near-quadratic-time approximation of the partial weak Fréchet similarity between polygonal curves, connecting geometric shortest-path techniques to curve similarity problems. Overall, the paper delivers near-linear-time, provably good geometric approximations for complex weighted-region path planning with practical implications for robotics and GIS systems.

Abstract

We consider the problem of computing an approximate weighted shortest path in a weighted subdivision, with weights assigned from the set $\{0, 1, \infty\}$. We present a data structure $B$, which stores a set of convex, non-overlapping regions. These include zero-cost regions (0-regions) with a weight of $0$ and obstacles with a weight of $\infty$, all embedded in a plane with a weight of $1$. The data structure $B$ can be constructed in expected time $O(N + (n/\varepsilon^3)(\log(n/\varepsilon) + \log N))$, where $n$ is the total number of regions, $N$ represents the total complexity of the regions, and $1 + \varepsilon$ is the approximation factor, for any $0 < \varepsilon < 1$. Using $B$, one can compute an approximate weighted shortest path from any point $s$ to any point $t$ in $O(N + n/\varepsilon^3 + (n/\varepsilon^2) \log(n/\varepsilon) + (\log N)/\varepsilon)$ time. In the special case where the 0-regions and obstacles are polygons (not necessarily convex), $B$ contains a $(1 + \varepsilon)$-spanner of the input vertices.

Figures (1)

• Figure 1: The sample points that are extreme in the directions of $r(k)$ and $r(k\theta + \pi)$ are marked with black dots. The sample points that are extreme in the directions of $r(k\theta \pm \pi/2)$ are marked with circles. Using these sample points, we generate a simplified polygon and construct $\mathcal{M}(k)$; the blue and green regions are examples of faces in $\mathcal{M}(k)$.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10
• Theorem 10